Artificial-Intelligence-munotes

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1 UNIT I
1
INTRODUCTION
Unit Structure
1.1 Objectives
1.2 Introduction
1.2.1 Introduction to AI
1.2.2 Objectives of AI
1.3 What is Artificial Intelligence?
1.3.1 What is AI.
1.3.2 Definitions of AI
1.4 Foundations of AI
1.4.1 Introduction
1.4.2 Turing Test
1.4.3 Weak AI ver sus Strong AI
1.4.4 Philosophy
1.5 History
1.6 The state of art AI today.
1.6.1 Current AI Innovations
1.6.2 Practical Applications of AI
1.7 Summary
1.8 Unit End Questions
1.9 References
1.1 OBJECTIVES After going through this unit, you will be able to:
 Define Artificial Intell igence
 Understand foundations of AI
 Explain how the AI was evolved
 Explain history of AI.
 Describe the Today’s AI, and Where and What research/ work is
going in AI field.
1.2 INTRODUCTION 1.2.1 Introduction to Ai:
Artificial Intelligence is to make com puters intelligent so that they can act
intelligently as humans. It is the study of making computers does things
which at the moment people are better at. munotes.in

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2 Introduction AI has made great effort in building intelligent systems and understanding
them. Another reason to un derstand AI is that these systems are interesting
and useful in their own right.
Many tools and techniques are used to construct AI systems. The AI
encompasses a huge variety of fields like computer science, mathematics,
logical reasoning, linguistics, neu ro-science and psychology to perform
specific tasks.
AI can be used in many areas like playing chess, proving mathematical
theorems, writing poetry, diagnosing diseases, creating expert systems,
speech recognition etc.
1.2.2 Objectives of Ai:
The field of artificial intelligence, or AI, attempts to understand intelligent
entities. Thus, one reason to study it is to learn more about ourselves.
AI strives to build intelligent entities as well as understand them.
The study of intelligence is also one of the ol dest disciplines. For over
2000 years, philosophers have tried to understand how seeing, learning,
remembering, and reasoning could, or should, be done.
AI currently encompasses a huge variety of subfields, from general -
purpose areas such as perception and logical reasoning, to specific tasks
such as playing chess, proving mathematical theorems, writing poetry, and
diagnosing diseases.
Often, scientists in other fields move gradually into artificial intelligence,
where they find the tools and vocabulary to systematize and automate the
intellectual tasks on which they have been working all their lives.
Similarly, workers in AI can choose to apply their methods to any area of
human intellectual endeavor. In this sense, it is truly a universal field.
1.3 ARTIF ICIAL INTELLIGENCE/ WHAT IS ARTIFICIAL INTELLIGENCE? 1.3.1 What Is Artificial Intelligence?:
Artificial Intelligence is the branch of computer science concerned with
making computers behave like humans.
Major AI textbooks define artificial intelligence as "the study and design
of intelligent agents," where an intelligent agent is a system that perceives
its environment and takes actions which maximize its chances of success.
John McCarthy, who coined the term in 1956, defines it as "the science
and engineer ing of making intelligent machines, especially intelligent
computer programs."

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3 Logical Agent Artificial Intelligence 1.3.2 Definitions of AI:
Artificial intelligence is “The science and engineering of making
intelligent machines, especially intelligent computer programs”.
Artificial Intellig ence is making a computer, a computer -controlled robot,
or a software think intelligently, as the intelligent humans think. AI is
accomplished by studying how the human brain thinks and how humans
learn, decide, and work while trying to solve a problem, an d then using the
outcomes of this study as a basis of developing intelligent software and
systems.
Some Definitions of AI:
There are various definitions of AI. They are basically categorized into
four categories:
i) Systems that think like humans
ii) Systems that think rationally
iii) Systems that act like humans
iv) Systems that act rationally
Category 1) Systems that think like humans:
An electronic machine or computer thinks like a human being.
“The exciting new effort to make computers think … machines with
minds, in t he full literal sense” (Haugeland, 1985)
“The automation of activities that we associate with human thinking,
activities such as decision making, problem solving, learning... ”
(Bellman, 1978)
Category 2) Systems that think rationally:
An electronic machin e or computer thinks rationally. If system thinks the
right thing, the system is rational.
"The study of mental faculties through the use of computational
models".(Charniak and McDermott, 1985).
"The study of the computations that make it possible to perce ive, reason,
and act". (Winston, 1992).
Category 3) Systems that act like humans:
An electronic machine or computer acts like human being. It acts same as
human being.
"The art of creating machines that performs functions that require
intelligence when pe rformed by people" (Kurzweil, 1990)
"The study of how to make computers do things at which, at the moment,
people are better" (Rich and Knight, 1991 ) munotes.in

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4 Introduction Category 4) Systems that act rationally:
An electronic machine or computer acts rationally.
"A field of study that seeks to explain and emulate intelligent behavior in
terms of computational processes" (Schalkoff, 1990)
"The branch of computer science that is concerned with the automation of
intelligent behavior" (Luger and Stubblefield, 1993)
1.4 FOUNDATION S OF AI 1.4.1 Introduction:
AI is a field which has inherited many ideas, viewpoints, and techniques
from other disciplines like mathematics, formal theories of logic,
probability, decision making, and computation.
From over 2000 years of tradition in phi losophy, theories of reasoning and
learning have emerged, along with the viewpoint that the mind is
constituted by the operation of a physical system. From over 400 years of
mathematics, we have formal theories of logic, probability, decision
making, and c omputation. From psychology, we have the tools with which
to investigate the human mind, and a scientific language within which to
express the resulting theories. From linguistics, we have theories of the
structure and meaning of language. Finally, from co mputer science, we
have the tools with which to make AI a reality.
1.4.2 Turing Test:
Acting humanly: The Turing Test Approach
 Test proposed by Alan Turing in 1950
 The computer is asked questions by a human interrogator.
The computer passes the test if a human interrogator, after posing some
written questions, cannot tell whether the written responses come from a
person or not. Programming a computer to pass , the computer need to
possess the following capabilities :
● Natural language processing to enable it to communicate successfully
in English.
● Knowledge representation to store what it knows or hears
 Automated reasoning to use the stored information to answer
questions and to draw new conclusions.
● Machine learning to adapt to new circumstances and to det ect and
extrapolate patterns
To pass the complete Turing Test, the computer will need
● Computer vision to perceive the objects, and munotes.in

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5 Logical Agent Artificial Intelligence ● Robotics to manipulate objects and move about .
Problem: Turing test is not reproducible, constructive, or amenable to
mathe matical analysis
1.4.3 Weak Ai Versus Strong Ai:
The definition of AI is along two dimensions, human vs. ideal and thought
vs. action. But there are other dimensions that are worth considering. One
dimension is whether we are interested in theoretical resu lts or in practical
applications. Another is whether we intend our intelligent computers to be
conscious or not.. The claim that machines can be conscious is called the
strong AI claim; the WEAK AI position makes no such claim. Artificial
intelligence is …
a. "a collection of algorithms that are computationally tractable,
adequate approximations of intractably specified problems"
(Partridge, 1991)
b. "the enterprise of constructing a physical symbol system that can
reliably pass the Turing Test" (Ginsbe rg, 1993)
c. "the field of computer science that studies how machines can be made
to act intelligently" (Jackson, 1986)
d. "a field of study that encompasses computational techniques for
performing tasks that apparently require intelligence when perfor med
by humans" (Tanimoto, 1990)
e. "a very general investigation of the nature of intelligence and the
principles and mechanisms required for understanding or replicating
it" (Sharpies et ai, 1989)
f. "the getting of computers to do things that seems t o be intelligent"
(Rowe, 1988)
Philosophy (428 B.C. -Present) :
Philosophers made AI conceivable by considering the ideas that the mind
is in some ways like a machine that operates on knowledge encoded in
some internal language and that thought can be used t o choose what
actions to take.
Philosophy is the implication of knowledge and understanding of
intelligence , ethics, logic, methods of reasoning, mind as physical system,
foundations of learnin g, language and rationality.
Mathematics:
Philosophers staked out most of the important ideas of AI, but to make the
leap to a formal science required a level of mathematical formalization in
three main areas: computation, logic,
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6 Introduction Algorithm and Probabilit y:
With logic a connection was made between probabilistic reasoning and
action.! DECISION THEORY Decision theory, pioneered by John Von
Neumann and Oskar Morgenstern (1944), combines! probability theory
with utility theory to give the first general theory that can distinguish
good! actions from bad ones. Decision theory is the mathematical
successor to utilitarianism, and provides the theoretical basis for many of
the agent designs.
Psychology (1879 – Present):
Psychology is to adopt the idea that humans an d animals can be
considered information processing machines.
It is all about adaptation, phenomena of perception and motor control,
experimental techniques (psychophysics, etc.)
Computer Engineering (1940 -Present):
For artificial intelligence to succeed, we need two things: intelligence and
an artifact. The computer has been the artifact of choice. AI also owes a
debt to the software side of computer science, which has supplied the
operating systems, programming languages, and tools needed to write
modern programs. Computer engineers provided the artifacts that make
AI applications possible.
To succeed in artificial intelligence, two things are required: intelligence
and an artifact. A computer machine is the best artifact to exhibit the
intelligence. Dur ing World War II, Some innovations are done in AI field.
In 1940, Alan Turing’s team had developed Heath Robinson to decode the
German messages. Then in 1943, Colossus was built from vacuum tubes
to decode complex code. In 1941, the first operational prog rammable
computer Z-3 was invented. In the US, ABC the first electronic computer
was assembled between 1940 and 1942. Mark I, II, and III computers were
developed at Harvard. ENIAC was developed; it is the first general -
purpose, electronic, digital compute r. Computing artillery firing tables is
one of its application.
In 1952, IBM 701 was built. This was the first computer to yield a profit
for its manufacturer. IBM went on to become one of the world's largest
corporations, and sales of computers have grow n.
Computer engineering has been remarkably successful, regularly doubling
performance every two years.
Linguistics (1957 -Present):
Linguists showed that language use fits into this model. Modern
linguistics and AI, then, were "born" at about the same tim e, and grew up
together, intersecting in a hybrid field called computational linguistics or
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7 Logical Agent Artificial Intelligence In 1957. B. F. Skinner published Verbal Behavior . It is related to
behaviorist approach to language learning. But curiously, a review of the
book became as well -known as the book itself, and served to almost kill
off interest in behaviorism. The author of the review was Noam Chomsky,
who had just published a book on his own theory. Syntactic Structures.
Chomsky showed how the behaviorist theory did not address the notion of
creativity in language —it did not explain how a child could understand
and make up sentences that he or she had never heard before. Chomsky's
theory —based on syntactic models going back to the Indian linguist
Panini (c . 350 B.C.) —could explain this, and unlike previous theories, it
was formal enough that it could in principle be programmed.
1.5 HISTORY OF AI The Gestation of Artificial Intelligence (1943 -1955) :
The first work that is now generally recognized as AI was done by Warren
McCulloch and Walter Pitts (1943). They drew on three sources:
knowledge of the basic physiology and function of neurons in the brain;
the formal analysis of propositional logic due to Russell and Whitehead;
and Turing's theory of com putation.
McCarthy convinced Minsky, Claude Shannon, and Nathaniel Rochester
to help him bring together U.S. researchers interested in automata theory,
neural nets, and the study of intelligence. They organized a two -month
workshop at Dartmouth in the su mmer of 1956. Perhaps the longest -
lasting thing to come out of the workshop was an agreement to adopt
McCarthy's new name for the field: artificial intelligence.
Early Enthusiasm, Great Expectations (1952 -1969) :
The early years of A1 were full of successes -in a limited way.
General Problem Solver (GPS) was a computer program created in 1957
by Herbert Simon and Allen Newell to build a universal problem solver
machine. The order in which the program considered sub goals and
possible actions was similar to t hat in which humans approached the same
problems. Thus, GPS was probably the first program to embody the
"thinking humanly" approach.
At IBM, Nathaniel Rochester and his colleagues produced some of the
first A1 programs. Herbert Gelernter (1959) construct ed the Geometry
Theorem Prover, which was able to prove theorems that many students of
mathematics would find quite tricky.
Lisp was invented by John McCarthy in 1958 while he was at the
Massachusetts Institute of Technology (MIT). In 1963, McCarthy start ed
the AI lab at Stanford.
Tom Evans's ANALOGY program (1968) solved geometric analogy
problems that appear in IQ tests.
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8 Introduction A Dose of Reality (1966 -1973) :
From the beginning, AI researchers were making predictions of their
coming successes. The following stat ement by Herbert Simon in 1957 is
often quoted: “It is not my aim to surprise or shock you -but the simplest
way I can summarize is to say that there are now in the world machines
that think, that learn and that create. Moreover, their ability to do these
things is going to increase rapidly until -in a visible future -the range of
problems they can handle will be coextensive with the range to which the
human mind has been applied.
Knowledge -Based Systems: The Key To Power? (1969 -1979) :
Dendral was an influenti al pioneer project in artificial intelligence (AI) of
the 1960s, and the computer software expert system that it produced. Its
primary aim was to help organic chemists in identifying unknown organic
molecules, by analyzing their mass spectra and using know ledge of
chemistry.
A1 Becomes An Industry (1980 -Present) :
In 1981, the Japanese announced the "Fifth Generation" project, a 10 -year
plan to build intelligent computers running Prolog. Overall, the A1
industry boomed from a few million dollars in 1980 to b illions of dollars
in 1988.
The Return of Neural Networks (1986 -Present) :
Although computer science had neglected the field of neural networks
after Minsky and Papert's Perceptrons book, work had continued in other
fields, particularly physics. Large colle ctions of simple neurons could be
understood in much the same way as large collections of atoms in solids.
Psychologists including David Rumelhart and Geoff Hinton continued the
study of neural -net models of memory.
Recent Events:
In recent years, approach es based on hidden Markov models (HMMs)
have come to dominate the area. Speech technology and the related field
of handwritten character recognition are already making the transition to
widespread industrial and consumer applications.
The Bayesian network formalism was invented to allow efficient
representation of, and rigorous reasoning with, uncertain knowledge.
1.6 THE STATE OF ART AI TODAY 1.6.1 Current Ai Innovations:
Chatbots, smart cars, IoT devices, healthcare, banking, and logistics all
use artifi cial intelligence to provide a superior experience. One AI that is
quickly finding its way into most consumer's homes is the voice assistant, munotes.in

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9 Logical Agent Artificial Intelligence such as Apple's Siri, Amazon's Alexa, Google's Assistant, and Microsoft's
Cortana. Some of them are listed below i n tabular format. Area Application/ Example/ Company Name Autonomous cars (Driverless car) Tesla Model S, Pony.ai, Waymo, Apple, Kia-Hyundai, Ford, Audi, Huawei. Speech Recognition Apple's Siri and Google's Alexa Chatbot Swelly, eBay, Lyft, Yes sire, 1-800-Flowers Web search engines ai.google Translator SYSTRAN Natural Language Processing IBM Watson API, Medical Diagnosis, Imaging Artificial Intelligence assistance in “keeping
well” Pattern Detection RankBrain by Google
1.6.2 Practical Applications of Ai:
Autonomous Planning And Scheduling:
A hundred million miles from Earth, NASA's Remote Agent program
became the first on -board autonomous planning program to control the
scheduling of opera tions for a spacecraft (Jonsson et al., 2000). Remote
Agent generated plans from high -level goals specified from the ground,
and it monitored the operation of the spacecraft as the plans were
executed -detecting, diagnosing, and recovering from problems as they
occurred.
Game Playing:
IBM's Deep Blue became the first computer program to defeat the world
champion in a chess match when it bested Garry Kasparov by a score of
3.5 to 2.5 in an exhibition match (Goodman and Keene, 1997).
Autonomous Control:
The ALVINN computer vision system was trained to steer a car to keep it
following a lane. It was placed in CMU's NAVLAB computer -controlled
minivan and used to navigate across the United States -for 2850 miles it
was in control of steering the vehicle 98% of t he time.
Diagnosis:
Medical diagnosis programs based on probabilistic analysis have been
able to perform at the level of an expert physician in several areas of
medicine.
Logistics Planning:
During the Persian Gulf crisis of 1991, U.S. forces deployed a Dynamic
Analysis and Replanning Tool, DART (Cross and Walker, 1994), to do
automated logistics planning and scheduling for transportation. This
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10 Introduction account for starting points, destinati ons, routes, and conflict resolution
among all parameters. The AI planning techniques allowed a plan to be
generated in hours that would have taken weeks with older methods. The
Defense Advanced Research Project Agency (DARPA) stated that this
single appli cation more than paid back DARPA's 30 -year investment in
AI.
Robotics:
Many surgeons now use robot assistants in microsurgery. HipNav
(DiGioia et al., 1996) is a system that uses computer vision techniques to
create a three -dimensional model of a patient 's internal anatomy and then
uses robotic control to guide the insertion of a hip replacement prosthesis.
Language Understanding And Problem Solving:
PROVERB (Littman et al., 1999) is a computer program that solves
crossword puzzles better than most human s, using constraints on possible
word fillers, a large database of past puzzles, and a variety of information
sources including dictionaries and online databases such as a list of
movies and the actors that appear in them.
1.7 SUMMARY Artificial intellig ence is “The science and engineering of making
intelligent machines, especially intelligent computer programs”.
Turing t est: To pass the complete Turing t est, the computer will need
Computer vision to perceive the objects, and Robotics to manipulate
object s and move about.
Artifici al intelligence is to make computers intelligent so that they can act
intelligently as humans.
Categorical definitions of AI are: Systems that think like humans, Systems
that think rationally, Systems that act like humans, System s that act
rationally.
AI comprises of Philosophy, Mathematics, Algorithm and Probability,
Psychology, Computer Engineering, Linguistics etc.
Work on Machine Intelligenc e started early. But the period which is
considered as History of AI started from The G estation of Artificial
Intelligence (1943 -1955), till The Return of Neural Networks (1986 -
Present) and Recent Events.
There are various current AI innovations such as Autonomous cars
(Driverless car), Speech Recognition, Chatbot, Web search engines,
Transl ator, Natural Language Processing, Medical Diagnosis, Imaging
Pattern Detection etc.
Various practical applications of AI such as Autonomous Planning And
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11 Logical Agent Artificial Intelligence Planning, Robotics, Language Un derstanding And Problem Solving etc.
1.8 UNIT END QUESTIONS 1.1 There are well -known classes of problems that are intractably difficult
for computers, and other classes that are provably undecidable by any
computer. Does this mean that AI is impossible?
1.2 Suppose we extend Evans's ANALOGY program so that it can score
200 on a standard IQ test. Would we then have a program more
intelligent than a human? Explain.
1.3 Examine the AI literature to discover whether or not the following
tasks can currently be so lved by computers:
a. Playing a decent game of table tennis (ping -pong).
b. Driving in the center of Cairo.
c. Playing a decent game of bridge at a competitive level.
d. Discovering and proving new mathematical theorems.
e. Writing an intentionally funny story.
f. Giving competent legal advice in a specialized area of law.
g. Translating spoken English into spoken Swedish in real time.
1.4 Find an article written by a lay person in a reputable newspaper or
magazine claiming the achievement of some intelligent capacity by a
machine, where the claim is either wildly exaggerated or false.
1.5 Fact, fiction, and forecast:
a. Find a claim in print by a reputable philosopher or scientist to the
effect that a certain capacity will never be exhibite d by computers,
where that capacity has now been exhibited.
b. Find a claim by a reputable computer scientist to the effect that a
certain capacity would be exhibited by a date that has since passed,
without the appearance of that capacity.
c. Compare the accuracy of these predictions to predictions in other
fields such as biomedicine, fusion power, nanotechnology,
transportation, or home electronics.
1.6 Some authors have claimed that perception and motor skills are the
most important part of intellig ence, and that "higher -level" capacities
are necessarily parasitic —simple add -ons to these underlying
facilities. Certainly, most of evolution and a large part of the brain
have been devoted to perception and motor skills, whereas AI has
found tasks such a s game playing and logical inference to be easier, in
many ways, than perceiving and acting in the real world. Do you think munotes.in

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12 Introduction that AI's traditional focus on higher -level cognitive abilities is
misplaced?
1.7 "Surely computers cannot be intelligent —they can only do what their
programmers tell them." Is the latter statement true, and does it imply
the former?
1.8 "Surely animals cannot be intelligent —they can only do what their
genes tell them." Is the latter statement true, and does it imply the
former?
1.9 REFERENCES Artificial Intelligence: A Modern Approach, 4th US ed.by Stuart Russell
and Peter Norvig .
Deepak Khemani, “A first course in Artificial Intelligence”, McGraw Hill
edition, 2013.
Patrick Henry Winston , “Artificial Intelligence”, Addison -Wesl ey, Third
Edition



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13 2
INTELLIGENT AGENTS
Unit Structure
2.1 Objectives
2.2 Introduction
2.3 Agents and environment
2.4 Good Behavior
2.4.1 Rational Agent
2.4.2 Mapping from percept sequences to actions
2.4.3 Performance Measure
2.4.4 PEAS
2.5 Nature of environment
2.6 The structure of agents
2.6.1 Structure
2.6.2 Softbots
2.6.3 PAGE
2.6.4 Types of Agent
2.7 Summary
2.8 Unit End Questions
2.9 References
2.1 OBJECTIVES After going through this unit, you will be able to :
 Define AI Agent,
 Understand the Agent and its Environment
 Understand the Rationality of agent, Performance measure
 Identify the PEAS, and PAGE
 Understand the nature of Environment
 Explain the Structure of Agents
2.2 INTRODUCTION An agent is anything that can be viewed as capturing its environment
through cameras, sensors or some other input devices and acting upon that
environment through effectors. A human agent has eyes, ears, and other
organs for sensors, and hands, legs, mouth, and other body parts for
effectors. A robotic agent substitutes cameras and infrared range finders
for the sensors and various motors for the effectors. A software agent has
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14 Intelligent Agents The focus is on design agents that do a good job of acting on their
environment,
What we mean by a good jo b, rationality, performance measure, then
different designs for successful agents, which things should be known to
the agent and several kinds of environments.
2.3 AGENT AND ENVIRONMENT Definition : An agent is anything that perceives its environment through
sensors and acting upon that environment through effectors.
An agent is split into architecture and an agent program.
An intelligent agent is an agent capable of making decisions about how it
acts based on experiences.

Sensors are used to receive percept s ignals. Percept signals can be
anything like audio, video, image, etc.
Effectors are used to make action. We can also call effectors as actuators.
The following are the sensors and actuators for Robot / Robotic agent
Sensors: Cameras, infrared range find ers, scanners, etc.
Actuators: Various motors, screen, printing devices.
A robotic agent substitutes cameras and infrared range finders for the
sensors and various motors for the effectors.
Environment is the surrounding of the robotic agent. It is the ar ea in which
agent does the work, performs some actions. In a vacuum cleaner robotic
agent, a room will be an environment. There are various types of
environment fully observable or partially observable, Static or dynamic,
Accessible or inaccessible, Known or unknown, Single agent or multi
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15 Artificial Intelligence Agent perceives the signals of other robotic agent actions, temperature,
humidity, pressure, air type, atmospheric conditions, climate, other
surrounding conditions and many more. In addition to this, in taxi d riving
example – road conditions, traffic conditions, weather conditions, rain or
smoke fog, wind, driving rules etc.
2.4 GOOD BEHAVIOR Here, the behavior of the AI agent is checked. For this rationality,
performance measure is considered.
2.4.1 Rationa l agent:
A rational agent is the one that does “right” things and acts rationally so as
to achieve the best outcome even when there is uncertainty in knowledge.
Rational agent can be defined as an agent who makes use of its percept
sequence, experience and knowledge to maximize the performance
measures of an agent for every possible action. Performance measures can
be any like Accuracy, Speed, Safety, total work done with respect to time,
efficiency, saving money and time, following Legal rules, etc.

What is rational at any given time depends on four things:
1) The performance measure defines the degree of success. Performance
is the measure of successful completion of any task.
2) Complete perceptual history is the percept sequence. This percept
sequence is a sequence of signals or inputs that the agent has
perceived so far.
3) What the agent knows about the environment. Especially about the
environment dimensions, structure, basic laws, task, user, other
agents, etc.
4) The actions that the agent can perform. It shows the capability of the
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16 Intelligent Agents Ideal rational agent:
For each possible percept sequence, an ideal rational agent should do
whatever action is expected to maximize its performance measure, on the
basis of the evidence provided by the p ercept sequence and whatever
built-in knowledge the agent has.
2.4.2 The ideal mapping from percept sequences to actions :
For each percept signal what will be the associated action of the agent.
This information can be described in tabular format.

Let’s Consider the Vacuum Cleaner robotic agent. It cleans the room. For
simplicity two tiles are considered. The Vacuum cleaner agent is present
on Tile A and observing the environment i.e. Tile A and Tile B both. Both
the tiles have some dirt. This agent has to clean the dirt by sucking it.

The following table shows the percept sequence and its associated action.
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17 Artificial Intelligence Autonomous Agent :
An agent is autonomous to the extent that i ts action choices depend on its
own experience, rather than on knowledge of the en vironment that has
been built -in by the designer.
There are various autonomous agent are in deve lopment stage. Driverless
car- Waymo, Google’s Alexa, etc.
2.4.3 How and When to evaluate the agent’s success:
Performance measure:
Performance measure is one of the major criteria for measuring success of
an agent’s performance.
As a general rule, it is better to design performance measures according to
what one actually wants in the environment, rather than according to how
one thinks the agent should behave .
Example: The performance measure of Vacuum -cleaner Robotic agent can
depend upon various factors like it’s dirt cleaning ability, time taken to
clean that dirt, consumption of electricity, etc.
2.4.4 PEAS Properties of Agent:
What is PEAS?
PEAS: Perform ance, Environment, Actuators, Sensors.
Performance Measures - used to evaluate how well an agent solves the
task at hand
Environment - surroundings beyond the control of the agent
Actuators - determine the actions the agent can perform
Sensors - provide in formation about the current state of the environment
Examples :
1) Automated Car Driving agent/ Driverless Car:
Performance measures: Safety, Optimum speed, Comfortable journey,
Maximize profits.
Environment: Roads, Traffic conditions, Clients.
Actuators: St eering wheel, Accelerator, Gear, Brake, Light signal, Horn.
Sensors: Cameras, Sonar system, Speedometer, GPS, Engine sensors, etc.
2) Medical Diagnosis system:
Performance measures: Healthy patient (Sterilized instruments), minimize
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18 Intelligent Agents Environment: Pa tients, Doctors, Hospital environment.
Actuators: Camera, Scanner (to scan patient’s report).
Sensors: Body scanner, Organs scanner e.g. MRI, CT SCAN etc, Screen,
Printer.
3) Soccer Player Robot:
Performance measures: Number of goals, speed, legal game.
Environment: Team players, opponent team players, playing ground, goal
net.
Actuators: Joint angles, motors.
Sensors: Camera, proximity sensors, infrared sensors.
2.5 NATURE OF ENVIRONMENT OR PROPERTIES OF ENVIRONMENT 1) Accessible vs. inaccessible :
If an agent can obtain complete and accurate information about the state's
environment, then such an environment is called an Accessible
environment else it is called inaccessible.
Example :
Accessible environment - An empty closed room whose state can be
defined by its temperature, air pressure, humidity.
Inaccessible environment - Information about an event occurs in the
atmosphere of the earth.
2) Deterministic vs. non deterministic :
If an agent's current state and selected action can completely determine the
next state of the environment, then such an environment is called a
deterministic environment.
In an accessible, deterministic environment an agent does not need to
worry about uncertainty. If the environment is inaccessible, however, then
it may appear to be nondeterministic. This is particularly true if the
environment is complex, making it hard to keep track of all the
inaccessible aspects. Thus, it is often better to think of an environment as
deterministic or nondeterministic from the point of view of the agent.
3) Episodic vs. non episodic :
In an episodic environment, the agent's experience is divided into
"episodes." Each episode consists of the agent perception and its
corresponding action. The quality of its action depends just on the episode
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19 Artificial Intelligence previous episodes. Episodic environments are much simpler because the
agent does not need to think ahead. However, in a non -episodic
environment, an agent requires memory of past actions to determine the
next best actions.
4) Static vs. dynamic :
If the environment can change while an agent is deciding on an action,
then we say the environment is dynamic for that agent; otherwise it is
static. Static environments are easy to deal with because the agent need
not keep looking at the world while it is deciding on an action, nor need it
worry about the passage of time. If the environment does not change with
the passage of time but the agent's performance score does, then we say
the environment is semi dynamic.
Taxi driving is an example of a dynamic environment whereas Crossword
puzzles are an example of a static environment.
5) Discrete vs. continuous :
If there are a limited number of distinct, clearly defined percepts and
actions we say that the environment is discrete. Chess is discrete —there
are a fixed number of possible moves on each turn. Taxi driving is
continuous —the speed and location of the taxi and the other vehicles
sweep through a range of continuous values.
Examples of environments an d their nature or characteristics are as
follows:

2.6 STRUCTURE OF INTELLIGENT AGENT 2.6.1 Structure :
Structure of the agent describes how the agent works.
The job of AI is to design the agent program . Agent program is a function
that implements the ag ent mapping from percepts to actions. An
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20 Intelligent Agents The relationship among agents, architectures, and programs can be
summed up as follows:
agent = architecture + program
2.6.2 Software Agents :
It is also re ferred to as softbots.
It lives in artificial environments where computers and networks provide
the infrastructure.
It may be very complex with strong requirements on the agent
e.g. softbot designed to fly a flight simulator, softbot designed to scan
online news sources and show the interesting items to its customers,
World Wide Web, real -time constraints,
Natural and artificial environments may be merged user interaction
sensors and actuators in the real world - camera, temperature, arms,
wheels, etc.
2.6.3 Page :
PAGE is used for high -level characterization of agents.
P- Percepts - Information is acquired through the agent’s sensory system.
A- Actions - Operations are performed by the agent on the environment
through its actuators.
G- Goals - Desired outcome of the task with a measurable performance.
E- Environment - Surroundings beyond the control of the agent.
Following are some examples of agents and their PAGE descriptions. Sr. No. Agent Type Percepts Actions Goals Environment 1. Medical diagnosis system Symptoms, findings, patient's answers Questions, tests, treatments Healthy patient, minimize costs Patient, hospital 2. VacBot – Vacuum Cleaner Bot tile properties like clean/dirt, empty/occupied movement and orientation pick up dirt, move desired outcome of the task with a measurable performance surroundings beyond the control of the agent 3. StudentBot images (text, comments, questions, mastery of the material classroom munotes.in

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21 Artificial Intelligence pictures, instructor, classmates) sound (language) gestures note-taking (?) performance measure: grade 4. Satellite image analysis system Pixels of varying intensity, color Print a categorization of scene Correct categorization Images from orbiting satellite 5. Part-
picking
robot Pixels of varying intensity Pick up parts and sort into bins Place parts in correct bins Conveyor belt with parts 6. Refinery
controlle
r Temperature, pressure readings Open, close valves; adjust temperature Maximize purity, yield, safety Refinery 7. Interactive English tutor Typed words Print exercises, suggestions, corrections Maximize student's score on test Set of students
2.6.4 Different types of Agent program :
Every agent program has some skeleton. Real agent program implements
the mapping from percepts to action. There are four types of agent
program as follows:
1) Simple reflex agents
2) Agents that keep track of the world
3) Goal -based agents
4) Utility -based agents
1) Type 1 : Simple Reflex Agent :
Reflex agents respond immediately to percepts.
They choose actions only based on the curren t percept.
They are rational only if a correct decision is made only on the basis of
current precept.
Their environment is completely observable.
In the simple reflex agent, the correct decision can be made on the basis of
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22 Intelligent Agents Let’s consid er the driverless car – Car A. If the car B is running in front of
car A, and suddenly if car B has applied brakes, and car B brake lights
come ON, then driverless car agent should notice that and initiate
breaking. Thus it can be specified in condition ac tion rule and can be
represented in the form as
“If condition Then Action”.
“if car-in-front -is-breaking then initiate -braking”.


Schematic Diagram: Simple Reflex Agent
Following is the agent program basically it is a function for a simple reflex
agen t. It works by finding a rule whose condition matches the current
situation (as defined by the percept) and then doing the action associated
with that rule.
Function Simple_Reflex_Agent( percept ) returns action
Static: rules, a set of condition -action r ules
state  Interpret_Input (percept )
rule  Rule_Match ( state, rules)
action  Rule_Action [ rule ]
return action
This agent program calls the Interpret_Input() function that generates an
abstracted description of the current state from the percept input, and the
Rule_Match() function returns the first rule in the set of rules that matches
the given state description. Such agents can be implemented very
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23 Artificial Intelligence 2) Type 2: Agents that keep track of the world :
It is a reflex agent with internal state.
Internal State is a representation of unobserved aspects of current state
depending on percept history.
Consider an example of a driverless car. If the car in front is a recent
model, and has the centrally mounted brake light, then it will be possible
to tell if it is braking from a single image. Unfortunately, older models
have different configurations of tail lights, brake lights, and turn -signal
lights, and it is not always possible to tell if the car is braking.
Thus, even for the simple braking rule, our driver will have to maintain
some sort of internal state in order to choose an action. Here, the internal
state is not too extensive —it just needs the previous frame from the
camera to detect when two r ed lights at the edge of the vehicle go on or off
simultaneously.
Consider one case in car driving: from time to time, the driver looks in the
rear-view mirror to check on the locations of nearby vehicles. When the
driver is not looking in the mirror, the vehicles in the next lane are
invisible (i.e., the states in which they are present and absent are
indistinguishable); but in order to decide on a lane -change maneuver, the
driver needs to know whether or not they are there.
It happens because the sensors do not provide access to the complete state
of the world. In such cases, the agent may need to maintain some internal
state information in order to distinguish between world states that generate
the same perceptual input but nonetheless are significantly d ifferent. Here,
"significantly different" means that different actions are appropriate in the
two states.
As time passes internal state information has to be updated. For this two
kinds of knowledge have to be encoded in the agent program. First, we
need s ome information about how the world evolves independently of the
agent —for example, that an overtaking car generally will be closer behind
than it was a moment ago. Second, we need some information about how
the agent's own actions affect the world —for exa mple, that when the agent
changes lanes to the right, there is a gap (at least temporarily) in the lane it
was in before.
Following Figure shows how the current percept is combined with the old
internal state to generate the updated description of the cur rent state. munotes.in

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24 Intelligent Agents

Diagram: Schematic Diagram for Agent that keep track of the world/
Reflex Agent with Internal State
Agent Program:
The interesting part is the function Update -State, which is responsible for
creating the new internal state description. As w ell as interpreting the new
percept in the light of existing knowledge about the state, it uses
information about how the world evolves to keep track of the unseen parts
of the world, and also must know about what the agent's actions do to the
state of the world.
Following is the reflex agent with internal state. It is an agent that keeps
track of the world. It works by finding a rule whose condition matches the
current situation (as defined by the percept and the stored internal state)
and then doing the a ction associated with that rule.
function Reflex -Agent -With -State(percept) returns action
static: state, a description of the current world state
rules, a set of condition -action rules
state  Update -State (state, percept)
rule  Rule -Match (state, rules)
action  Rule -Action [rule]
state  Update -State (state, action)
return action.
3) Type 3: Goal -based agents :
Goal -based agents act so that they will achieve their goal
If just the current state of the environment is known then it is not always
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25 Artificial Intelligence The agent tries to reach a desirable state, the goal may be provided from
the outside (user, designer, environment), or inherent to the agent itself
The agent performs the action by considering the Goal. To achieve the
goal, agent has to c onsider long sequences of actions. Here, Search &
Planning is required. Decision making is required. It involves
consideration of the future —both "What will happen if I do such -and-
such?" and "Will that make me happy?
Goal based agents can only differentia te between goal states and non goal
states. Hence, their performance can be 100% or zero.
Goal -based agent is more flexible than reflex agent since the knowledge
supporting a decision is explicitly modeled, thereby allowing for
modifications.
Limitation: Once the goal is fixed, all the actions are taken to fulfill it.
And the agent loses flexibility to change its actions according to the
current situation.
Example: Vacuum cleaning Robot agent whose goal is to keep the house
clean all the time. This agent will keep searching for dirt in the house and
will keep the house clean all the time.

Diagram: Schematic Diagram for Goal Based Agent
4) Type 4: Utility -based agents:
Utility -based agents try to maximize their own "happiness."
There are conflicting goals, o ut of which only few can be achieved. Goals
have some uncertainty of being achieved and you need to weigh likelihood
of success against the importance of a goal.
Goals alone are not really enough to generate high -quality behavior. Here,
the degree of happi ness is considered.
For example, there are many action sequences that will get the taxi to its
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26 Intelligent Agents reliable, or cheaper than others. Goals just provide a crude distinction
between "happ y" and "unhappy" states, whereas a more general
performance measure should allow a comparison of different world states
(or sequences of states) according to exactly how happy they would make
the agent if they could be achieved. Because "happy" does not so und very
scientific, the customary terminology is to say that if one world state is
preferred to another, then it has higher utility for the agent.
Utility is therefore a function that maps a state onto a real number, which
describes the associated degree of happiness. A complete specification of
the utility function allows rational decisions in two kinds of cases where
goals have trouble. First, when there are conflicting goals, only some of
which can be achieved (for example, speed and safety), the utilit y function
specifies the appropriate trade -off. Second, when there are several goals
that the agent can aim for, none of which can be achieved with certainty,
utility provides a way in which the likelihood of success can be weighed
up against the importanc e of the goals
Utility function is used to map a state to a measure of utility of that state.
We can define a measure for determining how advantageous a particular
state is for an agent. To obtain this measure a utility function can be used.
The term utili ty is used to depict how “HAPPY” the agent is to find out a
generalization.
Examples: 1) Google Maps – A route which requires Least possible time.
1) Automatic Car: Should reach to the target location within least possible
time safely and without consuming m uch fuel.

Diagram: Schematic Diagram for Utility Based Agent
2.7 SUMMARY An agent is anything that perceives its environment through sensors and
acting upon that environment through effectors.
A rational agent is the one that does “right” things and acts rationally so as
to achieve the best outcome even when there is uncertainty in knowledge. munotes.in

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27 Artificial Intelligence PEAS - Performance, Environment, Actuators, Sensors determines the
behavior of an agent.
There are va rious nature of environment or p roperties of environment ,
they are as Accessible vs. inaccessible, Deterministic vs. non
deterministic, Episodic vs. non episodic, Static vs. d ynamic, Discrete vs.
continuous, etc.
Structure of Intelligent Agent: agent = architecture + program.
PAGE is used for high -level characteri zation of agents.
Different types of Agent program are as Simple reflex agents, Agents that
keep track of the world, Goal -based agents, Utility -based agents.
2.8 UNIT END QUESTIONS 1. Describe is an agent, rational Agent, autonomous agent.
2. Which are the four things that is considered to check the agent is
rational at any given time?
3. What is the good behavior of agent. How performance measure is the
key factor for
4. Which are the four things that is considered to check the rationality of
agent at any given time.
5. How and When to evaluate the agent’s success?
6. Describe Performance measure is one of the major criteria for
measuring success of an agent’s performance.
7. What is PEAS Properties of Agent? Explain with example the PEAS
for any fo ur agents.
8. Explain Nature of Environment or Properties of Environment
9. What is softbot?
10. What is PAGE? Describe with any five examples.
11. Explain the structure of Agent.
12. Which are the types of agent? Explain all the types with schemati c
diagram.
13. Write the Agent program for Simplex Reflex Agent and Reflex Agent
with Internal State.


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28 Intelligent Agents 2.9 REFERENCES Artificial Intelligence: A Modern Approach, 4th US ed.by Stuart Russell
and Peter Norvig .
Deepak Khemani, “A first course in Artificial I ntelligence”, McGraw Hill
edition, 2013.
Patrick Henry Winston , “Artificial Intelligence”, Addison -Wesley, Third
Edition.

*****


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V 29 UNIT II
3
SOLVING PROBLEMS BY SEARCHING
Unit Structure
3.1 Objective
3.2 Introduction
3.3 Problem Solving Agents
3.4 Examples problems
3.5 Searching for solutions
3.6 Uninformed search
3.7 Informed search strategies
3.8 Heuristic functions
3.9 Summary
3.10 Exercis es
3.11 Bibliography
3.1 OBJECTIVES After this chapter, you should be able to understand the following
concepts:
1. Understand how to formulate a problem description.
2. Understand and solve some problems of Artificial Intelligence.
3. Know about uninformed search and based algorithms.
4. Understand informed search and based algorithms.
5. Know about heuristic functions and strategies.
3.2 INTRODUCTION A problem -solving agent firstly formulates a goal and a problem to solve.
Then the agent calls a search procedure to sol ve it. It then uses the solution
to guide its actions. It will do whatever the solution recommends as the
next thing to do and then remove that step from the sequence. Once the
solution has been executed, the agent will formulate a new goal. Searching
techniques like uninformed and informed or heuristic use in many areas
like theorem proving, game playing, expert systems, natural language
processing etc. In search technique, firstly select one option and leave
other option if this option is our final goal, else we continue selecting,
testing and expanding until either solution is found or there are no more
states to be expanded.
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30 Solving Problems by Searching 3.3 PROBLEM SOLVING AGENTS It is very important task of problem domain formulation. Problems are
always dealt with an Artificial In telligence. It is commonly known as
state. For solving any type of task (problem) in real world one needs
formal description of the problem. Each action changes the state and the
goal is to find sequence of actions and states that lead from the initial sta rt
state to a final goal state.
Goals help organize behavior by limiting the objectives that the agent is
trying to achieve and hence the actions it needs to consider. Goal
formulation, based on the current situation and the agent’s performance
measure, is the first step in problem solving.
Problem formulation is the process of deciding what actions and states to
consider, given a goal. the agent will not know which of its possible
actions is best, because it does not yet know enough about the state that
results from taking each action. If the agent has no additional
information —i.e., if the environment is unknown.
An agent with several immediate options of unknown value can decide
what to do by first examining future actions that eventually lead to states
of known value.
The agent’s task is to find out how to act, now and in the future so that is
reaches a goal state. Before it can do this, it needs to decide what sorts of
actions and states it should consider.
3.3.1 Well Defined problem:
A problem can be d efined formally by four components.
1) The initial state that the agent starts in:
For example, Consider a agent program Indian Traveller developed for
travelling Hyderabad to Mumbai travelling through different states. The
initial state for this agent can b e described as In (Hyderabad).
2) A description of the possible actions available to the agent:
Given a particular state s, ACTIONS(s) returns the set of actions that can
be executed in s. We say that each of these actions is applicable in s. For
example, fro m state In (Mumbai), the applicable actions are {Go
(Solapur), Go (Osmanabad), Go (Vijayawada)}
3) The goal test:
Which determines whether a given state is a goal state. Sometimes there is
an explicit set of possible goal states, and the test simply checks wh ether
the given state is one of them.
For example, In Indian traveller problem the goal is to reach Mumbai i.e it
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31 Artificial Intelligence Sometimes the goal is specified by an abstract property rather than an
explicitly enumerated set of states. For example, in chess, the goal is to
reach a state called “checkmate,” where the opponent’s king is under
attack and can’t escape.
4) A path cost function that assigns a numeric cost to each path. The
problem -solving agent chooses a cost function that reflec ts its own
performance measure. For the agent trying to get to Mumbai, time is of
the essence, so the cost of a path might be its length in kilometres. The
step cost of taking action a in state x to reach state y is denoted by c (x,
a, y).
A solution to a problem is an action sequence that leads from the initial
state to a goal state. Solution quality is measured by the path cost function,
and an optimal solution has the lowest path cost among all solutions.

Fig. 3. 2.1 Indian traveller map
3.4 EXAMPLES PR OBLEMS The problem -solving approach has been applied to a vast array of task
environments. We list some known problems like Toy problem and real -
world problem. A toy problem is intended to illustrate or exercise various
problem -solving methods. This proble m gives exact description so it is
suitable for researchers to compare the performance of algorithms. A real -
world problem is one whose solution people actually care about.
3.4.1 Toy Problems:
The first example we examine is the vacuum world.
This can be formulated as problem as follows: munotes.in

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32 Solving Problems by Searching 1) States:
The state is determined by both the agent location and the dirt locations.
The agent is in one of two locations, each of which might or might not
contain dirt.
2) Initial state:
Our vacuum can be in any state of the 8 states shown in the figure 3.2.
3) Actions:
Each state has just three actions: Left, Right, and Suck
4) Transition Model:
The actions have their expected effects, except that moving Left in the
leftmost square, moving Right in the rightmost square, and Sucking in a
clean square have no effect.
5) Goal test:
No dirt at all locations.
6) Path cost:
Each cost step is 1.

Fig. 3. 2 The state space for the vacuum world. Links denote actions: L
= Left, R= Right, S= Suck
The second example we examine is 8 puzzle problem.
The 8-puzzle consists of eight numbered, movable tiles set in a 3x3 frame.
Each tile has a number on it. A tile that is adjacent to the blank space can
slide into the space. A game consists of a starting position and a specified
goal position. The goal is to transform the starting position into the goal
position by sliding the tiles around.


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33 Artificial Intelligence This can be formulated as problem as follows:
1) States:
Location of eight tiles and blank.
2) Initial state:
Any state can be designed as the initial state.
3) Actions:
Move blank left, right, up or down.
4) Goal test:
This checks whether the states match the goal configuration.
5) Path cost:
Each step cost is 1.

Fig.3.3 An instance of 8 -puzzle problem
The third example is 8 -Queen problem :
The goal of the 8 -queens problem is to place e ight queens on a chessboard
such that no queen attacks any other. (A queen attacks any piece in the
same row, column or diagonal.)
This can be formulated as problem as follows:
1) States:
Any arrangement of 0 to 8 queens on the board is a state
2) Initial state:
No queens on board.
3) Actions:
Add a queen to any empty square.
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34 Solving Problems by Searching 4) Transition Model:
Returns the board with a queen added to the specified square.
5) Goal Test:
8 queens are on the board, none attacked.

Fig. 3.4 Solution to the 8 -queen problem
The fourth exampl e is Water Jug problem :
You are given two jugs, a 4 -gallon one and a 3 -gallon one, a pump which
has unlimited water which you can use to fill the jug, and the ground on
which water may be poured.
Neither jug has any measuring markings on it. How can you g et exactly 2
gallons of water in the 4 -gallon jug?
State Representation and Initial State – we will represent a state of the
problem as a tuple (x, y) where x represents the amount of water in the 4 -
gallon jug and y represents the amount of water in the 3 -gallon jug. Note 0
d” x d” 4, and 0 d” y d” 3. Our initial state: (0,0). The goal state is à (2, n).
Rules:
1. Fill the 4 -Gallon Jug (x,y)  (4,y)
2. Fill the 3 -Gallon Jug. (x,y)  (x,3)
3. Pour some water out of 4 Gallon jug (x,y) (x-d,y)
4. Pour some water out of 3 Gallon jug (x,y) (x,y-d)
5. Empty 4 Gallon jug on the ground. (x,y) (0,y)
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35 Artificial Intelligence 7. Pour some water from 3 Gallon jug into the 4 Gallon jug until the 4
gallon jug is full. (x,y) (4, y - (4 - x))
8. Pour some water from 4 Gallon jug into the 3 Gallon jug until the 3
gallon jug is full. (x,y) (x - (3-y), 3)
9. Pour all water from 3 to 4 gallon. (x,y) (x+y, 0)
10. Pour all water from 4 to 3 Gallon jug. (x,y) (0, x+y)
11. Pour 2 gallon from 3 G to 4 G. (0,2) (2,0) if x12. Empty the 2 G in 4 G on the ground. (2,y) (0,y) if x Gallons in the 4-Gallon Jug Gallons in 3-Gallon Jug Rules Applied 0 0 - 0 3 2 3 0 9 3 3 2 4 2 7 0 2 5 2 0 9
The fifth example is Missionari es and Cannibals :
Three missionaries and their three cannibals are present at one side of a
river and need to cross the river. There is only one boat available.
At any point of time the number of cannibals should not outnumber the
number of missionaries at that bank.
It is also known that only two persons can occupy the boat available at a
time. Let Missionary is denoted by ‘M’ and cannibal by ‘C’
Rules:
1. (0, M): One missionary sailing the boat from bank -1 to bank -2.
2. (M,0): One missionary sailing boa t from bank -2 to bank -1.
3. (M, M): Two missionaries sailing the boat from bank 1 to bank 2.
4. (M, M): Two missionaries sailing the boat from bank 2 to bank 1.
5. (M, C): One missionary & one cannibal sailing the boat from bank 1
to bank -2.
6. (C, M): One missionary & one cannibal sailing the boat from bank 2
to bank -1.
7. (C, C): Two cannibals sailing the boat from bank -1 to bank -2.
8. (C, C): Two cannibals sailing the boat from bank -2 to bank -1.
9. (0, C): One cannibal sailing the boat from bank -1 to bank -2. munotes.in

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36 Solving Problems by Searching 10. (C,0): One cannibal sailing the boat from bank -2 to bank -1.

SSS

After application of rules Person in the river bank -1 Person in the river bank -2 Boat Position Start state M, M, M, C, C, C 0 Bank-1 5 M, M, C, C C, M Bank-2 2 M, M, M, C, C C Bank-1 7 M, M, M C, C, C Bank-2 10 M, M, M, C C, C Bank-1 3 M, C C, C, M, M Bank-2 6 M, C, M, C C, M Bank-1 3 C, C C, M, M, M Bank-2 10 C, C, C M, M, M Bank-1 7 C C, C, M, M, M Bank-2 10 C, C C, M, M, M Bank-1 7 0 M, M, M, C, C, C Bank-2
3.4.2 Real -World Problems :
Example 1: Route finding problem :
A route -finding problem is defined in terms of specified locations and
transitions along links between them, such as web sites and in -car systems
that provide driving directions, are relative ly straightforward extensions of
the Indian map example. Others, such as routing video streams in
computer networks, military operations planning, and airline travel -
planning systems.
Ex. Airline travel problem specified as follows:
1) States:
Each state obv iously includes a location (e.g., an airport) and the current
time.
2) Initial state:
This is specified by the user’s query.
3) Actions:
Take any flight from the current location, in any seat class, leaving after
the current time, leaving enough time for within -airport transfer if needed.
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37 Artificial Intelligence 4) Transition Model:
The state resulting from taking a flight will have the flight’s destination as
the current location and the flight’s arrival time as the current time
5) Goal test:
Are we at the final destination specified by the user?
6) Path cost:
This depends on monetary cost, waiting time, flight time, customs and
immigration procedures, seat quality, time of day, type of airplane,
frequent -flyer mileage awards, and so on.
Touring Problem:
This is closely related to route -finding problems, but with an important
difference. Consider, for example, the problem visits every city in Fig. 3.1
at least once. Starting and ending at Solapur. The actions correspond to
trips between adjacent cities. Each state must include not just the curre nt
location but also the set of cities the agent has visited. So, the initial state
would be In (Solapur), visited (Solapur). and the goal test would check
whether the agent is in Bucharest and all 27 cities have been visited.
Example 2: Travelling salesm an problem (TSP problem)
It is a touring problem in which each city must be visited exactly once.
The aim is to find the shortest tour.
1) States:
Cities or locations
Each city must be visited exactly once. Visited cities must be kept as state
information.
2) Initial state:
Starting point, no cities visited.
3) Successor function:
Move from one city to another city.
4) Goal test:
All cities visited, Agent at the initial location.
5) Path cost:
Distance between cities.

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38 Solving Problems by Searching Example 3: VLSI Layout Problem :
problem requires posi tioning millions of components and connections on a
chip to minimize area, minimize circuit delays, minimize stray
capacitances, and maximize manufacturing yield.
1) States:
Position of components, wires on a chip.
2) Initial State:
Incremental: no components placed
Complete -state: All components place (randomly, manually)
3) Successor Function:
Incremental: placed components, route wire
Complete -state: move components, move wire
4) Goal test:
All components placed; components connected as specified
5) Path cost:
Compl ex.
Example 4: Robot Navigation
It is a generalization of the route -finding problem described earlier. Rather
than following a discrete set of routes, a robot can move in a continuous
space with (in principle) an infinite set of possible actions and states . For a
circular robot moving on a flat surface, the space is essentially two -
dimensional. When the robot has arms and legs or wheels that must also
be controlled, the search space becomes many -dimensional.
1) States:
Locations, Position of actuators.
2) Initia l state:
Start position
3) Successor Function:
Movement, action of actuators
4) Goal Test:
Task -dependent
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39 Artificial Intelligence 5) Path cost:
complex
Example 5: Automatic assembly sequencing
The aim is to find an order in which to assemble the parts of some object.
If the wrong order is chosen, there will be no way to add some part later in
the sequence without undoing some of the work already done. Checking a
step in the sequence for feasibility is a difficult geometrical search
problem closely related to robot navigation.
1) States:
Loca tion of components
2) Initial state:
No components assembled
3) Successor function:
Place component
4) Goal test:
System fully assembled
5) Path cost:
Number of moves
3.5 SEARCHING FOR SOLUTIONS A solution is an action sequence, so search algorithms work by
consideri ng various possible action sequences. The possible action
sequences starting at the initial state form a search tree with the initial
state at the root; the branches are actions and the nodes correspond to
states in the state space of the problem. A search problem where we aim
not only at reaching our goal but also at doing so at minimal cost is an
optimisation problem.
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40 Solving Problems by Searching 3.6 UNINFORMED SEARCH This topic covers several search strategies that come under the heading of
uninformed search or blind search. This search algorithm uses only initial
state, search operators and a test for a solution. It proceeds systematically
by exploring nodes either randomly or in some predetermined order.
3.6.1 Breadth First Search (BFS) :
It is a simple strategy in which the root node is expanded first, then all the
successors of the root node are expanded next, then their successors, and
so on.
All the nodes are expanded at a given depth in the search tree before any
nodes at the next level are expanded.
It starts at the tree roo t and explores all of the neighbor nodes at the
present depth prior to moving on to the nodes at the next depth level.

Fig. 3.5 Breadth -first search on a simple binary tree. At each stage,
the node to be expanded next is indicated by a marker
Function Br eadth First Search (start)
Begins
Open  start
Closed [ ];
While(open # [ ]) do
Begin
X  remove first [open]
If x is goal then return success
Else begin
Right end (open)  generated child(x)
Closed  x;
End
End
Return fail
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41 Artificial Intelligence The BFS algorithm, presented above, explore the space in a level -by-level
fashion. It maintains two lists namely open and closed. Open list contains
states which have been generated but whose children have not been
expanded and closed list records states tha t have already been examined.
Open is maintained as a queue First in first out (FIFO).
Time complexity: Equivalent to the number of nodes traversed in BFS
until the shallowest solution.
Space complexity: Equivalent to how large can the fringe get.
Completeness: BFS is complete, meaning for a given search tree, BFS
will come u p with a solution if it exists.
Optimality: BFS is optimal as long as the costs of all edges are equal .
3.6.2 Depth First Search (DFS) :
The algorithm starts at the root node and explores as far as possible along
each branch before backtracking.
DFS traverses the tree deepest node first. It would always pick the deeper
branch until it reaches the solution . The strategy can be implemented by
tree search with LIFO (Last in First Out) queue, most popularly known as
a stack.
Function Breadth First Search (start)
Begins
Open  start
Closed [ ];
While (open # [ ]) do
Begin
X  remove first [open]
If x is g oal then return success
Else begin
Left end (open)  generated child(x)
Closed  x;
End
End
Return fail
End
DFS always expands the deepest nodes in the current fringe of the search
tree. The search proceeds immediately to the deepest lev el of the search
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42 Solving Problems by Searching

Fig. 3.6 Depth -first search on a binary tree
3.6.3 Uniform Cost Search :
When all step costs are equal, breadth -first search is optimal because it
always expands the shallowest unexpanded node. By a simple extension,
we can find an algorithm that is optimal with any step -cost function.
Instead of expanding the shallowest node, uniform -cost search expands the
node n with the lowest path cost g(n).

Fig. 3.7 Uniform cost search on a binary tree
Uniform -cost search does not care abou t the number of steps a path has,
but only about their total cost. Therefore, it will get stuck in an infinite
loop if there is a path with an infinite sequence of zero -cost actions.
Uniform -cost search is guided by path costs rather than depths.
The prima ry goal of the uniform -cost search is to find a path to the goal
node which has the lowest cumulative cost.
Uniform -cost search expands nodes according to their path costs form the
root node.
It can be used to solve any graph/tree where the optimal cost i s in demand.
A uniform -cost search algorithm is implemented by the priority queue. It
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43 Artificial Intelligence Uniform cost search is equivalent to BFS algorithm if the path cost of all
edges is the same.
3.6.4 Depth Limited Search :
The problem of Depth First Search can be solved by supplying DFS with a
predetermined depth limit L. Nodes at depth L are treated as if they have
no successors. It shows the infinite path problem. This approach
introduces an additional source of i ncompleteness when Lshallowest goal is beyond the depth limit L.
Depth -limited search can be terminated with two Conditions of failure:
Standard failure value: It indicates that problem does not have any
solution.
Cutoff failure value: It defi nes no solution for the problem within a given
depth limit.

Fig. 3.8 Depth limited search tree
In the above example start node is S, Goal node is J and limit is 2.
Completeness: DLS search algorithm is complete if the solution is above
the depth -limit.
Time Complexity: Time complexity of DLS algorithm is O(bℓ).
Space Complexity: Space complexity of DLS algorithm is O (b×ℓ).
Optimal: Depth -limited search can be viewed as a special case of DFS,
and it is also not optimal even if ℓ>d.
3.6.9 Iterative Deepeni ng Depth -First :
The iterative deepening algorithm is a combination of DFS and BFS
algorithms. This search algorithm finds out the best depth limit and does it
by gradually increasing the limit until a goal is found.
This algorithm performs depth -first sear ch up to a certain "depth limit",
and it keeps increasing the depth limit after each iteration until the goal
node is found. munotes.in

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44 Solving Problems by Searching This Search algorithm combines the benefits of Breadth -first search's fast
search and depth -first search's memory efficiency.
The i terative search algorithm is useful uninformed search when search
space is large, and depth of goal node is unknown.
Function iterative deepening search (problem): returns solution or
failure
Begin
Inputs, problem, depth
For depth 0 to ∞ do
Result <- depth – limited – search (problem, depth)
If result # cut -off then return result.
End.

Fig. 3.9 Iterative deepening depth first search tree
1’st Iteration  A
2nd Iteration  A, B, C
3rd Iteration  A, B, D, E, C, F, G
4th Iteration  A, B, D, H, I, E, C, F, K, G
In the fourth iteration, the algorithm will find the goal node.
Completeness:
This algorithm is complete is if the branching factor is finite.
Time Complexity:
Let's suppose b is the branching factor and depth is d then the worst -case
time complexity is O(bd).
Space Complexity:
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45 Artificial Intelligence Optimal:
IDDFS algorithm is optimal if path cost is a non - decreasing function of
the depth of the node.
3.6.10 Bidirectional Search :
Bidirectional search algorithm runs t wo simultaneous searches, one form
initial state called as forward -search and other from goal node called as
backward -search, to find the goal node.
Bidirectional search substitutes one single search graph with two small
subgraphs in which one starts the search from an initial vertex and other
starts from goal vertex. The search stops when these two graphs intersect
each other. Bidirectional search can use search techniques such as BFS,
DFS, DLS, etc

Fig. 3.10 Bidirectional search tree
In the above searc h tree, bidirectional search algorithm is applied.
This algorithm divides one graph/tree into two sub -graphs. It starts
traversing from node 1 in the forward direction and starts from goal node
16 in the backward direction. The algorithm terminates at nod e 9 where
two searches meet.
Completeness: Bidirectional Search is complete if we use BFS in both
searches.
Time Complexity: Time complexity of bidirectional search using BFS
is O(bd).
Space Complexity: Space complexity of bidirectional search is O(bd).
Optimal: Bidirectional search is Optimal.

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46 Solving Problems by Searching 3.7 INFORMED SEARCH STRATEGIES In this topic we will see more efficient search strategy, the informed
search strategy or Heuristic Search. It is a search strategy that uses
problem -specific knowledge beyond the defin ition of the problem itself.
The term Heuristic is used for algorithms which find solutions among all
possible ones. Heuristic is a rule of thumb or judgement technique that
leads to a solution but provides no guarantee of success.
3.7.1 Greedy Best -First Search :
Best First Search is a combination of depth -first (DFS) and breadth -first
(BFS) search methods. It is used to select the single path at a time which is
more promising one than the current path. The most promising node is
chosen for expansion. It e valuates nodes by using just the heuristic
function; that is, f(n) = h(n).
Best first search algorithm:
Step 1: Put the initial node in OPEN.
Step 2: Pick the best node on OPEN, use heuristic method to find the
promising node, generate its successor.
Follo w the steps to each of its successor, do the step 2 till OPEN is not
empty and the goal is not achieved.
Step 3: Evaluate the non -generated nodes so far add them to the list of
OPEN and record their parents. If they have been generated before,
change the p arent if new path is better than previous one.
Advantages:
Best first search can switch between BFS and DFS by gaining the
advantages of both the algorithms.
This algorithm is more efficient than BFS and DFS algorithms.
Disadvantages:
It can behave as an u nguided depth -first search in the worst -case scenario.
It can get stuck in a loop as DFS.
This algorithm is not optimal. State H(n) S 13 A 12 B 4 C 7 D 3 E 8 munotes.in

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47 Artificial Intelligence F 2 H 4 I 9 G 0

Fig. 3.11 Greedy best first search graph.
Expand the nodes of S and p ut in the CLOSED list
Initialization: Open [A, B], Closed [S]
Iteration 1: Open [A], Closed [S, B]
Iteration 2: Open [E, F, A], Closed [S, B]
: Open [E, A], Closed [S, B, F]
Iteration 3: Open [I, G, E, A], Closed [ S, B, F]
: Open [I, E, A], Closed [S, B, F, G]
Hence the final solution path will be: S----> B----->F----> G
Time Complexity: The worst -case time complexity of Greedy best first
search is O(bm).
Space Complexity: The worst -case space complexity of Greedy best first
search is O(bm). Where, m is th e maximum depth of the search space.
Complete: Greedy best -first search is also incomplete, even if the given
state space is finite.
Optimal: Greedy best first search algorithm is not optimal.
3.7.2 A* search :
A* combines the value of the heuristic functio n h(n) and the cost to reach
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48 Solving Problems by Searching F(n)= g(n)+h(n)
Since g(n) gives the path cost from the start node to node n, and h(n) is the
estimated cost of the cheapest path from n to the goal, we have f(n) =
estimated cost of the cheapest solution thr ough n.
It has combined features of UCS and greedy best -first search, by which it
solves the problem efficiently. A * search algorithm finds the shortest path
through the search space using the heuristic function.
This search algorithm expands less search tree and provides optimal result
faster.
A* algorithm is similar to UCS except that it uses g(n)+h(n) instead of
g(n).
Algorithm of A* Algorithm:
Step1: Place the starting node in the OPEN list.
Step 2: Check if the OPEN list is empty or not, if the list is empty then
return failure and stops.
Step 3: Select the node from the OPEN list which has the smallest value
of evaluation function (g+h), if node n is goal node, then return success
and stop, otherwise
Step 4: Expand node n and generate all of its succ essors, and put n into
the closed list. For each successor n', check whether n' is already in the
OPEN or CLOSED list, if not then compute evaluation function for n' and
place into Open list.
Step 5: Else if node n' is already in OPEN and CLOSED, then it s hould be
attached to the back pointer which reflects the lowest g(n') value.
Step 6: Goto to Step 2 .
Advantages :
A* search algorithm is the best algorithm than other search algorithms.
A* search algorithm is optimal and complete.
This algorithm can solve v ery complex problems.
Disadvantages:
It does not always produce the shortest path as it mostly based on
heuristics and approximation.
A* search algorithm has some complexity issues.
The main drawback of A* is memory requirement as it keeps all generated
nodes in the memory, so it is not practical for various large -scale
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49 Artificial Intelligence Example: State H(n) S 5 A 3 B 4 C 2 D 6 G 0


Fig. 3.12 A * search graph
Initialization: {(S, 5)}
Iteration1: {(S--> A, 4), (S -->G, 10)}
Iteration2: {(S--> A-->C, 4), (S --> A-->B, 7), (S -->G, 10)}
Iteration3: {(S--> A-->C--->G, 6), (S --> A-->C--->D, 11), (S --> A-->B,
7), (S -->G, 10)}
Iteration 4 will give the final result, as S--->A--->C--->G it provides the
optimal path with cost 6.
Complete: A* algorithm is complete as long as:
Branching factor is finite.
Cost at every action is fixed.
Time complexity : It depends upon heuristic function. the time complexity
is O(b^d), where b is the branching factor.
Space Complexity: The space complexity of A* search algorithm
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50 Solving Problems by Searching Optimal: A* search algorithm is optimal. 1 2 3 8 4 7 6 5
3.8 HEURISTIC FUNCTIONS Heuristic function estimates the cost of an optimal path between a pair of
states in a single agent path -finding problem. Heuristic helps in solving
problems, even thou gh there is no guarantee that is will never lead in the
wrong direction.
A good heuristic function is determined by its efficiency. More is the
information about the problem, more is the processing time.
Some toy problems, such as 8 -puzzle, 8 -queen, tic -tac-toe, etc., can be
solved more efficiently with the help of a heuristic function .
Consider the following 8 -puzzle problem where we have a start state and a
goal state. Our task is to slide the tiles of the current/start state and place it
in an order foll owed in the goal state. There can be four moves either left,
right, up, or down . 1 2 3 8 6 7 5 4
Start state Goal state


Fig. 3 .13 8-puzzle solution munotes.in

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51 Artificial Intelligence 3.9 SUMMARY To solve Artificial Intelligence problem, we need to follow certain steps.
Firstly, define the problem precisely it means specify the problem space,
the operators for moving within the space and the starting and goal state.
Secondly, analyse the problem and finally, select one or more techniques
for representing knowledge and for problem solving and apply it to the
problem.
A problem consists of five parts: the initial state, a set of actions, a
transition model describing the results of those actions, a goal test
function, and a path cost function. The environment of the problem is
repres ented by a state space. A path through the state space from the initial
state to a goal state is a solution
Uninformed search methods have access only to the problem definition.
The basic algorithms are as follows:
 Breadth -first search expands the shallow est nodes first; it is complete,
optimal for unit step costs, but has exponential space complexity.
 Uniform -cost search expands the node with lowest path cost, g(n), and
is optimal for general step costs.
 Depth -first search expands the deepest unexpanded node first. It is
neither complete nor optimal, but has linear space complexity. Depth -
limited search adds a depth bound.
 Iterative deepening search calls depth -first search with increasing
depth limits until a goal is found. It is complete, optimal for unit step
costs, has time complexity comparable to breadth -first search, and has
linear space complexity.
 Bidirectional search can enormously reduce time complexity, but it is
not always applicable and may require too much space.
 Informed search methods m ay have access to a heuristic function h(n)
that estimates the cost of a solution from n.
 The generic best -first search algorithm selects a node for expansion
according to an evaluation function.
 Greedy best -first search expands nodes with minimal h(n). It is not
optimal but is often efficient.
 A∗ search expands nodes with minimal f(n) = g(n) + h(n). A ∗ is
complete and optimal, provided that h(n) is admissible (for TREE -
SEARCH) or consistent (for GRAPH -SEARCH).
3.10 EXERCISES 1. Compare uniformed search with informed search.
2. What are the problems with Hill climbing and how can they be solved munotes.in

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52 Solving Problems by Searching 3. State the best first search algorithm.
4. Explain advantages and disadvantages of DFS and BFS.
5. You have two jugs, measuring 8 gallons, 5 gallons and a wa ter faucet.
You can fill the jugs up or empty them out from one to another or
onto the ground. You need to measure out exactly three gallons.
6. Three towers labelled A, B and C are given in which tower A has
finite number of disks (n) with decreasing size. The task of the game
is to more the disks from tower A to tower C using tower B as an
auxiliary. The rules of the game are as follows:
1. Only one disk can be moved among the towers at any given time.
2. Only the “top” disk can be removed.
3. No large disk can site over a small disk.
Solve the problem and describe the following terms:
a) State b) Initial state c) Goal state d) Operators e) Solution
7 Write down the heuristic function for 8 -puzzle problem and solve the
problem.
8 What is the use of online search agent in unknown environment?
9 Define the following terms in reference to Hill Climbing
a) Local Maximum
b) Plateau
c) Ridge
10 Using suitabl e example, illustrate steps of A * search. Why is A*
search better than best first search?
11 Describe A* search algorithm. Prove that A* is complete and optimal
3.11 BIBLIOGRAPHY  Deva Rahul, Artificial Intelligence a Rational Approach, Shroff, 2014.
 Khem ani Deepak, A First course in Artificial Intelligence, McGraw
Hill, 2013.
 Chopra Rajiv, Artificial Intelligence a Practical Approach, S. Chand,
2012.
 Russell Stuart, and Peter Norvig Artificial Intelligence a Modern
Approach, Pearson, 2010.
 Rich Elaine, e t al. Artificial intelligence, Tata McGraw Hill, 2009.

*****
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53 4
BEYOND CLASSICAL SEARCH
Unit Structure
4.1 Objective
4.2 Introduction
4.3 Local search algorithms
4.4 Searching with non -deterministic action
4.5 Searching with partial observations
4.6 Online search agents and unknown environments
4.7 Summary
4.8 Unit End Questions
4.9 Bibliography
4.1 OBJECTIVES After this chapter, you should be able to:
 Understand the local search algorithms and optimization problems.
 Understand searching techniques of non -deterministic action.
 Familiar with partial observ ation.
 Understand online search agents and unknown environments.
4.2 INTRODUCTION This chapter covers algorithms that perform purely local search in the state
space, evaluating and modifying one or more current states rather than
systematically exploring p aths from an initial state. These algorithms are
suitable for problems in which all that matters is the solution state, not the
path cost to reach it. The family of local search algorithms includes
methods inspired by statistical physics (simulated anneali ng) and
evolutionary biology (genetic algorithms).
The search algorithms we have seen so far, more often concentrate on path
through which the goal is reached. But the problem does not demand the
path of the solution and it expects only the final configura tion of the
solution then we have different types of problem to solve.
Local search algorithm operates using single path instead of multiple
paths. They generally move to neighbours of the current state. Local
algorithms use very little and constant amount of memory. Such kind of
algorithms have ability to figure out reasonable solution for infinite state
spaces.
They are useful for solving pure optimization problems. For search where
path does not matter, Local search algorithms can be used, which operates munotes.in

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54 Beyond Classical Search by using a single current state and generally move only to neighbours of
the state.
4.3 LOCAL SEARCH ALGORITHMS • Algorithms that perform purely local search in the state space,
evaluating and modifying one or more current states rather than
systematicall y exploring paths from an initial state.
• These algorithms are suitable for problems in which all that matters is
the solution state, not the path cost to reach it.
• The family of local search algorithms includes methods inspired by
statistical physics (simulated annealing) and evolutionary biology
(genetic algorithms).
• This systematicity is achieved by keeping one or more paths in memory
and by recording which alternatives have been explored at each point
along the path.
• When a goal is found, the path to th at goal also constitutes a solution to
the problem.
• In many problems, however, the path to the goal is irrelevant.
• Local search algorithms operate using CURRENT NODE a single
current node (rather than multiple paths) and generally move only to
neighbors o f that node.
• Local Search Algorithm use very little memory —usually a constant
amount; and they can often find reasonable solutions in large or infinite
(continuous) state spaces for which systematic algorithms are
unsuitable.
• It is also useful for solving pure optimization problems, in which the
aim is to find the best state according to an objective function.
• Hill Climbing is a technique to solve certain optimization problems.
• In these techniques, we start with a sub -optimal solution and the
solution is im proved repeatedly until some condition is maximized.
• The idea of starting with a sub -optimal solution is compared to walking
up the hill, and finally maximizing some condition is compared to
reaching the top of the hill.
Algorithm:
1) Evaluate the initial state. If it is goal state then return and quit.
2) Loop until a solution is found or there are no new operators left.
3) Select & apply new operator.
4) Evaluate new state
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55 Artificial Intelligence If it is better than current state then makes it new current state.
If it is not better than current then go to step 2.
4.3.1 Hill Climbing search :
It is so called because of the way the nodes are selected for expansion. At
each point search path, successor node that appears to lead most quickly to
the top of the hill is selected for exploration. It terminates when it reaches
a “peak” where no neighbour has a higher value. This algorithm doesn’t
maintain a search tree so the data structure for the current node need only
record the state and the value of the objective function. Hill Climbing
Algorithm is sometimes called as a greedy local search because it grabs a
good neighbour state without thinking ahead about where to go next.
Unfortunately, hill climbing suffers from following problems:
This is shown in Figure 4. 2.1

Fig. 4.1 Hill Climbing
 Local maxima :
Local maximum is a state which is better than its neighbour states, but
there is also another state which is higher than it. They are also called as
foot-hills. It is shown in Fig. 4. 2.2

Local Maximum





Fig. 4. 2 Local maximum or Foot Hill
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56 Beyond Classical Search Solution to this problem :
Backtracking technique can be a solution of the local maximum in state
space landscape. Create a list of the promising path so that the algorithm
can backtrack the search space and explore other paths as well.
 Plateau :
It is a flat area of the search space in which a whole set of neighbouring
states (nodes) have the same heuristic value. A plateau is shown in Fig.
4.3.


Plateau/ Flat max imum



Fig 4 .3 Plateau
Solution to this problem :
The solution for the plateau is to take big steps or very little steps while
searching, to solve the problem. Randomly select a state which is far away
from the current state so it is possible that the alg orithm could find non -
plateau region.
Another solution is to apply small steps several times in the same
direction.
 Ridge :
It’s an area which is higher than surrounding states but it cannot be
reached in a single move. It is an area of the search space whi ch is higher
than the surrounding areas and that itself has a slope. Ridge is shown in
Fig. 4.4.
Ridge



Fig. 4.4 Ridge

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57 Artificial Intelligence Solution to this problem :
Trying different paths at the same time is a solution. With the use of
bidirectional search, or by mov ing in different directions, we can improve
this problem.
Advantages of Hill Climbing :
It can be used in continuous as well as discrete domains.
Disadvantages of Hill Climbing :
It is not an efficient method.
It is not suitable for those problems whose valu e of heuristic function
drops off suddenly when solution may be in sight.
It is local method as it looks at the immediate solution and decides about
the next step to be taken rather than exploring all consequences before
taking a move.
4.3.2 Simulated Anne aling :
A hill -climbing algorithm which never makes a move towards a lower
value guaranteed to be incomplete because it can get stuck on a local
maximum. And if algorithm applies a random walk, by moving a
successor, then it may complete but not efficient. Simulated Annealing is
an algorithm which yields both efficiency and completeness. In
mechanical term Annealing is a process of hardening a metal or glass to a
high temperature then cooling gradually, so this allows the metal to reach
a low -energy crystall ine state.
The same process is used in simulated annealing in which the algorithm
picks a random move, instead of picking the best move. If the random
move improves the state, then it follows the same path. Otherwise, the
algorithm follows the path which has a probability of less than 1 or it
moves downhill and chooses another path.
Algorithm :
1. Evaluate the initial state. If it is also goal state, then return it and quit.
Otherwise, continue with the initial state as the current state.
2. Initialize BE ST-SO-FAR to the current state.
3. Initialize T according to the annealing schedule.
4. Loop until a solution is found or until there are no new operators left
to be applied in the current state.
a) Select an operator that has not yet been applied to th e current
state and apply it to produce a new state.
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58 Beyond Classical Search  E = (value of current) - (value of new state)
 If the new state is a goal state, then return it and quit.
 If it is not goal state but is better than the current state , then make
it the current state. Also set BEST -SO-FAR to this new state.
 If it is not better than current state, then make it the current state
with probability p’ as defined above. This step is usually
implemented by invoking a random number generator to produce
a number in the range [0, 1]. If that number is less than p’ then
the move is accepted. Otherwise, do nothing.
c) Revise T as necessary according to the annealing schedule.
5. Return BEST -SO-FAR, as the answer.
To implement this revised algorithm, it is necessary to select an annealing
schedule, which has three components. The first is the initial value to be
used for temperature. The second is the criteria that will be used to decide
when the temperature of the system should be reduced. The third is a
amount by which the temperature will be reduced each time it is changed.
There may also be a fourth component of the schedule, namely, when to
quit. Simulated annealing is often used to solve problems in which the
number of moves from a given sate is very large. For such problems, it
may not make sense to try all possible moves.
4.3.3 Local beam search :
In this algorithm, it holds k number of states at any given time. At the
start, these states are generated randomly. The successors of these k states
are computed with the help of objective function. If any of these
successors is the maximum value of the objective function, then the
algorithm stops. Otherwise, the (initial k states and k number of successors
of the states = 2k) states are placed in a poo l. The pool is then sorted
numerically. The highest k states are selected as new initial states. This
process continues until a maximum value is reached. function Beam
Search (problem, k), returns a solution state.
Start with k randomly generated states
Loop
Generate all successors of all k states
If any of the states =solution, then return the state
Else select the k best successors
End
4.3.4 Genetic algorithms :
Genetic Algorithms (GAs) are adaptive heuristic search algorithms that
belong to the larger part of evolutionary algorithms. Genetic algorithms
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59 Artificial Intelligence intelligent exploitation of random search provided with historical data to
direct the search into the region of better performance in s olution space.
They are commonly used to generate high -quality solutions for
optimization problems and search problems.
Genetic algorithms simulate the process of natural selection which means
those species who can adapt to changes in their environment are able to
survive and reproduce and go to next generation. In simple words, they
simulate “survival of the fittest” among individual of consecutive
generation for solving a problem. Each generation consist of a population
of individuals and each individual represents a point in search space and
possible solution. Each individual is represented as a string of
character/integer/float/bits. This string is analogous to the Chromosome.
Steps of genetic algorithms:
Initial Population :
The process begins with a se t of individuals which is called a Population.
Each individual is a solution to the problem you want to solve. An
individual is characterized by a set of parameters (variables) known as
Genes. Genes are joined into a string to form a Chromosome (solution). In
a genetic algorithm, the set of genes of an individual is represented using a
string, in terms of an alphabet. Usually, binary values are used (string of
1s and 0s). We say that we encode the genes in a chromosome.
Fitness Function :
The fitness functi on determines how fit an individual is (the ability of an
individual to compete with other individuals). It gives a fitness score to
each individual. The probability that an individual will be selected for
reproduction is based on its fitness score.
Selec tion:
The idea of selection phase is to select the fittest individuals and let them
pass their genes to the next generation. Two pairs of individuals (parents)
are selected based on their fitness scores. Individuals with high fitness
have more chance to b e selected for reproduction.
Crossover :
It is the most significant phase in a genetic algorithm. For each pair of
parents to be mated, a crossover point is chosen at random from within the
genes. Offspring are created by exchanging the genes of parents amo ng
themselves until the crossover point is reached.
Mutation :
In certain new offspring formed, some of their genes can be subjected to a
mutation with a low random probability. This implies that some of the bits
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60 Beyond Classical Search

Fig. 4 .5 The genetic algorithm, illustrated for digit strings representing 8 -
queens states. The initial population in (a) is ranked by the fitness function
in (b), resulting in pairs for mating in (c). They produce offspring in (d),
which are subject to mutation in (e)
4.4 SEARCHING WITH NON -DETERMINISTIC ACTION When the environment is either partially observable or nondeterministic
(or both), precepts become useful. When the environment is
nondeterministic, precepts tell the agent which of the possible outcomes of
its actions has actually occurred. In both cases, the future precepts cannot
be determined in advance and the agent’s future actions will depend on
those future precepts. So, the solution to a problem is not a sequence but a
contingency plan (also known as a strategy) that specifies what to do
depending on what precepts are received.
4.4.1 The erratic vacuum world :
There are three actions - Left, right and Suck. And the goal is to clean up
all the dirt.

Fig. 4.6 The eight possible states of the vacuum world; states 7 and 8
are goal states.
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61 Artificial Intelligence In the erratic vacuum world, the Suck action works as follows:
• When applied to a dirty square the action cleans the square and
sometimes cleans up dirt in an adjacent square, too.
• When applied to a clean squa re the action sometimes deposits dirt on
the carpet.
• a contingency plan such as the following:
[Suck, if State = 5 then [Right, Suck] else [ ]]
If the environment is observable, deterministic, and completely known,
then the problem is trivially solvabl e by any of the algorithms.
For example, if the initial state is 1, then the action sequence [Suck, Right,
Suck] will reach a goal state, 8. Thus, solutions for nondeterministic
problems can contain nested if –then–else statements;
4.4.2 AND - OR search tree s:
A solution to a problem can be obtained by decomposing it into smaller
sub-problems. Each of this sub -problem can then be solved to get its
solution. These sub -solutions can then be recombined to get a solution as a
whole. OR graph finds a single path.
AND arc may point to any number of successor nodes, all of which must
be solved in order for an arc to point a solution. This is actually called as
an AND -OR graph or AND/OR tree.

Fig. 4.7 An algorithm for searching AND –OR graphs generated by
nondetermin istic environments.
Figure 4. 8 gives a recursive, depth -first algorithm for AND –OR graph
search. One key aspect of the algorithm is the way in which it deals with
cycles, which often arise in nondeterministic problems (e.g., if an action
sometimes has no e ffect or if an unintended effect can be corrected). If the
current state is identical to a state on the path from the root, then it returns
with failure. This doesn’t mean that there is no solution from the current
state; it simply means that if there is a noncyclic solution, it must be
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62 Beyond Classical Search incarnation can be discarded. With this check, we ensure that the
algorithm terminates in every finite state space, because every path must
reach a goal , a dead end, or a repeated state.
Example.








Fig. 4.8 AND/OR graph example
4.5 SEARCHING WITH PARTIAL OBSERVATIONS When an environment is partially observable, an agent can be in one of
several possible states. An action leads to one of several p ossible
outcomes.
To solve these problems, an agent maintains a belief state that represent
the agent’s current belief about the possible physical state it might be in,
given the sequence of actions and percepts up to that point.
Agents percepts cannot pin down the exact state the agent is in
Let Agents have Belief states
◦ Search for a sequence of belief states that leads to a goal
◦ Search for a plan that leads to a goal
Sensor -less vacuum world
Assume belief states are the same but no location or dust sensor s
Initial state = {1, 2, 3, 4, 5, 6, 7, 8}
Action: Right
Result = {2, 4, 6, 8}
Right, Suck
Result = {4, 8}
Right, Suck, Left, Suck
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63 Artificial Intelligence

Fig. 4. 2.9 Sensor less vacuum world
4.5.1 Searching with no observation :
It is instructive to see how the belief -state search problem is constructed.
Suppose the underlying physical problem P is defined by ACTIONSP,
RESULTP, GOAL -TESTP, and STEP -COSTP.
Then we can define the corresponding sensor less problem as follows:
• Belief states: The entire belie f-state space contains every possible set
of physical states. If P has N states, then the sensor less problem has up
to 2N states, although many may be unreachable from the initial state.
• Initial state: Typically, the set of all states in P, although in s ome cases
the agent will have more knowledge than this.
• Actions: This is slightly tricky. Suppose the agent is in belief state b =
{s1, s2}, but ACTIONS P(s1) ≠ ACTIONS (s2); then the agent is
unsure of which actions are legal.
 Transition model: The agent doesn’t know which state in the belief
state is the right one; so as far as it knows, it might get to any of the
states resulting from applying the action to one of the physical states in
the belief state.
 Goal test: The agent wants a plan that is sure to work, which means
that a belief state satisfies the goal only if all the physical states in it
satisfy GOAL -TEST P. The agent may accidentally achieve the goal
earlier, but it won’t know that it has done so.
 Path cost: This is also tricky. If the same acti on can have different
costs in different states, then the cost of taking an action in a given
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64 Beyond Classical Search

Fig. 4.10 (a) Predicting the next belief state for the sensorless vacuum
world with a deterministic action, right. (b) Prediction for the same
belief state and action in the slippery version of the sensorless vacuum
world.
4.5.2 Searching with observations :
For a general partially observable problem, we have to specify how the
environment generates percepts for the age nt. For example, we might
define the local -sensing vacuum world to be one in which the agent has a
position sensor and a local dirt sensor but has no sensor capable of
detecting dirt in other squares. The formal problem specification includes
a PERCEPT(s) function that returns the percept received in a given state.
(If sensing is nondeterministic, then we use a PERCEPTS function that
returns a set of possible percepts.) For example, in the local -sensing
vacuum world, the PERCEPT in state 1 is [A, Dirty]. Fu lly observable
problems are a special case in which PERCEPT(s) = s for every state s,
while sensorless problems are a special case in which PERCEPT(s) = null.
• The ACTIONS, STEP -COST, and GOAL -TEST are constructed from
the underlying physical problem just a s for sensorless problems, but
the transition model is a bit more complicated.
• The prediction stage is the same as for sensorless problems: given the
action a in belief state b, the predicted belief state is ˆb = PREDICT
(b, a).
• The observation prediction stage determines the set of percepts o that
could be observed in the predicted belief state: POSSIBLE -
PERCEPTS (ˆb) = {o : o = PERCEPT(s) and s ∈ ˆb} .
• The update stage determines, for each possible percept, the belief state
that would result from the per cept. The new belief state bo is just the
set of states in ˆb that could have produced the percept: bo =
UPDATE (ˆb, o) = {s : o = PERCEPT(s) and s ∈ ˆb}.
• Putting these three stages together, we obtain the possible belief states
resulting from a given acti on and the subsequent possible percepts:
RESULTS (b, a) = {bo : bo = UPDATE(PREDICT(b, a), o) and o ∈
POSSIBLE -PERCEPTS(PREDICT(b, a))} .

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65 Artificial Intelligence Fig.4.1 0 Two example of transitions in local -sensing vacuum worlds. (a)
In the deterministic world, Right is applie d in the initial belief state,
resulting in a new belief state with two possible physical states; for those
states, the possible percepts are [B, Dirty] and [B, Clean], leading to two
belief states, each of which is a singleton. (b) In the slippery world, Right
is applied in the initial belief state, giving a new belief state with four
physical states; for those states, the possible percepts are [A, Dirty], [B,
Dirty], and [B, Clean], leading to three belief states as shown.
4.5.3 Solving partially observab le problems :
The preceding section showed how to derive the RESULTS function for a
nondeterministic belief -state problem from an underlying physical
problem and the PERCEPT function.
To solve these problems, an agent maintains a belief state that represent
the agent's current belief about the possible physical state it might be in,
given the sequence of actions and percepts up to that point.
Agent’s percepts cannot pin down the exact state the agent is in
Let Agents have Belief states
 Search for a sequence of belief states that leads to a goal
 Search for a plan that leads to a goal
For such formulation, the AND -OR search algorithm can be applied
directly to derive a solution. Figure 4. 2.11 shows part of the search tree for
the local -sensing vacuum world, as suming an initial percept [A, Dirty].
The solution is the conditional plan [Suck, Right, if B state = {6} then
Suck else [ ]] .

Fig. 4. 11 AND -OR search tree
4.5.4 An agent for partially observable environments :
The design of a problem -solving agent for partially observable
environments is quite similar to the simple problem -solving agent. the
agent formulates a problem, calls a search algorithm (such as AND -OR-
GRAPH -SEARCH) to solve it, and executes the solution. There are two
main differences. First, t he solution to a problem will be a conditional plan munotes.in

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66 Beyond Classical Search rather than a sequence; if the first step is an if –then–else expression, the
agent will need to test the condition in the if -part and execute the then -part
or the else -part accordingly. Second, the agent will need to maintain its
belief state as it performs actions and receives percepts.

Fig. 4. 12 Two prediction –update cycles of belief -state maintenance in the
kindergarten vacuum world with local sensing.
4.6 ONLINE SEARCH AGENTS AND UNKNOWN ENVIRONMENTS So far, we have concentrated on agents that use offline search algorithms.
They compute a complete solution before setting foot in the real world and
then execute the solution. In contrast, an online search agent interleaves
computation and action: first it takes an action, then it observes the
environment and computes the next action. Online search is a good idea in
dynamic or semi dynamic domains —domains where there is a penalty for
sitting around and computing too long.
Online search is also helpful in nondeterministic domains because it
allows the agent to focus its computational efforts on the contingencies
that actually arise rather than those that might happen but probably won’t.
Of course, there is a trade -off: the more an agent plans ahead, the le ss
often it will find itself up the creek without a paddle.
Online search is a necessary idea for unknown environments, where the
agent does not know what states exist or what its actions do. In this state
of ignorance, the agent faces an exploration probl em and must use its
actions as experiments in order to learn enough to make deliberation
worthwhile.
The canonical example of online search is a robot that is placed in a new
building and must explore it to build a map that it can use for getting from
A to B.
4.6.1 Online search problem:
An online search problem must be solved by an agent executing actions,
rather than by pure computation. online search agents can build a map and
find a goal if one exists.
We assume a deterministic and fully observable env ironment but we
stipulate that the agent knows only the following:
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67 Artificial Intelligence 1. ACTIONS(s)
2. The step -cost function
3. GOAL -TEST(s).
For example, in the maze problem shown in Figure, the agent does not
know that going Up from (1,1) leads to (1,2); nor, having done that, d oes it
know that going Down will take it back to (1,1).
 Necessary in unknown environments
 Robot localization in an unknown environment (no map)
 Does not know about obstacles, where the goal is, that UP from (1,1)
goes to (1, 2)
 Once in (1, 2) does not know that down will go to (1, 1)
 Some knowledge might be available
 If location of goal is known, might use Manhattan distance heuristic
 Competitive Ratio = Cost of shortest path without exploration/Cost of
actual agent path
 Irreversible actions can lead to de ad ends and CR can become infinite.
 Hill- Climbing is already an online search algorithm but stops at local
optimal.

Fig. 4. 13 A simple maze problem
4.6.2 Online search agents :
After each action, an online agent receives a percept telling it what state i t
has reached; from this information, it can augment its map of the
environment. An online algorithm, on the other hand, can discover
successors only for a node that it physically occupies.
To avoid traveling all the way across the tree to expand the next node, it
seems better to expand nodes in a local order. Depth -first search has
exactly this property because (except when backtracking) the next node
expanded is a child of the previous node expanded.
An online depth -first search agent is shown in Figure 4.2.14
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Fig. 4. 14 An online search agent that uses depth -first exploration. The
agent is applicable only in state spaces in which every action can be
“undone” by some other action
It is fairly easy to see that the agent will, in the worst case, end up
traversing every link in the state space exactly twice. For exploration, this
is optimal; for finding a goal, on the other hand, the agent’s competitive
ratio could be arbitrarily bad if it goes off on a long excursion when there
is a goal right next to the initial state.
Because of its method of backtracking, ONLINE -DFS-AGENT works
only in state spaces where the actions are reversible. There are slightly
more complex algorithms that work in general state spaces, but no such
algorithm has a bounded competit ive ratio.
4.6.3 Online local search :
Like depth -first search, hill -climbing search has the property of locality in
its node expansions. In fact, because it keeps just one current state in
memory, hill -climbing search is already an online search algorithm!
Unfortunately, it is not very useful in its simplest form because it leaves
the agent sitting at local maxima with nowhere to go. Moreover, random
restarts cannot be used, because the agent cannot transport itself to a new
state.
Instead of random restart s, one might consider using a random walk to
explore the environment. A random walk simply selects at random one of
the available actions from the current state; preference can be given to
actions that have not yet been tried. It is easy to prove that a ra ndom walk
will eventually find a goal or complete its exploration, provided that the
space is finite. On the other hand, the process can be very slow.

Fig. 4.1 5 An environment in which a random walk will take exponentially
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69 Artificial Intelligence 4.7 SUMMARY This chapter has examined search algorithms for problems beyond the
“classical” case of finding the shortest path to a goal in an observable,
deterministic, discrete environment. Local search methods such as hill
climbing operate on complete -state formulations, keeping only a small
number of nodes in memory. Several stochastic algorithms have been
developed, including simulated annealing, which returns optimal solutions
when given an appropriate cooling schedule.
A genetic algorithm is a stochasti c hill -climbing search in which a large
population of states is maintained. New states are generated by mutation
and by crossover, which combines pairs of states from the population.
In nondeterministic environments, agents can apply AND –OR search to
gener ate contingent plans that reach the goal regardless of which outcomes
occur during execution.
When the environment is partially observable, the belief state represents
the set of possible states that the agent might be in. Standard search
algorithms can b e applied directly to belief -state space to solve sensor less
problems, and belief -state AND –OR search can solve general partially
observable problems.
4.8 UNIT END QUESTIONS 1 What do you mean by local maxima with respect to search technique?
2 Explain Hill Climbing Algorithm.
3 Explain AO* Algorithm.
4 Explain a searching with nondeterministic action.
5 Explain searching with partial observations.
6 Write a note on simulated annealing.
7 Explain a local search algorithm.
8 Write a note on online search agents.
9 Write a note on unknown environments.
10 Consider the sensorless version of the erratic vacuum world. Draw the
belief -state space reachable from the initial belief state {1, 2, 3, 4, 5,
6, 7, 8}, and explain why th e problem is unsolvable.
4.9 BIBLIOGRAPHY  Deva Rahul, Artificial Intelligence a Rational Approach, Shroff, 2014.
 Khemani Deepak, A First course in Artificial Intelligence, McGraw
Hill, 2013.
 Chopra Rajiv, Artificial Intelligence a Practical Approach, S. Ch and,
2012.
 Russell Stuart, and Peter Norvig Artificial Intelligence a Modern
Approach, Pearson, 2010.
 Rich Elaine, et al. Artificial intelligence, Tata McGraw Hill, 2009.

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70
UNIT III
5
ADVERSARIAL SEARCH
Unit Structure
5.0 Objective
5.1 Introduction
5.2 Games
5.3 Optimal Decision in Games
5.3.1 Minimax Algorithm
5.3.2 Optimal Decision in Multilayer Games
5.4 Alpha -Beta Pruning
5.4.1 Move Ordering
5.5 Imperfect Real -Time Decisi on
5.5.1 Evaluation Function
5.5.2 Cutting off Search
5.5.3 Forward Pruning
5.5.4 Search versus Lookup
5.6 Stochastic Games
5.6.1 Evaluation functions for games of chance
5.7 Partially Observable Games
5.7.1 Kriegspiel: Partially observable Chess
5.7.2 C ard Games
5.8 State -of-Art Game Programming
5.9 Summary
5.10 Exercise
5.11 Bibliography
5.0 OBJECTIVE The objective of this chapter is to study the different strategies used by
players to compete each other and try to win the game.
5.1 INTRODUCTION In this chapter we will see Adversarial search method in which we
examine the situation where agent compete with one another and try to
defeat one another in order to win the game. Adversarial search is a game
playing technique in which two or more players w ith conflicting goals are
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71 Artificial Intelligence 5.2 GAMES In single agent environment the solution is expressed in the form of
sequence of actions. But there might be a situation occur in a game
playing where more than one agent searching for a solution in the same
search space. The environment with more than one agent is called as
multi -agent environment, in which each agent has to consider the action of
other agent and how they affect its own welfare. Such conflicting goal
give rise to the adversarial search. So, searches in which two or more
players with conflicting goals are trying to explore the same search space
for the solution are known as Games.
Types of Games in AI :
Perfect Information Game : In this game agent a ct sequentially and
observe the state of the world before acting. Each agent acts to maximize
its own utility. The agent has all the information about the game and they
can monitor each other movements as well. E.g. Chess, Checkers etc.
Imperfect informati on Game : In this type of game agent don’t have all
information about the game and not aware of what’s going on. For e.g. tic -
toc-toe, blind bridge, battleship etc.
Deterministic game - Deterministic games are follow a strict pattern and
set of rules for the game. Change in state is fully determined by player
move. For e.g . chess, Go, Checkers, tic -tac-toe etc.
Non-Deterministic games : non-deterministic games have various
unpredictable events and it involves a factor of chance or luck. So, change
in state is partially determined s by chance. In this type of game each
action response is not fixed. Such type of game is also called as stochastic
games. E.g. Poker, Monopoly, Backgammon etc.
Game Problem formulation :
A game can be formally defined s a kind of search problem with various
elements as follows
 S0- initial state, which specifies how the game is set up at the start
 Player(s) -defines which player has the move in a state
 Actions(s) -returns the set of legal moves in a state.
 Result (s, a) -The transition model defines the result of a move.
 Terminal -Test(s) - A terminal test is true when the game is over and
false otherwise. Terminal state is sate where game has ended.
 Utility (s, p) - Utility function defines the final numeric value for a
game that ends in termi nal state s for player p. Let’s consider chess in
which outcome is a win, lose or draw, with values +1, 0 or ½.
In zero-sum game the total payoff to all players is the same for every
instance of the game. Chess is a zero some game because every game has
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72 Adversarial Search Game Tree :
A game tree is defined by the initial state, ACTION and RESULT
functions. A game tree is a tree where nodes indicate game states and
edges indicate moves.
We consider tic -tac-toe game with two player, one pl ayer call MAX and
another player call MIN. the following figure shows the part of game tree
for tic -tac-toe. Max has nine possible moves from initial state. The player
alternates between placing an X on MAX’s turn and O on MIN’s turn until
reach up to lea f node. Leaf nodes corresponding to terminal state such that
one plyer has three in a row or all the squares are filled. Both players will
continue each node, and trying to keep each other from winning. In game
tree there a layer for MIN and MAX called as ply. The numbers on leaf
node indicate the utility value of the terminal state. By considering MAX
point of view, high value is good for MAX and bad for MIN. In this either
MAX win or MIN win or it’s draw.

5.3 OPTIMAL DECISION IN GAMES The optimal solut ion in a normal search problem would be a sequence of
actions that leads to a goal state or a terminal state. In adversarial search,
MAX must find a contingent strategy, which specifies MAX’s move in the
initial state, then as a result from every possible response by MIN, Max
makes moves in the state, then MIN’s moves in the state resulting from
every possible response by MAX to those moves and so on.
Given a game tree, the optimal strategy can be determined from the
minimax value of each node, which can be written as MINIMAXX(n). if
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73 Artificial Intelligence game, then the minimax value of a node is the utility of being in the
corresponding state. The minimax value of a terminal state is just its
utility. MAX pr efers to move to a state of maximum value and MIN
prefers a state of minimum value then

5.3.1 Minimax Algorithm :
Minimax is a barracking algorithm used in decision making and game
theory. It is used to find the optimal move for a player by assuming the
opponent is also playing optimally. It uses recursion to search through the
game tree. A Minimax algorithm is often used to play games in AI such as
Chess, tic -tac-toe, Go etc. Algorithm computes the minimax decision for
the current state.
In this algorit hm there are two players, MAX and MIN. Both players play
the game as one tries to get the highest score while the opponent tries to
get the lowest score. In the MINIMAX algorithm, the complete game tree
is explored based on depth first search algorithm. A MINIMAX algorithm
proceeds down the tree until the terminal node, then backtracks the tree in
a recursive manner.
In the following diagram, the algorithm first recurses down to the three
lowest left nodes and uses UTILITY function on them to find their val ues
are 3,12 and 8 respectively. Then it takes the minimum vale 3 and return
that value as the backup for node B. A same process gives backup value of
2 for C and 2 for D. Finally, to get the backed -up value of 3 for the root
node, we take the maximum of 3 ,2, and 2.
If maximum depth of the tree is m and at each point legal moves are b then
the time complexity of the minimax algorithm is O(bm). The algorithm
that generates all action at once then, the space complexity is O(bm). The
algorithm that generates a ctions one at a time then space complexity is
O(m).

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74 Adversarial Search Example :
1 The algorithm generates the entire game tree and apply the utility
function to get the utility values for the terminal state. Now consider
in the following diagram A is the initial state.

1) Now, first find the utilities value for MAX, so we will compare
each value in terminal state with initial value for MAX and
determine the higher node values.
So, for node D max ( -1, 8) => 8
for node E max ( -3, -1) => -1
for node F max (2 , 1) => 2
for node G max ( -3, 4) => 4

2) Now, it’s a turn for minimizer to minimize the MAX utility, so it
will compare all node values and finds the third layer node values.
for node B min (8, -1) => -1
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75 Artificial Intelligence

3) Now its turn for MAX to choose the maximum of all node values
and find the maximum value for the root node
for node A max ( -1, 2) => 2

5.3.2 Optimal Decision in Multiplayer Game :
There are many popular games that allow more than two players. Let us
examine how we can use the minimax concept for multiplayer games.
From a technical perspective, this seems straightforward but it raises some
interesting new conceptual issues. First, its necessary to replace single
value of each node with a vector of value. Consider a three player game
with players A, B and C having a vector (V A,VB,VC) associated with each
node. This vector gives the utility of the state from each player’s point of
view for terminal states. This can be implemented using utility functions
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76 Adversarial Search Now, for nonterminal states, consider the node X in the game tree as
shown in below figure. In that state, player C decides what to do. The two
choices lead to terminal states with utility vectors (V A =1, V B =2, V C =6)
and (V A =4, V B =2, V C =3). C choose the first move because 6 is bigger
than 3, means if state X is reached, subsequently play will lead to a
terminal state with utilities (V A =1, V B =2, V C =6). So, in this vector it
backed -up the values of X.
Algorithm for calculating mi nimax decision

5.4 ALPHA -BETA PRUNING It is the modified version of minimax algorithm. In minimax search, the
number of game states it has to examine is exponential in the depth of the
tree. The exponent can’t be eliminated but we can cut it to half. The refore,
there is a way to compute the correct minimax decision without checking
every node in the game tree called as pruning. It involves two threshold
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77 Artificial Intelligence parameters such as Alpha and Beta for future expansion, so known as
alpha -beta pruning or alpha -beta algorithm. This can be applied at any
depth of a tree. The alpha -beta pruning to a standard minimax algorithm
returns the same moves but it removes all the nodes which are not
affecting the final decision. It makes the algorithm fast by pruning these
nodes .
α=the best or the highest value choice found so far at any point along the
path of maximizer. The initial value of beta is -∞.
β= the best or lowest value choice found so far at any point along the path
of minimizer. The initial value of beta is +∞.
During alpha -beta search, the value of alpha and beta are updated as they
go along, and prunes the remaining branches at a node as soon as the value
of current node is known to be worse than the current alpha and beta
values for MAX and MIN respectively. The m ain condition required for
alpha -beta pruning is α >= β.
Alpha -Beta Search Algorithm :

Example :
1) In the first step MAX player start with first move from node A
where α= -∞ and β= +∞. Now these values of α and β are passed to
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78 Adversarial Search

2) At node D the MAX player will play and calculate the α value. The
value of α compare with 2 and 3. So max (2, 3) =3. So, 3 will be the
value of α at node D.
3) Now backtracking is performed at node B. The value of β will
change as MIN player will play. Now β = +∞, will compare with the
available subsequent nodes value. So, min (∞, 3) = 3, hence at node
B now α= -∞, and β= 3.

4) Algorithm traverse the next successor of node B which is nothing but
node E. the value of α = -∞ and β =3. Now, MAX will play at node E
and value of α will change. The current value of α will be compared
with 5. So max ( -∞, 5) =5. At node E, α =5 and β =3. But α>= β so
the right successor of E will be pruned an algorithm stop traversing
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79 Artificial Intelligence

5) Algorithm again backt rack the tree from node B to A. The value of
alpha will be changes at node A. max ( - ∞,3) =3 and β=+∞. Both the
values are passed to right successor of A which is C. Now, at node
C, α=3 and β= +∞, and these same values will be passed on to node
F
6) At node F value of alpha is compared with 0 so max (3,0)= 3 and
compared with right child 1, so max (3,1) =3 but still α remains 3,
but the node value of F will become 1.

7) Node F return the node value 1 to node C. At node C the value of
beta is changed so min ( ∞, 1) =1. Now at C, α=3 and β= 1, and again
it satisfies the condition α>=β. Now G node will be pruned and
algorithm not traverse the entire sub tree of G. C now returns the
value of 1 to A here the best value for A is max (3, 1) = 3. And
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80 Adversarial Search

5.4.1 Move Ordering :
The effectiveness of alpha -beta pruning is highly dependent on the order
in which states are examined. It can be of two types
1) Worst ordering - Alpha beta pruning algorithm exactly work like
minimax algorithm and it does not prune any leaves of the tree in
some cases. So, in that case consumes more time because of alpha
beta factors. Such a move pruning is known as worst ordering. the
time complexity in for such case is O(bm).
2) Ideal ordering - it occurs when lots of pruning happens in the tree.
Best move is occurred at the left side of the tree. The time
complexity in ideal ordering is O(bm/2).
5.5 IMPERFECT REAL -TIME DECISION The minimax algorithm generates entire game search space while the
alpha beta pruning algorithm al lows to prune large path. The alpha beta
has to search all the way to terminal state for at least a portion of search
space. Due to the fact that moves have to be made within a reasonable
amount of time, this depth is not usually practical. In order to mak e a
move in reasonable amount of time, program should cut off the search
earlier and heuristic evaluation function is applied. It effectively turns
nonterminal nodes into terminal leaves.
Alter minimax or alpha beta in two ways, first is replace the utilit y
function by heuristic evaluation function EVAL. This estimates the
position’s utility. And second, replace terminal test by a cutoff test that
decides when to apply EVAL function. The following is the heuristic
minimax for state s and maximum depth d
H-MINIMAX (s, d)=
EVAL(s) if
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81 Artificial Intelligence max aЄAction(s)H-MINIMAX(RESULT(s,a),d+1 if
Player(s)=MAX min aЄ Action(s)
H-MINIMAX (RESULT(s,a),d+1 if
Player(s)=MIN
5.5.1 Evaluation Function :
This function returns an estimate of the expected utility of the game from
a given position. the performance of a game playing program strongly
depends on the quality of its evaluation function. It is possible that the
wrong evaluation function will be led an agent to position that turns out to
be lost.
Design good evaluati on function :
1) The evaluation function should order the terminal states same as true
utility function: win states must better than draws, which in turn
draw states must be better than loss states.
2) The computation must not take too long.
3) The evaluation func tion should be strongly correlated with the actual
chances of winning for non -terminal states.
The majority of evaluation functions work by calculating various features
of the state. For example, there are features for the number of white
pawns, black pawn and so on in chess. This all features define various
categories of state. The state in each category has same value for all the
features. Consider one category includes two pawns vs. one pawn. Any
given category contains some states that led to wins, some draw and some
led to losses. The evaluation function cannot determine which states are
which. It can return single value that reflect the proportion of states with
each outcome.
Suppose in two -pawns vs. one -pawn 72% of the states led to win, 20%
loss an d 8% draw. Then expected value=(0.72*
1)+(0.20*0)+(0.08*0.5)=0.76. the expected value can be found for each
category resulting in an evaluation function for each state. The evaluation
function for terminal not return actual expected values as long as the
ordering of the state is the same. Many evolutions function find separate
numerical contributions from each feature and combine them to find total
value. Weighted linear function is a kind of evaluation function that can be
expressed as the following
Eval(s ) = w 1 f1 (s) + w 2 f2 (s) + … + w n fn(s)
Where w i is a weight, fi is a feature of the position, the fi is the number of
each kind of piece and the w i is the value of the pieces in a chess.
5.5.2 Cutting off Search :
Alpha -beta search is modified so that it will call the heuristic EVAL
function when it is appropriate to cut off the search. So, we can modify munotes.in

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82 Adversarial Search alpha beta search algorithm and replace terminal state with cutoff -test and
utility is replace by EVAL.
if Cutoff -Test(state, depth) then return Eval(stat e)
by setting a fixed depth limit is the easiest way to control the amount of
searching so that cutoff return true for all depth greater than fixed depth d.
In order to select a move within the allocated time, the depth d is chosen
for it. Iterative deepen ing search can be applied and also helps with move
ordering. The program returns the move selected by the deepest completed
search as time runs out.
5.5.3 Forward Pruning :
Forward pruning is a technique in which some moves at a given node are
pruned immedi ately without further consideration. It reduces number of
nodes t be examined at each level in a search process. Beam search is a
one approach for forward pruning. Consider a beam of the n best moves
for each ply rather than considering all possible moves. But in this
approach, there is no guarantee that the best move will not be pruned
away.
The PROBCUT or probabilistic cut algorithm is a forward pruning version
of alpha -beat search. Same as alpha -beta search PROBCUT also prunes
node that are probably out side the window. This probability is computed
by doing a shallow search and find the backed -up value v of a node. Based
on past experience, estimat e the probability that a score of v at depth d in
the tree would be outside (α, β).
5.5.4 Search Versus Lookup :
Many games plying programs us table lookup for the opening and ending
of game rather than search. In order to play each opening, the best advice
of human expert is copied from books and input it in a table for the
computer’s use. Also, computer can collect statistics from a database of
previously played games to see which opening sequences are led to win.
After starting the game, in the early mov es there are few choices and thus
usually after 10 moves the program must switch form table lookup to
search. Near the end of the game there are few possible positions and thus
more chance to do lookup.
A human can tell the general strategy for playing a game while computer
on the other hand can completely solve the endgame by producing a
policy, which is mapping from every possible state to the best move in that
state.
5.6 STOCHASTIC GAMES Many games are unpredictable in nature, such as dice throw., such type of
games is called as stochastic games. The outcome of such games depends
on skills as well as luck. For e.g., Gambling game Golf ball game,
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83 Artificial Intelligence Backgammon game combine luck and skill of players. Dice are rolled at
the starting of pla yer’s turn to find the legal moves. Each player moves the
pieces according to the dice are thrown. For example, if a player who
plays with black piece roll the dice and dice shown 6 -5 and has four
possible moves.

The goal of the game is to move all one ’s pieces off the board. White
moves clockwise and black move anticlockwise. A piece can move to any
position until multiple opponents are pieces are there. White knows his or
her own legal moves but don’t know about the black legal moves. So
white cannot build a standard game tree. A game tee in a backgammon
must include chance node in addition with MIN and MAX node. The
possible dice node denoted by the branches leading from each chance nod.
Brach is labeled with the roll and its probability. There are 36 ways to
thrown the dice, but because a 6 -5 is the same as 5 -6, there are only 21
distinct rolls.

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84 Adversarial Search The next step is to understand how to make correct decision and pick up
the move that leads to the best position. Th e position does not define
minimax value, instead expected value of position has been calculated. For
chance node expected value is calculated by using the sum of the value
over all outcomes, weighted by the probability of each chance action
EXPECTIMINIMA X(s) =
UTILITY(s) if TERMINAL -TEST(s)
max a EXPECTIMINIMAX (RESULT(s, a)) if PLAYER(s) = MAX
mina EXPECTIMINIMAX (RESULT(s, a)) if PLAYER(s) = MIN
∑r P(r)EXPECTIMINIMAX (RESULT(s, r)) if PLAYER(s) =
CHANCE
Where r is possible dice roll, (RESULT(s, r) same state as s
5.6.1 Evaluation functions for games of chance
In the same way that heuristic function returns an estimate of the distance
to the goal, an evaluation function returns an estimate of the expected
utility of the game from a given position. The performance of the game
playing program depends strongly on the quality of its ev aluation function.
An inaccurate evaluation function will guide an agent towards position
that turns out to be lost. There are some ways to design good Evaluation
function.
First, the evaluation function should order the terminal states in the same
way as the true utility function: state that are wins must evaluate be tter
than draws, which in turn must be better than losses. Using the evaluation
function would otherwise cause an agent to make mistakes even if it can
see all the way to the end of the game. Second, the computation must not
take too long. Third, the evalu ation function must be strongly correlated
with the actual chances of winning for nonterminal states.
Most evaluation function works by calculating various features of the
state. For example, in chess, we would have features for the number of
white pawns, black pawns, white queen, black queen and so on. These
combine feature defines various categories or equivalent classes of states.
The state in each category has the same values for all the features. Any
given category will contain some state that lead to wins, some that lead to
draws and some that lead to losses. The evaluation function cannot know
which state are which, but it can return a single value that reflects the
proportion of states with each outcome. For example, suppose our
experience suggests t hat 72% of the states encountered in the two -pawns
vs. one -pawn category lead to a win 20% to a loss (0), and 8% to a draw
(1/2). Then a reasonable evaluation for states in the category is the
expected value: (0.72 × +1) + (0.20 × 0) + (0.08 × 1/2) = 0.76. An
evaluation function can be derived for every state by determining the
expected value for each category. As with terminal states, the evaluation munotes.in

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85 Artificial Intelligence function need not return actual expected values as long as the ordering of
the states is the same.
This kind of analysis required too many categories and hence too much
experience to estimate all t he probabilities of winning. Instead, most
evaluation functions compute separate numerical contributions for each
feature and then combine them to calculate the total value. For example,
introductory chess books give an approximate material value for each
piece: each pawn is worth 1, a knight or bishop is worth 3, a rook 5, and
the queen 9. Other features such as “good pawn structure” and “king
safety” might be worth ha lf a pawn, say. These feature values are then
simply added up to obtain the evaluation of the position.
A secure advantage equivalent to a pawn gives a substantial likelihood of
winning, and a secure advantage equivalent to three pawns should give
almost c ertain victory Mathematically, this kind of evaluation function is
called a weighted linear function and denoted as

where each w i is a weight and each f i is a feature of the position.
5.7 PARTIALLY OBSERVABLE GAMES Partially observable games are qualitat ively different from other games. In
these games various tricks are used to confused the opponent include the
use of scouts and spies together the information and use of concealment
and bluff too confuse the opponent etc. Kriegspiel is a partially observab le
chess.
5.7.1 Kriegspiel: Partially Observable Chess :
The uncertainty about the state of the board arises entirely from lack of
access to the choices made by the opponent in deterministic partially
observable games. For e.g., Battleships, Stratego etc. I n partially
observable chess the pieces can move but are completely invisible from
the opponent. Means White can see only his pieces and black can see only
his pieces. There is a referee who can see all the pieces of black and white
and who can judge the g ame. It also periodically makes announcement
that are heard by both players.
Any move that white chess piece would make if there were not any black
piece is proposed to the referee. If move is not legal then referee
announced illegal move. So, white piec e may keep proposing until legal
one is found and try to guess more about the location of black piece. If
legal move is proposed then the referee announces the following “capture
on square X”, “check by D”, Knight, Rank, File, Long Diagonal, Short
Diagonal etc. If Black piece is checkmated the referee says so; otherwise,
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86 Adversarial Search Kriegspiel state estimation directly map onto the partially observable, non -
deterministic framework. For partially observable game the notion of
strategy is alter ed. a move is making for every possible percept sequence
that might receive. For Kriegspiel a winning strategy is one that, for each
possible percept sequence leads to an actual checkmate foe every possible
board state in the current belief state, regardle ss of how opponent moves.
5.7.2 Card Games :
Many examples of partial observability are provided by card game, where
the missing information is generated randomly. In many games cards are
distributed randomly at the beginning of the game, where each plyer n ot
visible to opponent. For e.g., bridge, hearts etc.
It might seem that, these card games are the same as dice game. The cards
are distributed randomly and find the moves available for each player.
Consider all possible deals of invisible cards and solve each one as if it
were a fully observable game. Then select the move that has best outcome.
Suppose that each deal has probability P(s) then the moves are as follows
argmax ∑sa P(s) MINIMAX(RESULT(s, a))
In most of the card games the number of possible deals is larger. In bridge
game, each player sees two hands out of the four so there are two unseen
hands of 13 cards. Then the number of deals is 10,400,600. So inst ead of
solving 10 million we resort to Monte Carlo approximation. In this,
instead of taking all the deal, consider a random sample of N deals where
probability of deals s appearing in sample is propositional to P(s). even for
small N like 100 to 1000 the method gives good approximation.
5.8 STATE -OF-ART GAME PROGRAMMING State of art game programs are blindly fast, highly optimized machine that
incorporate the latest engineering. But this type of games is not much use
for doing shopping or driving off -road
Chess :
it is well known for defeating world champion Garry Kasparov. Deep Blue
examined 200 million positions per second, used very sophisticated
evaluation and undisclosed methods for extending some lines of search up
to 40 ply. The key to its success see ms to have been its ability to generate
singular extensions beyond the depth limit for sufficiently interesting lines
of forcing/forced moves.
Checkers:
Jonathan Schaeffer and colleagues developed CHINOOK which runs on
regular computers and it uses alpha beta search. It is used an endgame
database defining perfect play for all positions.
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87 Artificial Intelligence Othello :
it is also called as Reversi. It is more popular as a computer game than a
board game. It has a smaller search space than chess and 5 to 15 legal
moves.
Backgam mon:
The most of the work has gone into improving evaluation function. Gerry
Tesauro combined reinforcement learning with neural network to develop
accurate evaluator.
Go:
It is the most popular board game in Asia. It is 19 by 19 board game in
which move s are allowed into every empty square. The branching factor
starts at 361 which is too daunting for alpha -beta search method. The
evaluation function for this game is difficult to write because control
territory is often very unpredictable until the endgam e.
Bridge :
It is multiplayer game in which cards are hidden from the other player.
Players are paired into two teams for four player games.
5.9 SUMMARY  A game has five components such as initial state, legal actions, result
of action, terminal test and utility function.
 The minimax algorithm can select optimal moves by a depth -first
enumeration of the game tree in two player zero sum game.
 The alpha -beta search algorithm achieves much greater efficiency by
eliminating subtree that are irrelevant.
 Instead of considering whole game tree, cut the search off at some
point and apply evaluation function to calculate the utility of state.
 Game of chance can be handled by an extension to minimax
algorithm that calculate a chance node.
 Humans retain the edge in se veral games of imperfect information,
such as poker, bridge, and Kriegspiel.
5.10 EXERCISE 1) Explain various strategies of game playing.
2) Explain the components to formally defined the game.
3) Explain minimax algorithm with example.
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88 Adversarial Search 5) Explain alpha -beta pruning
6) Explain alpha -beta cut -offs in game playing with example.
7) Describe and implement state description, move generators, terminal
state, utility function and evaluation function for any of the following
stochastic game: Bridge, Monopoly
8) Prove that alpha -beta pruning takes time O(2m/2) with optimal move
ordering, where m is maximum depth of the game tree.
5.11 BIBLIOGRAPHY 1) Artificial Intelligence: A Modern Approach by Stuart Russell and
Peter Nor vig
2) Artificial Intelligence: javapoint.com


*****





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89 UNIT III
6
LOGICAL AGENT
Unit Structure
6.0 Objective
6.1 Introduction
6.2 Knowledge -Based Agent
6.3 The Wumps World
6.4 Logic
6.5 Propositional Logic: A Very Simple Logic
6.5.1 Syntax
6.5.2 Semantics
6.5.3 A Simple Knowledge Base
6.5.4 A Simple Infe rence Procedure
6.6 Propositional Theorem Proving
6.6.1 Inference and proofs
6.6.2 Proof by Resolution
6.6.3 Horn Clauses and Definite Clauses
6.6.4 Forward and Backward Chaining
6.7 Effective Propositional Model Checking
6.7.1 A Complete Backtracking Al gorithm
6.7.2 A Local Search Algorithm
6.7.3 The Landscape of random SAT Problems
6.8 Agent Based on Propositional Logic
6.9 Summary
6.10 Bibliography
6.11 Unit End Exercise
6.0 OBJECTIVE In this chapter we design agent -based system that can represen t a complex
world. The new representation about the world can be derived by using
process of inference and these new representations to deduce what to do.
Also develop a logic as a general class of representation to support
knowledge -based agent.
6.1 INTRO DUCTION Knowledge can be a particular path towards the solution. The knowledge
must be represented in a particular format before using it. In 8 puzzle munotes.in

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90 Logical Agent transition model, the knowledge of what the actions do is hidden inside the
domain specific code of the R ESULT function. I observable environment
as agent can represent what it knows about the current state is to list all
possible concrete states. In this chapter logic is develop as a general class
of representations to support knowledge -based agent. The know ledge -
based agent can accept new tasks in the form of explicitly described gals.
The agent can achieve ability quickly by learning new knowledge about
the environment and also update the relevant knowledge so they can adapt
to changes in the environment.
6.2 KNOWLEDGE -BASED AGENT The main component of knowledge -based agent is its knowledge base. A
knowledge base is a set of sentences and each sentence expressed in a
language known as knowledge representation language.it is also known as
KB. It represents s ome assertion about the world. Sometimes a sentence is
dignified with axioms, taking the sentence as given without deriving it
from another sentence.

The above diagram represents the architecture of knowledge based -agent.
This agent takes the input by pe rceiving the environment. The knowledge
base use TELL and ASK operation. TELL is add new sentence to the
knowledge base perceive from the environment whereas, ASK is way to
query what is known. The new sentence is derived from old is called as
inference. B oth ASK and TELL operation involve inference and it obey
the requirements that when one ASKs a question of the knowledge base.
The answer should follow from what has been told previously to
knowledge base. The agent maintains a knowledge base which initial ly
contain some background knowledge. The following figure shows a
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91 Artificial Intelligence

MAKE -PERCEPT -SENTENCE generate a sentence asserting that the
agent perceived the given percept at a given time. MAKE -ACTION -
QUERY generate a sentence that a sks what action should be taken at the
current time. MAKE -ACTION -SENTENCE generate a sentence asserting
that the selected action was executed.
The knowledge -based agent has three different levels :
1) Knowledge level: this is the first level of knowledge -based agent in
which need to specify only what the agent knows and what its goals
are, in order to fix its behavior. For example, an automated taxi might
have to go from Location A to location B and one bridge is the only
link between the two locations. An agen t crosses the bridge because it
knows that, that will achieve its goal.
2) Logical level: At this level it is necessary to understand how
knowledge representation of knowledge is stored. The sentences are
encoded into different logics. For example, at logica l level we expect
that automatic taxi agent reach to the destination.
3) Implementation level: this level agent take action as per logic and
knowledge level. So, it is representation of logic and knowledge. For
example, at this level an automated taxi agent i mplements his
knowledge and logic to reach to destination.
6.3 THE WUMPUS WORLD The Wumpus world is an example of a world which describe an
environment in which knowledge -based agent can show their worth. It is a
cave consisting of 4X4 rooms connected by passageways. So, there are
total 16 rooms connected with each other. The cave has a room contain
beast and the beast can eat anyone who enter into the room. The Wumpus
can be killed by the agent if he is facing it. There are some rooms contains
pits. And i f agent falls in pits, then he will be stuck there forever. In this
cave, there is a possibility of finding a heap of gold. So, the goal of agent
is to find the gold and climb out the cave without fallen into pits or eaten
by Wumpus. The agent will get a r eward if he come out with gold. If agent
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92 Logical Agent There are some points which helps the agent to navigate through the cave
safely such as the rooms adjacent to the Wumpus are smelly, so it would
have som e stench. The room adjacent to pits has a breeze, so when agent
reach near pits, he feels the breeze. If the room is glitter, then it Contin
gold. The Wumpus can be killed by the agent if is facing it.

The PEAS for the Wumpus world are described as follo w
 Performance measure - +1000 if the agent come out from the cave
with gold. -1000 if agent falls into the pits or eaten by the Wumpus. -1
for each action taken and -10 for using an arrow.
 Environment :
 4 X 4 grid of rooms connected to each other.
 The agen t always starts from the position [1,1] facing to right.
 The location of gold and the Wumpus are selected randomly from the
squares.
 Each square of the cave other than the start can be pit with probability
0.2.
 Actuators :
 the agent can move forward
 turn left and turn right by 900.
 The grab action can be used to pick up the gold.
 The action shoot can be used to fire an arrow.
 The arrow continues until it either hits the Wumpus or wall. The agent
has only one arrow.
 The action climb can be used to clim b out of the cave.
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93 Artificial Intelligence  Sensors : The agent has five sensors as follows.
 The agent will perceive the stench if he is in the room which is adjacent
to Wumpus.
 It perceives breeze if the room directly adjacent to pit.
 The glitter is perceived by agent when room contains gold.
 It perceives bump if agent walk into the wall.
 When the Wumpus is shot, it releases a horrible scream which can be
perceived anywhere in the cave.
The percept will be given to agent program in the form of five symbols if
there is a stench and breeze but no glitter, bump or scream the agent
program will get [Stench, Breeze, None, None, None]
Consider the following example, the agent’s initially starts from the
location [1,1] and [1,1] is a safe square. Suppose it can be denoted by “A”
or “OK” respectively in square [1,1]. So, the first percept is [None, None,
None, None, None] from which agent can conclude about neighboring
squares such as [1,2] and [2,1] are out of dangers. They are OK.

Let’s suppose agent supposed to move in [2,1], so t here must be a pit in
[2,2] or [3,1] or both. Now at this point only one square is OK [1,2]. So,
agent is turn around and go back to [1,1] and then [1,2]
The agent perceives a stench in [1,2] means there must be a Wumpus
nearby. But nearby of [1,2] is [1,1 ] and [2,2] and both are OK. So, agent
can guess that Wampus is in [1,3]. The agent feels that there is a lack of a
breeze in [1,2] means there is no pit in [2,2]. The agent already inferred
that there must be a pit in either [2,2] or [3,1]. So this means it must be in
[3,1] .
Now at room [2,2] there is neither pit nor Wumpus so agent moves to [2,3]
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94 Logical Agent

6.4 LOGIC The proof or validation of a reason can be defined as Logic. Knowledge
base consist of sentence and these sentences are expressed according to
the syntax. Syntax is a formal description of the structure of program in a
given language. The notion of syntax must be in well forms for example
“x+y=10” is a well -formed but “xyz+=” i s not in well formed.
A logic can also define the semantic or meaning of sentence. Semantic is a
formal description of the programs in a given language. The truth of each
sentence with respect to each possible world defined by semantics. For
example, consi der the sentence “x+y=10” is true in a world where possible
value of x is 5 and y is 5, but false where x is 1 and y is1. In standard logic
each sentence must be either true or false only. The term Model is used
instead of possible world when need to be mo re precise.
If a sentence α is true in a model then m is a model of α and M(α) notation
is used to mean the set of all models of α. Logical reasoning is the process
through which any assertion is verified. It is relation of logical entailment
between sent ences. It can be denoted by α |=β and mean that the sentence
α entails the sentence β. The entailment is defined as α |=β if and only if,
in every model in which α is true then β is also true and can be written as
α |=β iff M(α) ⊆ M(β)
this analysis can be applied on Wumpus world reasoning example.
6.5 PROPOSITIONAL LOGIC: A VERY SIMPLE LOGIC Propositional logic is a simplest form of logic in which knowledge is
represented as proposition. A proposition is a declarative statement that’s
either true or false. It is also called as a Boolean logic because it works on
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95 Artificial Intelligence Propositional logic consists of object, relation or functions and logical
connectivity. In tautology propositional formul a is always true whereas in
contradiction propositional formula is always false.
6.5.1 Syntax :
Syntax of propositional logic defines allowable sentences in the model.
The atomic sentences can be made up of single propositional symbol.
Each symbol stands fo r proposition that can be either true or false. The
uppercase symbols are used for example, A,P,R etc. Complex sentences
are constructed from simple sentences by using parentheses and logical
connectives. There are five connectives are as follows
1) Negation or not - A sentence such as ¬ P is called as negation of P. A
literal is either a positive or negative literal.
For example, Sonu did not read the Novels can be represented as ¬READ
(Sonu, Novels)
2) and – it is called as conjunction used to represent compoun d statement.
Denoted by ∧.
For example, Sonu is intelligent and hardworking can be represented as
follows
Suppose P - Sou is intelligent
Q- Sonu is Hardworking then P ∧Q
3) or – A sentence such as P ∨Q is called as disjunction.
For example, Sonu is Doctor or Engineer
Suppose P - Sou is Doctor
Q- Sonu is Engineer then P ∨Q
4) implies - it is used to represent if then statement and denoted by P=>Q
for example, if it’s raining, then streets are wet
Suppose P -if it’s raining
Q-Street is wet then P=>Q
5) if and only if - it is also called as bidirectional connective. P ⇔ Q
show that it is true whenever both P=>Q and Q=>P are true.
For example, Sonu eat the fruit if and only if it’s an Apple
Suppose P -Sonu eats fruit
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96 Logical Agent

A BNF( Backus Naur -Form) - A BNF grammar of sentences in
propositional logic is described as follows
Sentence ->Atomic sentence| Complex sentence
Atomic sentence ->True | False | Symbol
Symbol -> P | Q | R….
Complex sentence ->¬ Sentence
|(Sentence ∧ Sentence)
|(Sentence ∨ Sentence)
|(Sentence => Sentence)
|(Sentence ⇔ Sentence)
Operator precedence - ¬, ∧, ∨, =>, ⇔
Every sentence constructed with binary connectives must be closed in
parenthesis because grammar is very strict about parenthesis.
6.5.2 Semantics :
The semantic defines the rules for determining the truth of a sentence with
respect to particular model. In a propositional logic, model fixes the truth
value as true or false for every proposition symbol. Suppose if the
sentence in the kn owledge base makes use of the proposition symbols
such as P 1,2, P2,2 and P 3,1 then model is as follows
M1 = {P 1,2=false, P 2,2=false, P 3,1=false}
With these three symbols there are 23=8 possible models. The semantic
must specify how to compute the truth val ue of any sentence and it done
recursively. By using atomic sentences and five connectives all sentences
are constructed. So, it’s necessary to specify how to compute the truth of
atomic sentence and sentence which are formed with each of the five
connecti ves. The atomic sentences are easy
 True is true and False is false in every model.
 The truth value of every other proposition symbol must be specified
directly in the model.
There are five rules for complex sentence, that hold any substances P and
Q in any model m:
 ¬ P is true iff P is false in m.
 P∧ Q is true iff both P and Q are true in m.
 P ∨ Q is true iff either P or Q is true in m.
 P=>Q is true unless P is true and Q is false in m.
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97 Artificial Intelligence 6.5.3 A Si mple Knowledge Base :
Now consider immutable aspects of the Wumpus world with the following
symbol for each location [x,y] such as
Px,y is true if there is a pit in [x,y]
Wx,y is true if there is Wumpus in [x,y] dead or alive
Bx,y is true if the agent perce ives a breeze in [x,y]
Sx,y is true if the agent perceive a stench in {x,y \
Now consider the following situations and label each sentence R i so it can
be referred further.
1) there is no pit in [1,1]
R1: ¬ P 1,1
2) A square is breeze if and only if there is pit in a neighboring square
R2: B1,1 ⇔ (P1,2 V P 2,1)
R3: B2,1 ⇔ (P1,1 V P 2,2 V P 3,1)
3) Now the breeze percept has been included for the first two squares
R4: ¬ B 1,1
R5: ¬ B 2,1
6.5.4 A Simple Inference Procedure :
The first algorithm for inference is model check ing approach which is
direct implementation of entailment. It checks that α is true in every model
in which KB is true. Models are assignments of true or false to every
symbol. Now consider Wumpus world example, the proposition symbols
are B 1,1, B 2,1, P1,1, P1,2, B 2,1, B 2,2, an B 3,1 with these seven symbols there
are 27=128 possible models.
A general algorithm for deciding entailment in propositional logic is as
shown below. The algorithm directly implements the definition of
entailments so algorithm is sou nd. The algorithm is complete because it
works for any KB and α and always terminate. There are 2n models if KB
and α contains n symbols in all. The algorithm has time complexity O(2n).
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6.6 PROPOSITIONAL THEOREM PROVING To construct the proof of the de sired sentence the rules can be apply
directly to the sentence in knowledge base without consulting the model.
The entailment can be done by theorem proving. If the number of models
is large but the length of the proof is short then theorem proving can be
more efficient as compare to model checking.
Logical equivalence - two sentences α and β are logically equivalent if
they re true in the same set of models. It can be denoted by α ≡ β. Or any
two sentences α and β are equivalent only if each of them entails the other
and denoted by α ≡ β if and only if α |= β and β |= α .
Validity - a sentence is true in all model then it is valid.
Tautology - A valid sentence is called as tautology.
Deduction theorem - it is derived from the definition of entailment, which
was known to the ancient Greeks for any sentences α and β, α |= β if and
only if the sentence (α ⇒ β) is valid
Satisfiability - A sentence is satisfiable if it is true in or satisfied by some
model. It can be checked by enumerating the possible models until one is
found that satisfies the sentence.
6.6.1 Inference and proo fs:
Inference rules can be applied to derive a proof.
1) Modus Pones :
It is represented as follows

It indicates that any sentence of the form α ⇒ β and α are given, then the
sentence β can be inferred
For example, if (Wumpus Ahead ∧ Wumpus Alive) => Shoot and
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99 Artificial Intelligence 2) And-elimination : It says that from conjunction any conjuncts can be
inferred

for example, from the sentence (Wumpus Ahead ∧ Wumpus Alive) the
inference, “Wumpus Alive” can be drawn.
3) Monotonicity : it says that the set of entailed sentences can only
increases as information is added to the knowledge base. For any
sentence α and β
if KB|= α then KB ∧ β|= α
Monotonicity means inference rules can be applied whenever suitable
premises ar e found in the knowledge base. the conclusion of the rule must
follow regardless of what else is in the knowledge base.
6.6.2 Proof by Resolution :
The single inference rule called resolution that produce a complete
inference algorithm when coupled with an y complete search algorithm.
Unit resolution rule takes a clause of a disjunction of literals and produce a
new clause. A single literal can be viewed as a disjunction of one literal
known as a unit clause. the unit resolution rule can be generalized to th e
full resolution rule

Where l i and m j are complementary literals. So, resolution takes two
clauses and generate a new clause. The new clause containing all the
literals of the two original clause except two complementary literals. The
resulting clause s hould contain only one copy of each literal is the
technical aspect of resolution rule. The removal of multiple copies of
literals is called factoring.
Conjunctive Normal Form :
Every sentence of propositional logic is logically equivalent toa
conjunction of clauses. A sentence can be expressed as a conjunction of
clause is called as Conjunctive normal form or CNF. The following s
procedure for converting sentence B1,1 (P1,2 ∨ P2,1 ) into CNF
1) Eliminate , by replacing α ⇔ β with (α ⇒ β) ∧ (β ⇒ α)
(B1,1 ⇒ (P1,2 ∨ P2,1)) ∧ ((P1,2 ∨ P2,1) ⇒ B1,1) .
2) Eliminate ⇒, by replacing α ⇒ β with ¬α ∨ β
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100 Logical Agent 3) CNF requires ¬ to appears only in literals, so it is necessary to move ¬
inwards by repeated application of the following e quivalence.
¬(¬α) ≡ α (double -negation elimination)
¬(α ∧ β) ≡ (¬α ∨ ¬β) (De Morgan)
¬(α ∨ β) ≡ (¬α ∧ ¬β) (De Morgan)
We require only one application of the last rule in the example.
(¬B1,1 ∨ P1,2 ∨ P2,1) ∧ (¬(P1,2 ∧ P2,1) ∨ B1,1)
4) Now there is a sent ence containing nested ∧ and ∨ operators applied to
literals. Now by applying distributive law, distributing ∨ over ∧
whenever possible
(¬B1,1 ∨ P1,2 ∨ P2,1) ∧ (¬(P1,2 ∨ B1,1) ∧ (¬P2,1 ∨ B1,1)
The original sentence is now converted in CNF as a conjunction of three
clauses. It can be use as input to resolution procedure.
A resolution Algorithm :
It takes the input as a knowledge base and α and its output is either true or
false. First (KB ∧ ¬ α) is converted into CNF. Then the resolution rule is
applied to th e resulting clauses. Each pair in which complementary literals
are there is resolved to produce anew clause and it is added to the new set
if it is not there. The procedure continues until one of the two things
happens first, there are no new clauses that can be added in which KB
does not entails α or second two clauses resolve to yield the empty clause
in which case KB entails α. Th resolution algorithm is as shown belove.

6.6.3 Horn Clauses and Definite Clauses :
Some real -world knowledge base uses more restrictions on the form of
sentences to enables them to use more restricted and efficient inference
algorithm. The definite clause is one such restricted form, which is a
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101 Artificial Intelligence For example, (¬L 1,1 ∨ ¬Breeze ∨ B1,1) is a definite clause whereas (¬B 1,1 ∨
P1,2 ∨ P2,1) is not definite clause.
Horn clause is a disjunction of literals with at most one positive literal. So,
all definite clauses are horn clauses. Goal cluses are those in which there is
no positi ve literals.
6.6.4 Forward and Backward Chaining :
Forward chaining is an example of the data driven reasoning concept that
is reasoning in which the focus attention starts with the known data.it is
called as data driven as it reach the goal state using av ailable data.
Forward chaining is the process of making conclusion based on known
fact by starting from initial state to goal state. It can be used within an
agent to derive conclusion from incoming percepts. While doing this, it
cannot be kept any query in mind. The forward chaining is used in expert
system. For example, the Wumpus agent might TELL its percept to
knowledge base using an incremental forward chaining algorithm.
Now consider example following example,
As per the law, it is crime for B count ry to sell weapons to hostile nations.
Country A is an enemy of Country B, has some missiles, and all the
missiles were sold to it by John. John is a B country citizen now prove that
John is criminal. Now convert all facts into first order definite clause.
It is a crime for a country B to sell weapons to hostile nation. Let’s
consider variables p, q and r.
Rules:
1) Country B(p ) ∧ weapon ( q) ∧ sells (p,q,r) ∧ hostile (r ) ->
criminal (p)
2) Country A has some missiles. ?powns(A,p) ∧ missile(p) or Owns(
A,T1) Missiles(T1)
3) All missiles are sold to country A by John. ?p Missiles(p) ∧
Owns(A, p) -> Sells (John, p, A)
4) Missiles are weapons. Missile (p) -> Weapons (p)
5) Enemy of country B is hostile. Enemy (p, B) -> Hostile( p)
6) Country A is an enemy of Country B. Ene my (A , B)
7) John is from country B. Country B(John)
Steps :
1) First start with the known facts and select the sentences which do not
have implications such as B(Roberts), Enemy(A, B), Owns( A, T1) and
Missile(T1) can be represented as below
2) The facts thar are infer from available facts and with satisfied premises
Rule 1 - does not satisfy premises, so it will not be added in the first
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102 Logical Agent Rule 2 and 3 are already added.
Rule 4 -satisfy the substitution {p/T1}, so Sells (John, T1, A) is added. It
infers fr om conjunction of rue 2 and 3
Rule 6 - is satisfy with substitution (p/A) so Hostile (A) is added. It infers
from rule 7.

3) Check rule 1 is satisfied with the substitution {p/John, q/T1, r/A}, so
add Criminal(John). So, we reached at goal state and proved t hat Jhon
is Criminal.

Backward Chaining : In backward chaining it starts with the goal and
works backward from the query. If the query is known to be rule, then no
work is needed. Otherwise, the algorithm finds implications that can be
proved true then q is true. It is form of goal -directed reasoning. In this the
goal is broken into sub -goals to prove the facts true. In backward chaining
same example is consider.
Rules :
1) Country B(p ) ∧ weapon ( q) ∧ sells (p,q,r) ∧ hostile (r ) ->
criminal (p)
2) Country A has some missiles. ?powns(A,p) ∧ missile(p) or Owns(
A,T1) Missiles(T1)
3) All missiles are sold to country A by John. ?p Missiles(p) ∧
Owns(A, p) -> Sells (John, p, A)
4) Missiles are weapons. Missile (p) -> Weapons (p) munotes.in

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103 Artificial Intelligence 5) Enemy of country B is hostile. Enemy (p , B)-> Hostile( p)
6) Country A is an enemy of Country B. Enemy (A , B)
7) John is from country B. Country B(John)
Steps - in backward chaining start with goal state which is Criminal ( Jhon)
1) We will take the goal fact and from this infer other facts. At the end
prove that those facts true. So, now goal fact is Jhon is Criminal as
shown below

2) We will infer other facts from all fact which satisfies the rules. In rule 1
goal predicate is present with substitution {Jhon/p}. So, add all the
conjunctive facts below the first level and replace p with Jhon.

3) now extract further fact Missile(q) which infer from Weapon(q) as it
satisfies rule 5 Weapon(q) is also true with the substitution of a
constant T1 at q

4) now infer facts Missile(t1) and Owns (A, T1) from Sells (J hon, T1, r)
which satisfies the rule 4 with substitute of A in place of r. so two
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104 Logical Agent

5) Infer the fact Enemy (A, B) from Hostile( A) which satisfies rule 6 an
hence all the statements are proved true using backward chaining.

6.7 E FFECTIVE PROPOSITIONAL MODEL CHECKING For better inferences efficient algorithm is needed that based on model
checking. Algorithm approach can be backtracking search or hill -climbing
search.
6.7.1 A Complete Backtracking Algorithm :
This algorithm is also c alled as Davis -Putnam algorithm or DPLL
algorithm. It takes as input a sentence in conjunction normal form, which
is set of clauses. It is recursive depth first enumeration of possible models.
The algorithm is improvement over earlier general inferencing a lgorithm.
There are three improvements over the simple scheme TT -ENTAILS.
1) Early termination : The algorithm finds whether the sentence must
be true or false. A clause is true if any literal is true. A sentence is
false if any clause is false, which occurs w hen each of its literal is
false.
2) Pure symbol heuristic: it is a symbol that always appears with same
“sign” in all clauses. If a sentence has a model, then it has a model
with pure symbol assigned so as to make literal true.
3) Unit clause heuristic : it is c lause with just one literal. One unit
clause can create another unit clause. The only literal in unit clause is
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105 Artificial Intelligence

6.7.2 A Local Search Algorithm :
The local search algorithm like hill -climbing can be applied directly to
satisfiable problem. These algo rithms are worked properly if correct
evaluation function is provided. The task of algorithm is to find an
assignment that satisfies every clause. Therefore, those evaluation function
counts the number of unsatisfied clauses will be a good.
WALKSAT algorit hm picks up an unsatisfied clause and pick a symbol in
the clause to flip. It randomly chooses one of the ways to pick which
symbol to flip is min -conflict and random walk. Min - conflict is steps that
minimizes the number of unsatisfied clauses in the new state. Random
walks that pick the symbol randomly.
When the input sentence is indeed satisfiable WALKSAT returns a model.
But when it is return failure then there are two possible causes. First, the
sentence is unsatisfiable and second is need to give the algorithm more
time. WALKSAT algorithm cannot always detect unsatisfiability, which is
very necessary for deciding entailment.


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106 Logical Agent 6.7.3 The Landscape of random SAT Problems
There are some SAT problems that are harder than others. Simple
problems can be s olved with any old algorithm. Because SAT is NP -
complete, at least some problem instances will require exponential running
time. The n -queen problem quite tricky for backtracking search algorithm
but trivially easy for local search methods such as min conf lict. This is
because solutions are very densely distributed in the space of assignments,
and any initial assignment is guaranteed to have a nearby solution.
Therefore, n -queens is easy because it is underconstrained.
In the case of satisfiability problems in conjunctive normal form, an
underconstrained problem is one with relatively few clauses constraining
the variables. Consider the following example with a randomly generated
3-CNF sentence with five symbols and five clauses
(¬D ∨ ¬B ∨ C) ∧ (B ∨ ¬A ∨ ¬C) ∧ (¬C ∨ ¬B ∨ E) ∧ (E ∨ ¬D ∨ B) ∧ (B ∨
E ∨ ¬C) .
For this sentence sixteen of the 32 possible assignments are model, so, it
would take just two random guesses to find a model. In general,
underconstrained problems like this are ea sily satisfiable. In contrast, an
overconstrained problem will have many clauses in relation to the number
of variables and is likely to have no solution.
Beyond these basic intuitions, we have to define how random sentences
are generated. The notation CNF k(m,n) denotes a k -CNF sentence with m
cluses and n symbols, where the clauses are chosen uniformly,
independently and without replacement from all clauses with k different
literals, which are positive or negative at random.
The probability of satisfiab ility can be measured by examining the sources
of random sentences. As shown in following figure (a), plots the
probability for CNF 3(m,50), that is, sentences with 50 variables and 3
literals per clause, as a function of clause/ symbol ratio, m/n. For smal l
m/n the probability of satisfiability is close to 1, whereas for large m/n, the
probability is close to 0. The probability drops fairly sharply around
m/n=4.3. so, as n increases “cliff” gets sharper and sharper and stay in
roughly the same place for k=3 . Theoretically, the satisfiability threshold
conjecture says that for every k ≥ 3, there is a threshold ratio r k such that,
as n goes to infinity, the probability that CNF k(n, rn) is satisfiable becomes
1 for all values of r below the threshold, and 0 for all values above the
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107 Artificial Intelligence

(a) Graph showing the probability that a random 3 -CNF sentence with n = 50
symbols is satisfiable, as a function of the clause/symbol ratio m/n. (b) Graph of
the median run time on random 3-CNF sentences
6.8 AGENT BASED ON PROPOSITIONAL LOGIC Inference based agent -it requires the separate copies of its knowledge
base for every time step. Inference based agent takes exponential time for
inferencing and huge time requirements for completenes s. It is easy to
represent in propositional logic language. It requires large memory as it
stores previous percepts or states.
Circuit based agent - it doesn’t require separate copy of its KB for every
time step. Circuit based agent takes linear time for in ferencing which is
dependent on circuit size. For a complete circuit CBA will take
exponential time with respect to circuit size so it is incomplete. It is easy
to describe and construct knowledge base.
6.9 SUMMARY  Intelligent agent require knowledge abou t the world in order to take
good decision. Knowledge is contained in agent in the form of
sentences.
 Sentences in a knowledge representation language are stored in a
knowledge base.
 Basic concept of logic is represented by syntax, semantics, entailment,
inference, soundness and completeness.
 Syntax is a formal structure of sentences. Semantics is truth of
sentences based on models.
 Entailments is necessary truth of one sentence given another.
Inference is deriving sentences from other sentences
 In sound in ference algorithm, derivations produce only entailed
sentences. In complete algorithm, derivations can produce all entailed
sentences.
 Propositional logic isa simple language consisting of proposition
symbols and logical connectives.
 Resolution is complete for propositional logic, forward, backward
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108 Logical Agent 6.11 UNIT END QUESTIONS 1) Explain knowledge -based agent? Explain role and importance.
2) What is knowledge based
3) Write a short note on Wumpus world problem.
4) Describ e PEAS for Wumpus world problem.
5) What is logic?
6) Explain propositional logic.
7) Explain resolution algorithm.
8) Explain CNF
9) Write a short note on propositional theorem proving.
10) Write the connectives used to form complex sentence of propositional
logic with exam ple.
11) Explain backward and forward chaining with example.
6.10 BIBLIOGRAPHY 1) Artificial Intelligence: A Modern Approach by Stuart Russell and Peter
Norvig
2) Artificial Intelligence: javapoint.com



*****

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109 UNIT IV
7
FIRST -ORDER LOGIC
Unit Structure
7.0 Objectives
7.1 Introduction
7.2 First-Order Logic
7.3 Syntax and semantics of First -Order Logic
7.3.1 Models for First -Order Logic
7.3.2 Symbols and Interpretations
7.3.3 Terms
7.3.4 Atomic Sentences
7.3.5 Complex Sentences
7.3.6 Quantifiers
7.3.7 Equality
7.3.8 An alternative semantics
7.4 Using First -Order Logic
7.4.1 Assertions and Queries in First -Order Logic
7.4.2 The kinship domain
7.4.3 Numbers, Sets
7.4.4 The Wumpus World
7.5 Knowledge Engineering in First -Order Logic
7.5.1 Knowledge Engineering process
7.5.2 Electronic Circuit Domain
7.6 Summary
7.7 Unit End Exercises
7.8 Bibliography
7.9 List of References
7.0 OBJECTIVES After going through this chapter, you will be able to:
 Represent knowledge in knowledge base using First -Order logic.
 Understand syntax and semantics of First -Order logic.
 Make use of Quantifiers for representation of knowledge.
 Learn use of First -Order logic in kinship domain, number and sets,
and in Wumpus world.
 Understand Knowledge -Engineering process.
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110 First-Order Logic 7.1 INTRODUCTION Statements can be represen ted using propositional logic. But
unfortunately, in propositional logic, we can only represent the facts,
which are either true or false. Propositional logic is not sufficient to
represent the complex sentences or natural language statements. The
proposit ional logic has very limited expressive power. Propositional logic
lacks the expressive power to concisely describe an environment with
many objects. Consider the following sentence, which cannot be
represented using propositional logic.
 "Some humans are i ntelligent"
 "Sachin likes cricket."
To represent the above statements, propositional logic is not sufficient, so
we need some more powerful logic such as first -order logic. We can adopt
the foundation of propositional logic that is a declarative, compositi onal
semantics which is context -independent and unambiguous and build a
more expressive logic on that foundation, by borrowing representational
ideas from natural language while avoiding its drawbacks. Important
elements in natural language are:
 Objects ( squares, pits, Wumpus )
 Relations among objects (is adjacent to, is bigger than) or unary
relations or properties (is red, round)
 Functions (father of, best friend of)
First-order logic (FOL) is built around the above 3 elements that is
objects, relation s & functions.
7.2 FIRST -ORDER LOGIC  First-order logic is another way of knowledge representation in
artificial intelligence. It is an extension to propositional logic.
 First-Order logic is sufficiently expressive to represent the natural
language statements i n a concise way.
 First-order logic is also known as Predicate logic or First -order
predicate logic. First -order logic is a powerful language that develops
information about the objects in an easier way and can also express the
relationship between those ob jects.
 First-order logic (like natural language) does not only assume that the
world contains facts like propositional logic but also assumes the
following things in the world:
o Objects: A, B, people, numbers, colors, wars, theories, squares, pits,
wumpus, etc.
o Relations: It can be unary relation such as: red, round, is adjacent, or
n-any relation such as: the sister of, brother of, has color, comes
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111 Artificial Intelligence o Function: Father of, best friend, third inning of, end of, etc.
 As a natural language, first -order lo gic also has two main parts:
o Syntax
o Semantics
7.3 SYNTAX & SEMANTICS OF FIRST -ORDER LOGIC We begin this section by specifying the way in which the possible worlds
of first -order logic reflect the ontological commitment to objects and
relations. Then we introdu ce the various elements of the language,
explaining their semantics as we go along.
7.3.1 Models for First -Order Logic:
Models of a logical language are the formal structures that constitute the
possible worlds under consideration. Models for propositional logic link
proposition symbols to predefined truth values. Models for first -order
logic have objects. Every possible world must contain at least one object.
Below figure shows a model with five objects:
1. Richard the Lionheart, King of England;
2. his young er brother, the evil King John;
3. the left legs of Richard;
4. the left legs of John;
5. and a crown.

Figure: A model with five objects
The objects in the model may be related in various ways. In the figure,
Richard and John are brothers. Thus, the brotherhoo d relation in this
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112 First-Order Logic The “brother” and “on head” relations are binary relations that is, they
relate pairs of objects. The model also contains unary relations, or
properties: the “person” property is true of both Richard and John; and the
“crown” property is true only for the crown.
Certain kinds of relationships are considered as functions, in that a given
object must be related to exactly one object. For example, ea ch person has
one left leg, so the model has a unary “left leg” function that includes the
following mappings: → Richard’s left leg → John’s left leg. 7.3.2 Symbols and Interpretations:
The syntax determines which collections of symbols are legal expressions
in first -order logic. The basic syntactic elements of first -order logic are the
symbols that stand for objects, relations, and functions. The symbols,
therefore, come in three kinds:
1. Constant symbols:
Constants symbols refer to objects in a universe of discourse. Objects can
be anything like integers, people, real numbers and geometric shapes etc.
Example: Richard, John
2. Predicate symbols:
A predicate symbol represents a property of or relation between terms that
can be true or false. Each predicate symbol has an associated arity
indicating its number of arguments. Brother, OnHead, Perso n, King, and
Crown, etc. are predicate symbols.
Example:
King(John) is a unary predicate which is having one arguments.
Brother(John, Richard) is a Binary predicate which is having two
arguments.
3. Function symbols :
Function constants refer to functions lik e mother, age, LeftLeg, plus and
times. Each function symbol has an associated arity indicating its number
of arguments. Example:
mother(Jane) - here, mother has arity one (unary) while in times(3,2) -
times has arity two(binary). Sentence → AtomicSentence | ComplexSentence AtomicSentence → Predicate | Predicate(Term, . . .) | Term = munotes.in

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113 Artificial Intelligence Term ComplexSentence → ( Sentence ) | [ Sentence ] | ¬Sentence | Sentence ∧ Sentence | Sentence ∨ Sentence | Sentence ⇒ Sentence | Sentence ⇔ Sentence | Quantifier Variable, . . . Sentence Term → Function(Term, . . .) | Constant | Variable Quantifier → ∀| ∃ Constant → A | X1 | John | · · · Variable → a | x | s | ··· Predicate → True | False | After | Loves | Raining | ··· Function → Mother | LeftLeg | · · · OPERATOR PRECEDENCE : ¬,=, ∧, ∨,⇒,⇔ Figure: The syntax of first -order logic, specified in Backus –Naur form
and Operato r precedences are specified from highest to lowest.
Like proposition symbols, the choice of names is entirely up to the user &
every model must provide the information required to determine if any
given sentence is true or false.
In addition to its object s, relations, and functions, each model includes an
interpretation that specifies exactly which objects, relations and functions
are referred to by the constant, predicate, and function symbols.
One possible interpretation for our example is as follows:
Interpretation:
 Richard refers to Richard the Lionheart and John refers to the evil King
John.
 Brother refers to the brotherhood relation; OnHead refers to the “on
head” relation that holds between the crown and King John; Person,
King, and Crown refer to th e sets of objects that are persons, kings, and
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114 First-Order Logic  LeftLeg refers to the “left leg” function.
7.3.3 Terms:
Objects are represented by terms. Terms are simply names for objects. A
term is a logical expression that refers to an object. Constant symbols a re
therefore terms, but it is not always convenient to have a distinct symbol
to name every object.
For example, in English we might use the expression “King John’s left
leg” rather than giving a name to his leg. At that time instead of using a
constant s ymbol, we use LeftLeg(John). A complex term is formed by a
function symbol followed by a parenthesized list of terms as arguments to
the function symbol.
Complex term is not a “subroutine call” that “returns a value.” There is no
LeftLeg subroutine that ta kes a person as input and returns a leg. We can
reason about left legs (e.g., stating the general rule that everyone has one
and then deducing that John must have one) without ever providing a
definition of LeftLeg.
For example, suppose the LeftLeg functio n symbol refers to the function
and John refers to King John, then LeftLeg(John) refers to King John’s
left leg.
7.3.4 Atomic Sentences:
A sentence can either be an atomic sentence or a complex sentence. An
atomic sentence is simply a predicate applied to a set of terms. Atomic
sentences are the most basic sentences of first -order logic. These sentences
are formed from a predicate symbol followed by a parenthesis with a
sequence of terms. We can represent atomic sentences as
Predicate (term1, term2, ..... ., term n)
Example:
1. Ravi and Ajay are brothers: Brothers(Ravi, Ajay).
2. Chinky is a cat: cat (Chinky).
3. John owns a car: Owns(John, Car)
Atomic sentences can also have complex terms as arguments. Such as,
Married(Father (Ravi),Mother (Ajay))
states that Ravi’s father is married to Ajay’s mother.
7.3.5 Complex Sentences:
Logical connectives can be used to construct more complex sentences,
with the same syntax and semantics as in propositional calculus. Logical
connectives like ￢, ∧, ∨, ⇒, ⇔ can be u sed.
( Sentence ) | [ Sentence ] munotes.in

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115 Artificial Intelligence ￢Sentence
Sentence ∧ Sentence
Sentence ∨ Sentence
Sentence ⇒ Sentence
Sentence ⇔ Sentence
Quantifier Variable, . . . Sentence
Here we have four sentences that are true in the model that we have
considered previously.
¬Broth er (LeftLeg(Richard), John)
Brother (Richard , John) ∧ Brother (John,Richard)
King(Richard ) ∨ King(John)
¬King(Richard) ⇒ King(John)
7.3.6 QUANTIFIERS A quantifier is a language element which generates quantification, and
quantification specifies the qua ntity of specimen in the universe of
discourse. These are the symbols that permit to determine or identify the
range and scope of the variable in the logical expression. There are two
types of Quantifiers .
1. Universal Quantifier, (for all, everyone, everythi ng)
2. Existential quantifier, (for some, at least one).
1. Universal Quantification ( ∀):
Universal quantifier is a symbol of logical representation, which specifies
that the statement within its range is true for everything or every instance
of a particular thing. The Universal quantifier is represented by a symbol
∀, which resembles an in verted A.
Note: In universal quantifier we use implication "→".
If x is a variable, then ∀x is read as:
For all x
For each x
For every x.
Example 1:
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116 First-Order Logic ∀ x King (x) ⇒ Person (x)
Thus, the se ntence will be read as, “For all x, if x is a king, then x is a
person.” The symbol x is called variable. By convention, variables are
lowercase letters.
Variable:
 A variable is a term by itself.
 It can also serve as the argument of a function for exampl e, LeftLeg (x).
 A term with no variables is called a ground term.
Consider the model that we have used previously and the intended
interpretation that goes with it. We can extend the interpretation in five
ways by asserting all possible values of x:
x → Richard the Lionheart,
x → King John,
x → Richard’s left leg,
x → John’s left leg,
x → the crown.
The universally quantified sentence ∀ x King (x) ⇒ Person (x) is true if the
sentence King (x) ⇒ Person (x) is true under each of the five extended
interpretati ons. That is, the universally quantified sentence is equivalent to
asserting the following five sentences:
i. Richard the Lionheart is a king ⇒ Richard the Lionheart is a person.
ii. King John is a king ⇒ King John is a person.
iii. Richard’s left leg is a king ⇒ Rich ard’s left leg is a person.
iv. John’s left leg is a king ⇒ John’s left leg is a person.
v. The crown is a king ⇒ the crown is a person.
Since, in our model, King John is the only king, the second sentence
asserts that he is a person and remaining four assertions are not valid
because Richard is not king and left leg of a person and crown object
cannot be a person.
Example 2:
All man drink coffee.
Let a variable x which refers to a man so all x can be represented as
below:
x1 drinks coffee.
x2 drinks coffee.
x3 dr inks coffee.
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117 Artificial Intelligence .xn drinks coffee.
We can write it as:
∀x man(x) → drink (x, coffee).
It will be read as: For all x, if x is a man, then x drinks coffee.
2. Existential Quantification:
Existential quantifiers are the type of quantifiers, which express that the
statement within its scope is true for at least on e instance of something. It
is denoted by the logical operator ∃, which resembles as inverted E. When
it is used with a predicate variable then it is called as an existential
quantifier. The sentence ∃x P says that P is true for at least one object x.
Note: In Existential quantifier we always use AND or Conjunction symbol
(∧).
If x is a variable, then existential quantifier will be ∃x or ∃(x). And it will
be read as:
There exists a 'x.'
For some 'x.'
For at least one 'x.'
Example 1:
King John has a crown on his head, we write
∃ x Crown (x) ∧ OnHead (x, John )
That is, at least one of the following is true:
i. Richard the Lionheart is a crown ∧ Richard the Lionheart is on John’s
head;
ii. King John is a crown ∧ King John is on John’s head;
iii. Richard’s lef t leg is a crown ∧ Richard’s left leg is on John’s head;
iv. John’s left leg is a crown ∧ John’s left leg is on John’s head;
v. The crown is a crown ∧ the crown is on John’s head.
The fifth assertion is true in the model.
Example 2:
Some boys are intellige nt.
∃x: boys(x) ∧ intelligent(x)
It will be read as: There are some x where x is a boy who is intelligent.
Properties of Quantifiers:
 In universal quantifier, ∀x∀y is similar to ∀y∀x.
 In Existential quantifier, ∃x∃y is similar to ∃y∃x.
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118 First-Order Logic Nested quantifiers:
More complex sentences can be expressed using multiple quantifiers. The
simplest case is where the quantifiers are of the same type. For example,
“Brothers are siblings” can be written as
∀ x ∀ y Brother (x, y) ⇒ Sibling(x, y)
Siblinghood is a symmetric relationship, we can write
∀ x, y Sibling(x, y) ⇔ Sibling(y, x)
In other cases, we will have mixtures. “Everybody read few books” means
that for every person, there is some book that person read:
∀ x ∃ y Read(x, y)
On the othe r hand, to say “There is someone who is liked by everyone,”
we write
∃ y ∀ x Likes(x, y)
The order of quantification is therefore very important. It becomes clearer
if we insert parentheses.
Connections between ∀ and ∃:
The two quantifiers are actually co nnected with each other, through
negation. Asserting that everyone dislikes parsnips is the same as asserting
there does not exist someone who likes them, and vice versa:
∀ x ¬Likes(x, Parsnips ) is equivalent to ¬ ∃ x Likes(x, Parsnips)
We can go one step further: “Everyone likes ice cream” means that there
is no one who does not like ice cream:
∀ x Likes(x, IceCream)
is equivalent to
¬∃ x ¬Likes(x, IceCream)
Because ∀ is a conjunction over the universe of objects and ∃ is a
disjunction, quantifiers obe y De Morgan’s rules. The De Morgan rules for
quantified and unquantified sentences are as follows:
∀ x ¬P ≡ ¬∃x P ¬(P ∨ Q) ≡ ¬P ∧¬Q
¬∀x P ≡ ∃x ¬P ¬(P ∧ Q) ≡ ¬P ∨¬Q
∀x P ≡ ¬∃x ¬P P∧ Q ≡ ¬(¬P ∨¬Q)
∃x P ≡ ¬∀x ¬P P∨ Q ≡ ¬(¬P ∧¬Q)
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119 Artificial Intelligence Examples of First -Order Logic:
Some Examples of First -Order Logic using quantifiers:
1. All birds fly.
In this statement the predicate is fly(bird).
And since there are all birds who fly so it will be represented as follows.
∀x bird(x) →fly(x)
2. Every man respe cts his parent.
In this statement, the predicate is respect(x, y), where x=man, and y=
parent.
Since there is every man so will use ∀, and it will be represented as
follows:
∀x man(x) → respects (x, parent)
3. Some boys play cricket.
In this statement, the predicate is play(x, y), where x= boys, and y= game.
Since there are some boys so we will use ∃, and it will be represented as:
∃x boys(x) → play(x, cricket)
4. Not all students like both Mathematics and Science.
In this statement, the predicate is like(x , y), where x= student, and y=
subject.
Since there are not all students, so we will use ∀ with negation, so
following representation for this:
¬∀ (x) [ student(x) → like(x, Mathematics) ∧ like(x, Science)]
5. Only one student failed in Mathematics.
In th is statement, the predicate is failed(x, y), where x= student, and y=
subject.
Since there is only one student who failed in Mathematics, so we will use
following representation for this:
∃(x) [ student(x) → failed (x, Mathematics) ∧∀ (y) [¬(x==y) ∧ stude nt(y)
→ ¬failed (x, Mathematics)]
7.3.7 Equality:
Equality symbol can be used to signify that two terms refer to the same
object. For example, Father (John)=Henry says that the object referred to
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120 First-Order Logic The equality symbol can be used to state facts about a given function, as
we just did for the Father symbol. It can also be used with negation to
insist that two terms are not the same object . To say that Richard has at
least two brothers, we would write
∃ x, y Brother (x,Richard ) ∧ Brother (y,Richard ) ∧¬(x=y)
The sentence ∃ x, y Brother (x,Richard ) ∧ Brother (y,Richard ) does not
have the intended meaning. If x is John & y is also John the sentence will
not be valid because Richard’s both the brothers will not be having the
same name as John, therefore x and y both should be different. For this
purpose, ¬(x=y) is added. The notation x ≠ y is sometimes used as an
abbreviation for ¬(x=y).
7.3.8 An Alternative Semantic:
Continuing the example from the pre vious section, suppose that we
believe that Richard has two brothers, John and Geoffrey. If we assert
Brother (John,Richard ) ∧ Brother (Geoffrey,Richard)
First, this assertion is true in a model where Richard has only one brother,
we need to add John ≠ G eoffrey. Second, the sentence doesn’t rule out
models in which Richard has many more brothers besides John and
Geoffrey. Thus, the correct translation of “Richard’s brothers are John and
Geoffrey” is as follows:
Brother (John,Richard ) ∧ Brother (Geoffrey, Richard) ∧ John ≠ Geoffrey
∧ ∀x Brother (x,Richard ) ⇒ (x=John ∨ x=Geoffrey)
7.4 USING FIRST -ORDER LOGIC Now we have defined an expressive logical language, it is time to learn
how to use it. In this section, we provide representations of some simple
domains. In knowledge representation, a domain is just some part of the
world about which we wish to express some knowledge. We begin with a
brief description of the TELL/ASK interface for first -order knowledge
bases. Then we look at the domains of
1. Family relatio nships
2. Numbers
3. Sets and lists
4. The Wumpus world
7.4.1 Assertions and Queries In First -Order Logic:
1. Assertions:
Sentences are added to a knowledge base using TELL, exactly as in
propositional logic. Such sentences are called assertions. For example, we
can assert that John is a king, Richard is a person, and all kings are
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121 Artificial Intelligence TELL(KB, King(John)) .
TELL(KB, Person(Richard)) .
TELL(KB, ∀ x King(x) ⇒ Person(x)) .
2. Queries:
We can ask questions from the knowledge base using ASK. For example,
ASK(KB, King(John))
Above query returns true. Questions asked with ASK are called queries or
goals.
ASK(KB, Person(John))
This query also returns true using above assertions. We can ask quantified
queries, such as
ASK(KB, ∃ x Person(x)) .
The answer is true, but we do not know the value of x makes the sentence
true. If we want to know what value of x makes the sentence true, we will
use a different function which is called ASKVARS.
ASKVARS(KB, Person(x))
Above function yields a stream of answers. In this case the re will be two
answers: {x/John} and {x/Richard}. Such an answer is called a
substitution or binding list. Which means in above query x can be
substituted with John or Richard.
7.4.2 The Kinship Domain:
The first example we consider is the domain of family relat ionships, or
kinship. This domain includes facts such as “Elizabeth is the mother of
Charles” and “Charles is the father of William” and rules such as “One’s
grandmother is the mother of one’s parent”. We have following in Kinship
domain according to First -Order logic.
1. Objects : objects in kinship domain are people.
2. Unary predicates: Male and Female.
3. Binary Predicates: Kinship relations parenthood, brotherhood,
marriage, and so on are represented by binary predicates: Parent,
Sibling, Brother, Sister, Chil d, Daughter, Son, Spouse, Wife,
Husband, Grandparent, Grandchild, Cousin, Aunt, and Uncle.
4. Functions: Mother and Father, because every person has exactly one
of each of these (at least according to nature’s design).
We can go through each function and pre dicate, writing down what we
know in terms of the other symbols.
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122 First-Order Logic Example:
1. One’s mother is one’s female parent:
∀ m, c Mother (c)=m ⇔ Female(m) ∧ Parent(m, c)
2. One’s husband is one’s male spouse:
∀ w, h Husband(h,w) ⇔ Male(h) ∧ Spouse(h,w)
3. Male and female are disjoint categories:
∀ x Male(x) ⇔ ¬Female(x)
4. Parent and child are inverse relations:
∀ p, c Parent(p, c) ⇔ Child (c, p)
5. A grandparent is a parent of one’s parent:
∀ g, c Grandparent (g, c) ⇔ ∃p Parent(g, p) ∧ Parent(p, c)
6. A sibling is another child of one’s parents:
∀ x, y Sibling(x, y) ⇔ x = y ∧ ∃p Parent(p, x) ∧ Parent(p, y)
We could go on for several e xamples like this. Each of these sentences can
be viewed as an axiom of the kinship domain. The axioms define the
Mother function and the Husband, Male, Parent, Grandparent, and Sibling
predicates. This is a natural way in which we can build up the
represe ntation of a domain.
Axioms can also be “just plain facts,” such as Male(Jim) and Spouse(Jim,
Laura) etc. Not all logical sentences about a domain are axioms. Some are
theorems that is, they are entailed by the axioms. For example, consider
the assertion that siblinghood is symmetric:
∀ x, y Sibling(x, y) ⇔ Sibling(y, x)
7.4.3 Numbers, Sets:
I. Numbers:
Numbers are the most vivid example of how a large theory can be built up
from a tiny kernel of axioms. We describe here the theory of natural
numbers or non -negative integers. We will use a predicate NatNum that
will be true of natural numbers; we need one constant symbol, 0; and we
need one function symbol, S (successor). The Peano axioms define natural
numbers and addition. Natural numbers are defined recursive ly:
NatNum(0) .
That is, 0 is a natural number and
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123 Artificial Intelligence Means for every object n, if n is a natural number, then S(n) is a natural
number. So the natural numbers are 0, S(0), S(S(0)), and so on.
We also need axioms to constrain the successor function:
∀ n 0 ≠ S(n)
∀ m, n m ≠ n ⇒ S(m) ≠ S(n)
Now we can define addition in terms of the successor function:
∀m NatNum(m) ⇒ + (0,m) = m
Above axiom says that adding 0 to any natural number m gives m itself.
∀ m, n NatNum(m) ∧ NatNum(n) ⇒ + (S(m), n) = S(+(m, n))
We can also write S(n) a s n+ 1, so the second axiom becomes
∀ m, n NatNum(m) ∧ NatNum(n) ⇒ (m + 1) + n = (m + n) + 1
This axiom reduces addition to repeated application of the successor
function.
Once we have addition, it is straightforward to define multiplication as
repeated a ddition, exponentiation as repeated multiplication, integer
division and remainders, prime numbers, and so on. Thus, the whole of
number theory (including cryptography) can be built up from one
constant, one function, one predicate and four axioms.
ii. Set :
The domain of sets is also fundamental to mathematics as well as to
commonsense reasoning. We want to be able to represent individual sets,
including the empty set. We need a way to build up sets by adding an
element to a set or taking the union or inter section of two sets. We will
want to know whether an element is a member of a set and we will want to
distinguish sets from objects that are not sets.
1. The empty set is a constant written as {}.
2. There is one unary predicate, Set, which is true of sets.
3. The binary predicates are x ∈ s (x is a member of set s) and s1 ⊆ s2 (set
s1 is a subset of set s2).
4. The binary functions are s1 ∩ s2 (the intersection of two sets), s1 ∪ s2
(the union of two sets), and {x|s} (the set resulting from adjoining
element x to set s).
One possible set of axioms is as follows:
1. The only sets are the empty set and those made by adjoining something
to a set:
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124 First-Order Logic 2. The empty set has no elements adjoined into it. In other words, there is
no way to decompose {} into a smaller set and an element:
¬∃ x, s {x|s}={}
3. Adjoining an element already in the set has no effect:
∀ x, s x ∈ s ⇔ s={x|s}
4. The only members of a set are the elements that were adjoined into it.
We express this recursively, saying that x is a member of s if and only if s
is equal to some set s2 adjoined with some element y, where either y is the
same as x or x is a member of s2:
∀ x, s x ∈ s ⇔ ∃y, s2 (s={y|s2} ∧ (x=y ∨ x∈ s2))
5. A set is a subset of another set if and only if all of the first set’s
members are members of the second set:
∀ s1, s2 s1 ⊆ s2 ⇔ (∀x x∈ s1 ⇒ x∈ s2)
6. Two sets are equal if and only if each is a subset of the other:
∀ s1, s2 (s1 =s2) ⇔ (s1 ⊆ s2 ∧ s2 ⊆ s1)
7. An object is in the intersection of two sets if and only if it is a member
of both sets:
∀ x, s1, s2 x ∈ (s1 ∩ s2) ⇔ (x∈ s1 ∧ x∈s2)
8. An object is in the union of two sets if and only if it is a member of
either set:
∀ x, s1, s2 x ∈ (s1 ∪ s2) ⇔ (x∈ s1 ∨ x∈s2)
7.4.4 The Wumpus World:
The first -order axioms for Wumpus wo rld are much more concise,
capturing in a natural way exactly what we want to say. Wumpus agent
receives a percept sequence with five elements. The corresponding first -
order sentence stored in the knowledge base must include both the percept
and the time a t which it occurred; otherwise, the agent will get confused
about when it saw what.
We use integers for time steps. A typical percept sentence would be
Percept ([Stench, Breeze, Glitter, None, None], 5)
Here, Percept is a binary predicate, and Stench and so on are constants
placed in a list.
The actions in the wumpus world can be represented by logical terms:
Turn(Right ), Turn(Left ), Forward , Shoot , Grab, Climb
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125 Artificial Intelligence ASKVARS( ∃ a BestAc tion(a, 5))
which returns a binding list such as {a/Grab}. The agent program can then
return Grab as the action to take.
The raw percept data implies certain facts about the current state. For
example:
∀ t, s, g, m, c Percept ([s, Breeze, g,m, c], t) ⇒ Breeze(t) ,
∀ t, s, b, m, c Percept ([s, b, Glitter,m, c], t) ⇒ Glitter (t) ,
and so on. Notice the quantification over time t. In propositional logic, we
would need copies of each sentence for each time step.
Simple “reflex” behavior can also be implemente d by quantified
implication sentences.
For example,
∀ t Glitter (t) ⇒ BestAction(Grab, t)
Given the percept and rules from the preceding paragraphs, this would
yield the desired conclusion
BestAction(Grab, 5) that is, Grab is the right thing to do.
We ha ve represented the agent’s inputs and outputs; now it is time to
represent the environment itself.
Objects are squares, pits, and the wumpus. We could name each square
Square 1,2 and so on but then the fact that Square 1,2 and Square 1,3 are
adjacent would h ave to be an “extra” fact, and we would need one such
fact for each pair of squares. It is better to use a complex term in which the
row and column appear as integers; for example, we can simply use the
list term [1, 2]. Adjacency of any two squares can be defined as
∀ x, y, a, b Adjacent ([x, y], [a, b]) ⇔(x = a ∧ (y = b − 1 ∨ y = b + 1)) ∨ (y
= b ∧ (x = a − 1 ∨ x = a + 1))
We will use a unary predicate Pit that is true of squares containing pits.
Since there is exactly one wumpus, we will use a constant Wumpus.
The agent’s location changes over time, so we write At(Agent, s, t) to
mean that the agent is at square s at time t. We can fix the wumpus’s
location with
∀t At(Wumpus, [2,2], t)
We can then say that objects can only be at one location at a time:
∀ x, s1, s2, t At(x, s1, t) ∧ At(x, s2, t) ⇒ s1 = s2
Given its current location, the agent can infer properties of the square from
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126 First-Order Logic and perceives a breeze, then that square is bree zy:
∀ s, t At(Agent, s, t) ∧ Breeze(t) ⇒ Breezy(s)
Having discovered which places are breezy, the agent can deduce where
the pits are. Whereas propositional logic necessitates a separate axiom for
each square and would need a different set of axioms for e ach
geographical layout of the world, first -order logic just needs one axiom:
∀ s Breezy(s) ⇔ ∃r Adjacent (r, s) ∧ Pit(r)
Similarly, in first -order logic we can quantify over time, so we need just
one successor -state axiom for each predicate, rather than a different copy
for each time step. For example, the axiom for the arrow becomes
∀ t HaveArrow(t + 1) ⇔ (HaveArrow(t) ∧¬Action(Shoot, t))
7.5 KNOWLEDGE ENGINEERING IN FIRST -ORDER LOGIC The process of constructing a knowledge -base in first -order logic is c alled
as knowledge - engineering. In knowledge -engineering, someone who
investigates a particular domain, learns important concept of that domain,
and generates a formal representation of the objects, is known as
knowledge engineer.
7.5.1 Knowledge Engineer ing Process:
Knowledge engineering projects vary widely in content, scope, and
difficulty, but all such projects include the following steps:
1. Identify the task:
The knowledge engineer must delineate the range of questions that the
knowledge base will supp ort and the kinds of facts that will be available
for each specific problem instance. For example, does the wumpus
knowledge base need to be able to choose actions or is it required to
answer questions only about the contents of the environment? Will the
sensor facts include the current location? The task will determine what
knowledge must be represented in order to connect problem instances to
answers. This step is analogous to the PEAS process for designing agents.
2. Assemble the relevant knowledge:
The kno wledge engineer might already be an expert in the domain, or
might need to work with real experts to extract what they know by a
process called knowledge acquisition. At this stage, the knowledge is not
represented formally. The idea is to understand the s cope of the
knowledge base, as determined by the task, and to understand how the
domain actually works. For the wumpus world, which is defined by an
artificial set of rules, the relevant knowledge is easy to identify. For real
domains, the issue of relevan ce can be quite difficult.
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127 Artificial Intelligence 3. Decide on a vocabulary of predicates, functions, and constants:
Translate the important domain -level concepts into logic -level names.
This involves many questions of knowledge -engineering style. Like
programming style, this can have a significant impact on the eventual
success of the project. For example, should pits be represented by objects
or by a unary predicate on squares? Should the agent’s orientation be a
function or a predicate? Should the wumpus’s location depend on ti me?
Once the choices have been made, the result is a vocabulary that is known
as the ontology of the domain. The word ontology means a particular
theory of the nature of being or existence.
4. Encode general knowledge about the domain:
The knowledge engineer writes down the axioms for all the vocabulary
terms. This pins down the meaning of the terms, enabling the expert to
check the content. Often, this step reveals misconceptions or gaps in the
vocabulary that must be fixed by returning to step 3 and iterati ng through
the process.
5. Encode a description of the specific problem instance:
If the ontology is well thought out, this step will be easy. It will involve
writing simple atomic sentences about instances of concepts that are
already part of the ontology. For a logical agent, problem instances are
supplied by the sensors, whereas a “disembodied” knowledge base is
supplied with additional sentences in the same way that traditional
programs are supplied with input data.
6. Pose queries to the inference procedure and get answers:
We can let the inference procedure operate on the axioms and problem -
specific facts to derive the facts we are interested in knowing. Thus, we
avoid the need for writing an application -specific solution algorithm.
7. Debug the knowledge base :
The answers to queries will be correct on the first try. More precisely, the
answers will be correct for the knowledge base as written, assuming that
the inference procedure is sound, but they will not be the ones that the user
is expecting. For example, if an axiom is missing, some queries will not be
answerable from the knowledge base. A considerable debugging process
could ensure that no axioms are missing.
7.5.2 The Electronic Circuit Domain:
In this topic, we will understand the Knowledge engineerin g process in an
electronic circuit domain. This approach is mainly suitable for creating
special -purpose knowledge base. Following are some main steps of the
knowledge -engineering process. Using these steps, we will develop a
knowledge base which will allo w us to reason about digital circuit (One -
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128 First-Order Logic

1. Identify the task:
The first step of the process is to identify the task, and for the digital
circuit, there are various reasoning tasks. At the first level or highest lev el,
we will examine the functionality of the circuit:
i. Does the circuit add properly?
ii. What will be the output of gate A2, if all the inputs are high?
At the second level, we will examine the circuit structure details such as:
i. Which gate is connected to the first input terminal?
ii. Does the circuit have feedback loops?
2. Assemble the relevant knowledge:
In the second step, we will assemble the relevant knowledge which is
required for digital circuits. So, for digital circuits, we have the following
required kno wledge:
i. Logic circuits are made up of wires and gates.
ii. Signal flows through wires to the input terminal of the gate, and each
gate produces the corresponding output which flows further.
iii. In this logic circuit, there are four types of gates used: AND, OR,
XOR, and NOT.
iv. All these gates have one output terminal and two input terminals
(except NOT gate, it has one input terminal).
3. Decide on vocabulary:
The next step of the process is to select functions, predicate, and constants
to represent the circuits, te rminals, signals, and gates. Firstly, we will
distinguish the gates from each other and from other objects. Each gate is
represented as an object which is named by a constant, such as, Gate(X1).
The functionality of each gate is determined by its type, whi ch is taken as
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129 Artificial Intelligence i. Circuits will be identified by a predicate: Circuit (C1).
ii. For the terminal, we will use predicate: Terminal(x).
iii. For gate input, we will use the function In(1, X1) for denoting the first
input terminal of the gate, and for output terminal we will use Out (1,
X1). The function Arity(c, i, j) is used to denote that circuit c has i
input, j output.
iv. The connectivity between gates can be represented by predicate
Connect(Out(1, X1), In(1, X 1)).
v. We use a unary predicate On(t), which is true if the signal at a
terminal is on.
4. Encode general knowledge about the domain:
To encode the general knowledge about the logic circuit, we need some
following rules:
i. If two terminals are connected then they have the same input signal, it
can be represented as
∀ t1, t2 Terminal (t1) ∧ Terminal (t2) ∧ Connect (t1, t2) → Signal
(t1) = Signal (t2)
ii. Signal at every terminal will have either value 0 or 1, it will be
represented as:
∀ t Termina l (t) →Signal (t) = 1 ∨Signal (t) = 0
iii. Connect predicates are commutative:
∀ t1, t2 Connect(t1, t2) → Connect (t2, t1)
iv. Representation of types of gates:
∀ g Gate(g) ∧ r = Type(g) → r = OR ∨r = AND ∨r = XOR ∨r = NOT
v. Output of AND gate will be zero if and only if any of its input is zero.
∀ g Gate(g) ∧ Type(g) = AND →Signal (Out(1, g))= 0 ⇔ ∃n Signal
(In(n, g))= 0
vi. Output of OR gate is 1 if and only if any of its input is 1:
∀ g Gate(g) ∧ Type(g) = OR → Signal (Out(1, g)) = 1 ⇔ ∃n Signal
(In(n, g))= 1
vii. Output of XOR gate is 1 if and only if its inputs are different:
∀ g Gate(g) ∧ Type(g) = XOR → Signal (Out(1, g)) = 1 ⇔ Signal
(In(1, g)) ≠ Signal(In(2, g))
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130 First-Order Logic ∀ g Gate(g) ∧ Type(g) = NOT → Signal (In(1, g)) ≠ Signal (Out(1,
g))
ix. All the gates in the above circuit have two inputs and one output
(except NOT gate).
∀ g Gate(g) ∧ Type(g) = NOT → Arity(g, 1, 1)
∀ g Gate(g) ∧ r =Type(g) ∧ (r= AND ∨r= OR ∨r= XOR) → Arity
(g, 2, 1)
x. All gates are logic circuits:
∀ g Gate(g) → Circuit (g)
5. Encode a description of the problem instance:
Now we encode problem of circuit C1, firstly we categorize the circuit and
its gate components. This step is easy i f ontology about the problem is
already thought. This step involves the writing simple atomics sentences
of instances of concepts, which is known as ontology.
For the given circuit C1, we can encode the problem instance in atomic
sentences as below: Since in the circuit there are two XOR, two AND, and
one OR gate so atomic sentences for these gates will be:
For XOR gate: Type(x1)= XOR, Type(X2) = XOR
For AND gate: Type(A1) = AND, Type(A2)= AND
For OR gate: Type (O1) = OR.
And then represent the connecti ons between all the gates.
6. Pose queries to the inference procedure and get answers:
In this step, we will find all the possible set of values of all the terminal
for the adder circuit. The first query will be:
What should be the combination of input whi ch would generate the first
output of circuit C1, as 0 and a second output to be 1?
∃ i1, i2, i3 Signal (In(1, C1))=i1 ∧ Signal (In(2, C1))=i2 ∧ Signal (In(3,
C1))= i3 ∧ Signal (Out(1, C1)) =0 ∧ Signal (Out(2, C1))=1
7. Debug the knowledge base:
Now we will debug the knowledge base, and this is the last step of the
complete process. In this step, we will try to debug the issues of
knowledge base. In the knowledge base, we may have omitted assertions
like 1 ≠ 0.

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131 Artificial Intelligence 7.6 SUMMARY  First-order logic a represent ation language that is far more powerful
than propositional logic.
 First-order logic makes use of objects and relations and thereby it gains
more expressive power.
 The syntax of first -order logic is built upon propositional logic. It adds
terms to represe nt objects, and has universal and existential quantifiers.
 A possible world, or model, for first -order logic includes a set of
objects and an interpretation that maps constant symbols to objects,
predicate symbols to relations among objects, and function s ymbols to
functions on objects.
 Use of First -Order logic is various domain such as kinship domain,
number and sets and in Wumpus world.
 Developing a knowledge base in first -order logic requires a careful
process of analyzing the domain, choosing a vocabul ary, and encoding
the axioms required to support the desired inferences.
7.7 UNIT END QUESTIONS 1. Why is First -Order Logic used over Propositional Logic?
2. What is First -Order Logic? Discuss the different elements used in first
order logic.
3. Explain Universal a nd Existential quantifier with example.
4. Define following terms.
a. Predicate
b. Function (in logic)
c. Model in First -order Logic
d. First -order Logic
5. Explain Knowledge Engineering process in detail.
6. Translate the following sentences into first -order logic :
a. Understanding leads to friendship.
b. Friendship is transitive.
Define all predicates, functions, and constants you use.
7. Write the following assertions in first -order logic:
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132 First-Order Logic b. Joe is an actor
c. All surgeons are doctors.
d. Joe does not have a lawyer (i.e., is not a customer of any lawyer).
e. Emily has a boss who is a lawyer.
f. Every surgeon has a lawyer.
7.8 LIST OF REFERENCES 1. A First Course in Artificial Intelligence , First Edition, Deepak
Khemani, Tata McGraw Hill Publisher
2. Artificial Intelligence: A Modern Approach, Third Edition, Stuart
Russel and Peter Norvig, Pearson Publisher
7.9 BIBLIOGRAPHY 1. Ayorinde, I. and Akinkunmi, B. (2013). Application of First -Order
Logic in Knowledge Based Systems. Afri can Journal of Computing &
ICT
2. https://www.javatpoint.com/ai -knowledge -engineering -in-first-order -
logic


*****



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133 8
INFERENCE IN FIRST -ORDER LOGIC
Unit Structure
8.0 Objectives
8.1 Introduction
8.2 Propositional vs. First -order inference
8.2.1 Inference rules for quantifiers
8.2.1.1 Universal Instantiation
8.2.1.2 Existential Instantiation
8.2.1.3 Universal Generalization
8.2.1.4 Existential Generalization
8.2.2 Reduction to propositional inference
8.3 Unification and lifting
8.3.1 A first -order inference rule
8.3.2 Unification
8.3.3 Storage and retrieval
8.4 Forward and Backward chaining
8.4.1 Forw ard chaining
8.4.2 Backward chaining
8.4.2.1 Backward chaining Example
8.4.2.2 Logic Programming
8.4.3 Forward chaining vs. Backward chaining
8.5 Resolution
8.5.1 Conjunctive Normal Form
8.5.2 The resolution inference rule
8.5.3 Example proof
8.5.4 Equalit y
8.5.5 Resolution strategies
8.5.6 Practical uses of resolution theorem provers
8.6 Summary
8.7 Unit End Exercises
8.8 List of References
8.9 Bibliography
8.0 OBJECTIVES After going through this chapter, you will learn:
 Inference rule for quantifiers.
 First-Order logic Inference rule. munotes.in

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134 Inference in First-Order Logic  Unification for finding substitutions that make different logical
expressions look identical.
 Inference using forward chaining & backward chaining.
 Translation of sentences into Conjunctive Normal Form (CNF) for
first-order log ic.
 Resolution inference rule using which two clauses that share no
variables, can be resolved if they contain complementary literals.
 Resolution strategies that help find proofs efficiently.
8.1 INTRODUCTION Inference in First -Order Logic is used to deriv e new facts or sentences
from existing sentences. In inference we define effective procedures for
answering questions posed in first - order logic . In propositional logic we
studied how inference can be achieved for propositional logic. In this
chapter, we extend those results to obtain algorithms that can answer any
answerable question stated in first -order logic. We will introduce inference
rules for quantifiers and show how to reduce first -order inference to
propositional inference. Then we introduce the idea of unification,
showing how unification can be used to construct inference rules that
work with first -order sentences. Forward chaining, backward chaining and
logic programming systems are also covered in this chapter. Forward and
backward chaining ca n be very efficient, but those are applicable only to
knowledge bases that can be expressed as sets of Horn clauses. General
first-order sentences require resolution -based theorem proving , which is
also discussed in this chapter.
8.2 PROPOSITIONAL Vs. FIRST -ORDER INFERENCE There are some simple inference rules that can be applied to sentences with
quantifiers to obtain sentences without quantifiers. These rules lead to the
idea that first-order inference can be done by converting the knowledge
base to propositional logic and then using propositional inference .
8.2.1 Inference Rules for Quantifiers :
As propositional logic we also have inference rules in first -order logic.
Following are some basic inference rules in First -Order logic:
1. Universal Instantiation
2. Existential Instantiation
3. Universal Generalization
4. Existential Generalization
8.2.1.1 Universal Instantiation :
 Universal instantiation is also called as universal elimination or UI is
a valid inference rule. It can be ap plied multiple times to add new
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135 Artificial Intelligence  The new knowledge base is logically equivalent to the previous
knowledge base.
 As per universal instantiation, we can infer any sentence obtained by
substituting a ground term (a term without variables) for the va riable.
 The universal instantiation rule state that we can infer any sentence
P(c) by substituting a ground term c (a constant within domain x)
from ∀ x P(x) for any object in the universe of discourse.
 It can be represented as

Example:1
IF "Every person like ice -cream"=> ∀x P(x). So we can infer that
"John likes ice -cream" => P(c)
Example: 2
Let's take a famous example,
"All kings who are greedy are Evil." So let our knowledge base contain
this detail as in the form of First -Order Logic:
∀x king(x) ∧ greedy (x) → Evil (x),
So, from this information, we can infer any of the following statements
using Universal Instantiation:
King(John) ∧ Greedy (John) → Evil (John),
King(Richard) ∧ Greedy (Richard) → Evil (Richard),
King(Father(John)) ∧ Greedy (Father(John)) → Evil (Father(John)),
Here, the substitutions are {x/John }, {x/Richard }, and {x/Father (John
)}.
8.2.1.2 Existential Instantiation :
 Existential instantiation is also called as Existential Elimination, which
is a valid inference rule i n first -order logic.
 It can be applied only once to replace the existential sentence.
 The new knowledge base is not logically equivalent to old knowledge
base, but it will be satisfiable if old knowledge bas e was satisfiable.
 This rule states that one ca n infer P(c) from the formula given in the
form of ∃x P(x) for a new constant symbol c.
 The restriction with this rule is that c used in the rule must be a new
term for which P(c) is true.
 It can be represented as:
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136 Inference in First-Order Logic Example:
From the given sentence:
∃x Crown(x) ∧ OnHead(x, John),
We can infe r:
Crown(K) ∧ OnHead( K, John), as long as K does not appear in the
knowledge base.
The above used K is a constant symbol, which is called Skolem constant.
The Existential instantiation is a special case of Skolemization process.
8.2.1.3 Universal General ization :
 Universal generalization is a valid inference rule which states that if
premise P(c) is true for any arbitrary element c in the universe of
discourse, then we can have a conclusion as

 It can be represented as:

 This rule can be used if we want to show that every element has a
similar property.
 In this rule, x must not appear as a free variable.
Example:
Let's represent, P(c): "A byte contains 8 bits", so for ∀ x P(x) "All bytes
contain 8 bits.", it will also be true.
8.2.1.4 Existential Generalization :
 An existential introduction is also known as an existential
generalization, which is a valid inference rule in first -order logic.
 This rule states that if there is some element c in the universe of
discourse which has a property P, then we can infer that there exists
something in the universe which has the property P.
 It can be represented as:

Example:
Let's say that,
"Priyanka got good marks in English."
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137 Artificial Intelligence 8.2.2 Reduction to Propositional Inference :
Once we have defined rules for inferring non -quantified sentences from
quantified sentences, it is possible to reduce first -order inference to
proposition al inference.
The first idea is that, just as an existentially quantified sentence can be
replaced by one instantiation, a universally quantified sentence can be
replaced by the set of all possible instantiations. For example, suppose our
knowledge base c ontains just the sentences
∀ x King (x) ∧ Greedy (x) ⇒ Evil (x)
King (John )
Greedy (John )
Brother (Richard , John )
Then we apply UI to the first sentence using all possible ground -term
substitutions from the vocabulary of the knowledge base. In this case,
{x/John} and {x/Richard }. We obtain
King (John) ∧ Greedy (John) ⇒ Evil (John)
King (Richard) ∧ Greedy (Richard) ⇒ Evil (Richard)
We discarded the universally quantified sentence. Now, the knowledge
base is propositionalized. We have King (John), Greedy (John),
Evil(John), King( Richard) as proposition symbols. Now, we can apply any
of the propositional algorithms to obtain conclusions such as Evil(John).
Every First -Order Logic knowledge base (KB) can be propositionalized so
as to preserve entailment. A ground sentence is entail ed by new KB if it is
entailed by the original KB.
Idea of the inference is that first propositionalize knowledge base (KB)
and query. After that apply resolution and then return result.
Problems with propositionalization:
Propositionalization generates l ots of irrelevant sentences.
Example, from:
∀x King(x) ∧ Greedy(x) ⇒ Evil(x)
King(John)
King(John)
∀y Greedy(y)
Brother(Richard,John)
It seems obvious that Evil(John), but propos itionalization produces lots of
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138 Inference in First-Order Logic 8.3 UNIFICATION AND LIFTING The inference of Evil(John) from the below sentences seems completely
obvious to a human being. We now show how to make it completely
obvious to a computer using inference rules.
∀ x King(x) ∧ Greedy(x) ⇒ Evil(x)
King(John)
Greedy(John)
8.3.1 A First -Order Inference Rule - Generalized Modus Ponens Rule :
For the inference process in FOL, we have a single inference rule which is
called Generalized Modus Ponens. It is lifted version of Modus ponens. It
raises Modus Ponens from g round (variable -free) propositional logic to
first-order logic. The key advantage of lifted inference rules over
propositionalization is that they make only those substitutions that are
required to allow particular inferences to proceed.
Generalized Modus Ponens can be summarized as, " P implies Q and P is
asserted to be true, therefore Q must be True."
According to Modus Ponens, for atomic sentences pi, pi', q. Where there is
a substitution θ such that SUBST (θ, pi',) = SUBST(θ, pi), it can be
represented as:

Example:
We will use this rule for Kings are evil, so we will find some x such that x
is king, and x is greedy so we can infer that x is evil.
Here let say,
p1' is king(John) p1 is king(x)
p2' is Greedy(y) p2 is Greedy(x)
θ is {x/John, y/John} q is Evil(x)
SUBST(θ, q) is Evil(John) .
8.3.2 Unification :
Unification is a process of making two different logical atomic
expressions identical by finding a substitution. Unification depends on the
substitution process. It takes two l iterals as input and makes them identical
using substitution.
Lifted inference rules require finding substitutions that make different
logical expressions look identical. This process is called unification and is munotes.in

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139 Artificial Intelligence a key component of all first -order inferen ce algorithms. The algorithm
takes two sentences and returns a unifier for them if one exists:
UNIFY(p, q)= θ where SUBST(θ, p)= SUBST(θ, q) .
Let us look at some examples of how UNIFY should behave. Suppose
we have a query
AskVars (Knows (John , x)): whom does John know?
Answers to this query can be found by finding all sentences in the
knowl edge base that unify with Knows(John , x). Here are the results of
unification with four different sentences that might be in the knowledge
base:
UNIFY(Knows (John, x), Knows (John, Jane)) = {x/Jane }
UNIFY(Knows (John, x), Knows (y, Bill )) = {x/Bill, y/J ohn }
UNIFY(Knows (John, x), Knows (y, Mother (y))) = {y/John , x/Mother
(John )}
UNIFY(Knows (John, x), Knows (x, Elizabeth )) = fail .
The last unification fails because x cannot take on the values John and
Elizabeth at the same time. Now, remember that Knows(x, Elizabeth)
means “Everyone knows Elizabeth,” so we should be able to infer that
John knows Elizabeth. The problem arises only because the two sentences
happen to use the same variable name, x. The problem can be avoided by
standardizing apart one of the two sentences being unified, which means
renaming its variables to avoid name clashes. For example, we can rename
x in Knows(x, Elizabeth) to x17 (a new variable name) without changing
its meaning. Now the unification will work:
UNIFY(Knows(John, x) , Knows(x17, Elizabeth)) = {x/Elizabeth,
x17/John} .
There is one more complication: we said that UNIFY should return a
substitution that makes the two arguments look the same. But there could
be more than one such unifier.
For example, UNIFY(Knows(John, x), Knows(y, z)) could return {y/John,
x/z} or {y/John, x/John, z/John}. The first unifier gives Knows(John, z) as
the result of unification, whereas the second gives Knows(John, John).
The second result could be obtained from the first by an additional
substitution {z/John}; we say that the first unifier is more general than the
second, because it places fewer restrictions on the values of the variables.
It turns out that, for every unifiable pair of expressions, there is a single
most general unifier (or MGU) that is unique up to renaming and
substitution of variables. (For example, {x/John} and {y/John} are
considered equivalent, as are {x/John, y/John} and {x/John, y/x}.) In this
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140 Inference in First-Order Logic 8.3.3 Storage and Retrieval :
The TELL and ASK fun ctions used to inform and interrogate a knowledge
base are the more primitive STORE and FETCH functions. STORE(s)
stores a sentence s into the knowledge base and FETCH(q) returns all
unifiers such that the query q unifies with some sentence in the knowledg e
base. The problem we used to illustrate unification for finding all facts that
unify with Knows(John, x) is an example of FETCH.
The simplest way to implement STORE and FETCH is to keep all the
facts in one long list and unify each query against every el ement of the
list. Such a process is inefficient, but it works. We can make FETCH more
efficient by ensuring that unifications are attempted only with sentences
that have some chance of unifying. For example, there is no point in trying
to unify Knows(John , x) with Brother (Richard, John). We can avoid such
unifications by indexing the facts in the knowledge base. A simple scheme
called predicate indexing puts all the Knows facts in one bucket and all the
Brother facts in another. The buckets can be stored in a hash table for
efficient access. Predicate indexing is useful when there are many
predicate symbols but only a few clauses for each symbol.
Sometimes, however, a predicate has many clauses. For example, suppose
that the tax authorities want to keep t rack of who employs and we use a
predicate Employs(x, y). This would be a very large bucket with millions
of employers and tens of millions of employees. Answering a query such
as Employs(x,Richard ) with predicate indexing would require scanning
the entir e bucket. For this particular query, it would help if facts were
indexed both by predicate and by second argument, perhaps using a
combined hash table key. Then we could simply construct the key from
the query and retrieve exactly those facts that unify wi th the query.
For other queries, such as Employs(IBM , y), we would need to have
indexed the facts by combining the predicate with the first argument.
Therefore, facts can be stored under multiple index keys, rendering them
instantly accessible to various queries that they might unify with.
Given a sentence to be stored, it is possible to construct indices for all
possible queries that unify with it. For the fact Employs(IBM ,Richard),
the queries are
Employs(IBM ,Richard) Does IBM employ Richard?
Employs (x,Richard ) Who employs Richard?
Employs(IBM , y) Whom does IBM employ?
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141 Artificial Intelligence

Figure (a) The subsumption lattice whose lowest node is
Employs(IBM, Richard).
Figure(b) The subsumption lattice for the sentence Employs (John , John).
These queries form a subsumption lattice, as shown in Figure(a). The
lattice has some interesting properties. For example, the child of any node
in the lattice is obtained from its parent by a single substitution and the
“highest” common descendan t of any two nodes is the result of applying
their most general unifier. The portion of the lattice above any ground fact
can be constructed systematically. A sentence with repeated constants has
a slightly different lattice, as shown in Figure(b).
The sc heme we have described works very well whenever the lattice
contains a small number of nodes. If function symbols are allowed, the
number of nodes is also exponential in the size of the terms in the sentence
to be stored. This can lead to a huge number of indices. We can respond
by adopting a fixed policy, such as maintaining indices only on keys
composed of a predicate plus each argument, or by using an adaptive
policy that creates indices to meet the demands of the kinds of queries
being asked.
8.4 FORWA RD AND BACKWARD CHAINING In artificial intelligence, forward and backward chaining is one of the
important topics, but before understanding forward and backward chaining
lets first understand that from where these two terms came.
Inference engine:
The infe rence engine is the component of the intelligent system in
artificial intelligence, which applies logical rules to the knowledge base to
infer new information from known facts. The first inference engine was
part of the expert system. Inference engine comm only proceeds in two
modes, which are:
1. Forward chaining
2. Backward chaining
Horn Clause and Definite clause:
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142 Inference in First-Order Logic knowledge base to use a more restricted and efficient inference algorithm.
Logical inference algorithms use forward and backward chaining
approaches, which require knowledge base in the form of the first -order
definite clause.
 Definite clause: A clause which is a disjunction of literals with exactly
one positive literal is known as a definite clause or strict horn clause. A
definite clause either is atomic or is an implication whose antecedent is
a conjunction of positive literals and whose conclusion is a single
positive literal. The following are first -order definite clauses:
King(x) ∧ Greedy(x) ⇒ Evil(x)
King(John)
Greedy(y)
 Horn clause: A clause which is a disjunction of literals with at most
one positive literal is known as horn clause. Hence all the definite
clauses are horn clauses.
Example: (¬ p V ¬ q V k). It has only one pos itive literal k.
It is equivalent to p ∧ q → k.
8.4.1 Forward Chaining :
Forward chaining is also known as a forward deduction or forward
reasoning method when using an inference engine. Forward chaining is a
form of reasoning which start with atomic senten ces in the knowledge
base and applies inference rules (Modus Ponens) in the forward direction
to extract more data until a goal is reached.
The Forward -chaining algorithm starts from known facts, triggers all rules
whose premises are satisfied, and add the ir conclusion to the known facts.
This process repeats until the problem is solved.
Properties of Forward -Chaining:
 It is a bottom -up approach, as it moves from bottom to top.
 It is a process of making a conclusion based on known facts or data,
by starting from the initial state and reaches the goal state.
 Forward -chaining approach is also called as data -driven as we reach
to the goal using available data.
 Forward -chaining approach is commonly used in the expert system,
such as CLIPS, business, and product ion rule systems.
Consider the following famous example which we will use in both
approaches.
Example:
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143 Artificial Intelligence nations. Country A, an enemy of America, has some missiles, and all the
missile s were sold to it by Robert, who is an American citizen."
Prove that "Robert is criminal."
To solve the above problem, first, we will convert all the above facts into
first-order definite clauses, and then we will use a forward -chaining
algorithm to reach the goal.
Facts Conversion into First -Order Logic:
1. It is a crime for an American to sell weapons to hostile nations. (Let's
say p, q, and r are variables)
American (p) ∧ weapon(q) ∧ sells (p, q, r) ∧ hostile(r) ⇒ Criminal(p)
...(1)
2. Country A has some missiles.
∃p Owns(A, p) ∧ Missile(p)
It can be written in two definite clauses by using Existential
Instantiation, introducing new Constant T1.
Owns(A, T1) ......(2)
Missile(T1) .......(3)
3. All of the missiles were sold to country A by Robert.
Missiles(p) ∧ Owns (A, p) ⇒ Sells (Robert, p, A) ......(4)
4. Missiles are weapons.
Missile(p) ⇒ Weapons (p) .......(5)
5. Enemy of America is known as hostile.
Enemy(p, America) ⇒Hostile(p) ........(6)
6. Country A is an enemy of America.
Enemy (A, America) .........(7)
7. Robert is American.
American(Robert) ..........(8)
This knowledge base contains no function sym bols and therefore it is an
instance of the class Datalog knowledge bases. Datalog is a language that
is restricted to first -order definite clauses with no function symbols.
Datalog gets its name because it can represent the type of statements
which are ty pically made in relational databases.
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144 Inference in First-Order Logic Forward chaining proof:
Step -1:
In the first step we will start with the known facts and will choose the
sentences which do not have implications, such as: American(Robert),
Enemy(A, America), Owns(A, T1), and Missile( T1). All these facts will
be represented as below.

Step -2:
At the second step, we will see those facts which infer from available facts
and with satisfied premises.
Rule -(1) does not satisfy premises, so it will not be added in the first
iteration.
Rule -(2) and (3) are already added.
Rule -(4) satisfy with the substitution {p/T1}, so Sells (Robert, T1, A) is
added, which infers from the conjunction of Rule (2) and (3).
Rule -(6) is satisfied with the substitution(p/A), so Hostile(A) is added and
which infer s from Rule -(7).

Step -3:
At step -3, as we can check Rule -(1) is satisfied with the substitution
{p/Robert, q/T1, r/A}, so we can add Criminal(Robert) which infers all the
available facts. And hence we reached our goal statement.
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145 Artificial Intelligence Hence it is proved that Robert is Criminal using forward chaining
approach.
8.4.2 Backward Chaining :
Backward -chaining is also known as a backward deduction or backward
reasoning method when using an inference engine. A backward chaining
algorithm is a form of reasoning, which s tarts with the goal and works
backward, chaining through rules to find known facts that support the
goal. In this section we will study backward chaining example and then
we describe how it is used in logic programming, which is the most widely
used form o f automated reasoning.
Properties of backward chaining:
 It is known as a top -down approach.
 Backward -chaining is based on modus ponens inference rule.
 In backward chaining, the goal is broken into sub -goal or sub -goals to
prove the facts true.
 It is called a goal -driven approach, as a list of goals decides which
rules are selected and used.
 Backward -chaining algorithm is used in game theory, automated
theorem proving tools, inference engines, proof assistants, and various
AI applications.
 The backward -chain ing method mostly used a depth -first search
strategy for proof.
8.4.2.1 Backward Chaining Example :
In backward -chaining, we will use the same above example, and will
rewrite all the rules.
1. It is a crime for an American to sell weapons to hostile nations. ( Let's
say p, q, and r are variables)
American (p) ∧ weapon(q) ∧ sells (p, q, r) ∧ hostile(r) ⇒ Criminal(p)
...(1)
2. Country A has some missiles.
∃p Owns(A, p) ∧ Missile(p)
It can be written in two definite clauses by using Existential
Instantiation, introducing new Constant T1.
Owns(A, T1) ......(2)
Missile(T1) .......(3)
3. All of the missiles were sold to country A by Robert.
Missiles(p) ∧ Owns (A, p) ⇒ Sells (Robert, p, A) ......(4) munotes.in

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146 Inference in First-Order Logic 4. Missiles are weapons.
Missile(p) ⇒ Weap ons (p) .......(5)
5. Enemy of America is known as hostile.
Enemy(p, America) ⇒Hostile(p) ........(6)
6. Country A is an enemy of America.
Enemy (A, America) .........(7)
7. Robert is American.
American(Robert) ........ ..(8)
Backward -Chaining proof:
In Backward chaining, we will start with our goal predicate, which is
Criminal(Robert), and then infer further rules.
Step -1:
At the first step, we will take the goal fact. And from the goal fact, we will
infer other facts, a nd at last, we will prove those facts true. So, our goal
fact is "Robert is Criminal," so following is the predicate of it.

Step -2:
At the second step, we will infer other facts form goal fact which satisfies
the rules. As we can see in Rule -1, the goal predicate Criminal(Robert) is
present with substitution {Robert/P}. So, we will add all the conjunctive
facts below the first level and will replace p with Robert.
Here we can see American (Robert) is a fact, so it is proved here.

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147 Artificial Intelligence Step -3:
At step -3, we will extract further fact Missile(q) which infer from
Weapon(q), as it satisfies Rule -(5). Weapon (q) is also true with the
substitution of a constant T1 at q.

Step -4:
At step -4, we can infer facts Missile(T1) and Owns(A, T1) form
Sells(Robert, T1, r) w hich satisfies the Rule - 4, with the substitution of A
in place of r. So these two statements are proved here.

Step -5:
At step -5, we can infer the fact Enemy(A, America) from Hostile(A)
which satisfies Rule - 6. And hence all the statements are proved tru e using
backward chaining.
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148 Inference in First-Order Logic

8.4.2.2 Logic Programming :
In Logic programming the system is constructed by expressing knowledge
in a formal language and that problem is solved by running inference
processes on that knowledge. Prolog is t he most widely used logic
programming language. It is used primarily as a Rapid prototyping
language and for symbol -manipulation tasks such as writing compilers and
parsing natural language. Many expert systems have been written in
Prolog for legal, medica l, financial, and other domains.
Prolog programs are sets of definite clauses written in a notation which is
different from standard first -order logic. Prolog uses uppercase letters for
variables and lowercase for Constants which is the opposite of our
convention for logic. Commas separate conjuncts in a clause, and the
clause is written “backwards”; instead of
A ∧ B ⇒ C
In Prolog we have
C :- A, B.
Here is a typical example:
criminal(X) : - american(X), weapon(Y), sells(X,Y,Z), hostile(Z).
The notation [E|L] denotes a list whose first element is E and whose rest is
L. Here is a Prolog program for append(X,Y,Z), which succeeds if list Z is
the result of appending lists X and Y:
append([],Y,Y).
append([A|X],Y,[A|Z]) : - append(X,Y,Z).
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149 Artificial Intelligence 1. appending an empty list with a list Y produces the same list Y and
2. [A|Z] is the result of appending [A|X] onto Y, provided that Z is the
result of appending X onto Y.
We can ask the query
append(X,Y,[1,2]): what two lists can be appended to give [1,2]?
We get the solutions
X=[] Y=[1,2];
X=[1] Y=[2];
X=[1,2] Y=[]
The execution of Prolog programs is done through depth -first backward
chaining. Some aspects of Prolog fall outside standard logical inference:
 Prolog uses the database semantics rather than first -order semantics,
and this is apparent in its treatment of equality and negation.
 There is a set of built -in functions for arithmetic. Literals using these
function symbols are “proved” by executing code rather than doing
furthe r inference. For example, the goal “X is 4+3” succeeds where X
bound to 7. On the other hand, the goal “5 is X+Y” fails, because the
built-in functions do not do arbitrary equation solving.
 There are built -in predicates that have side effects when executed .
These include input -output predicates and the assert/retract predicates
for modifying the knowledge base. Such predicates have no
counterpart in logic and can produce confusing results.
 Prolog uses depth -first backward -chaining search with no checks for
infinite recursion. This makes it very fast when given the right set of
axioms, but incomplete when given the wrong ones.
 Dynamic programming is the one in which the solutions to
subproblems are constructed incrementally from smaller subproblems
and are ca ched to avoid re -computation. We can obtain a similar
effect in a backward chaining system using memoization that is,
caching solutions to subgoals as they are found and then reusing those
solutions when the subgoal recurs, rather than repeating the previo us
computation.
 This is the approach taken by tabled logic programming systems,
which use efficient storage and retrieval mechanisms to perform
memoization. Tabled logic programming combines the goal -
directedness of backward chaining with the dynamic prog ramming.
 Prolog uses database semantics. There is no way to assert that a
sentence is false in Prolog. This makes Prolog less expressive than
first-order logic, but it is part of what makes Prolog more efficient and munotes.in

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150 Inference in First-Order Logic more concise. If given problem can be de scribed with database
semantics, it is more efficient to reason with Prolog or some other
database semantics system, rather than translating into FOL and
reasoning with a full FOL theorem prover.
8.4.3 Forward Chaining Vs. Backward Chaining :
Sr. No Forward Chaining Backward Chaining 1 Forward chaining starts from known facts and applies inference rule to extract more data unit it reaches to the goal. Backward chaining starts from
the goal and works backward
through inference rules to find
the required fac ts that support
the goal. 2 It is a bottom-up approach It is a top -down approach 3 Forward chaining is known as data-driven inference technique as we reach to the goal using the available data. Backward chaining is known as
goal-driven technique as we
start from the goal and divide
into sub -goal to extract the
facts. 4 Forward chaining reasoning applies a breadth-first search strategy. Backward chaining reasoning applies a depth-first search strategy. 5 Forward chaining tests for all the available rules Backward chaining only tests for few required rules. 6 Forward chaining is suitable for the planning, monitoring, control, and interpretation application. Backward chaining is suitable for diagnostic, prescription, and debugging application. 7 Forward chaining can generate an infinite number of possible conclusions. Backward chaining generates a finite number of possible conclusions. 8 It operates in the forward direction. It operates in the backward direction. 9 Forward chaining is aimed for any conclusion. Backward chaining is only aimed for the required data.
8.5 RESOLUTION Resolution is a valid inference rule producing a new clause implied by two
clauses containing complementary literals. A literal is an atomic symbol or
its negation, i.e., P, ~P. Resolution is a theorem proving technique that
proceeds by building refutation proofs, i.e., proofs by contradictions. It
was invented by a Mathematician John Alan Robinson in the year 1965.
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151 Artificial Intelligence a conjunction of sentences. To use resolution, translate knowledge base
into conjunctive normal form (CNF), where each sentence is written as a
disjunction of (one or more) literals.
Resolution is used, if there are various statements are giv en, and we need
to prove a conclusion of those statements. Unification is a key concept in
proofs by resolutions. Propositional resolution using refutation is a
complete inference procedure for propositional logic. Here, we describe
how to extend resolutio n to first -order logic. Resolution is a single
inference rule which can efficiently operate on the conjunctive normal
form or clausal form.
8.5.1 Conjunctive Normal Form for First -Order Logic :
First-order resolution requires that sentences be in conjunctiv e normal
form (CNF) that is, a conjunction of clauses, where each clause is a
disjunction of literals.
Clause: Disjunction of literals is called a clause. It is also known as a unit
clause.
Literals can contain variables, which are assumed to be universal ly
quantified. For example, the sentence
∀ x American(x) ∧Weapon(y) ∧ Sells(x, y, z) ∧ Hostile(z) ⇒ Criminal (x)
in CNF form it can be written as
¬American(x) ∨ ¬Weapon(y) ∨ ¬Sells(x, y, z) ∨ ¬Hostile(z) ∨ Criminal
(x)
Every sentence of first -order logic can be converted into an inferentially
equivalent CNF sentence.
The procedure for conversion to CNF is similar to the propositional case.
The principal difference arises from the need to eliminate existential
quantifiers. We illustrate the procedure by tr anslating the sentence
“Everyone who loves all animals is loved by someone,” or
∀ x [∀ y Animal(y) ⇒ Loves(x, y)] ⇒ [∃ y Loves(y, x)]
The steps for translating the sentences in CNF are as follows:
1. Eliminate implications:
∀ x [¬∀ y ¬Animal(y) ∨ Loves(x, y)] ∨ [∃ y Loves(y, x)]
2. Move ¬ inwards:
In addition to the usual rules for negat ed connectives, we need rules for
negated quantifiers. Thus, we have
¬∀x p becomes ∃ x ¬p
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152 Inference in First-Order Logic Our sentence will go through the following transformations:
∀ x [∃ y ¬(¬Animal(y) ∨ Loves(x, y))] ∨ [∃ y Loves(y, x)]
∀ x [∃ y ¬¬Animal(y) ∧ ¬Loves(x, y)] ∨ [∃ y Loves(y, x)]
∀ x [∃ y Animal (y) ∧¬Loves(x, y)] ∨ [∃ y Loves(y, x)]
Notice how a universal quantifier ( ∀ y) in the premise of the implication
has become an existential quantifier. The sentence will now be read as
“Either there is some animal that x doesn’t love, or someone loves x.” The
meaning of the original sentence has been preserved.
3. Standardize variables:
For sentences like ( ∃xP(x)) ∨(∃xQ(x)) which use the same variable name
twice, change the name of one of the variables. This avo ids confusion later
when we drop the quantifiers. Thus, we have
∀ x [∃ y Animal (y) ∧¬Loves(x, y)] ∨ [∃ z Loves(z, x)]
4. Skolemize:
Skolemization is the process of removing existential quantifiers by
elimination. In the simple case, it is just like the Exi stential Instantiation
rule: translate ∃x P(x) into P(A), where A is a new constant. However, we
can’t apply Existential Instantiation to our sentence above because it
doesn’t match the pattern ∃x P(x); only parts of the sentence match the
pattern.
Thus, w e want the Skolem entities to depend on x and z:
∀ x [Animal (F(x)) ∧¬Loves(x, F(x))] ∨ Loves(G(z), x)
Here F and G are Skolem functions. The general rule is that the arguments
of the Skolem function are all the universally quantified variables in
whose s cope the existential quantifier appears.
5. Drop universal quantifiers:
At this point, all remaining variables must be universally quantified and
the previous sentence is equivalent to one in which all the universal
quantifiers have been moved to the left. We can now drop the universal
quantifiers:
[Animal (F(x)) ∧ ¬Loves(x, F(x))] ∨ Loves(G(z), x)
6. Distribute ∨ over ∧:
[Animal (F(x)) ∨ Loves(G(z), x)] ∧ [¬Loves(x, F(x)) ∨ Loves(G(z), x)]
This step may require flattening out nested conjunctions and disjunctions.
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153 Artificial Intelligence 8.5.2 The Resolution Inference Rule :
The resolution rule for first -order clauses is simply a lifted version of the
propositional resolution rule. Resolution can resolve two clauses if they
contain complementary literals, which are assumed to be standa rdized
apart so that they share no variables.
Propositional literals are complementary if one is the negation of the other.
First-order literals are complementary if one unifies with the negation of
the other. Thus, we have

Where, UNIFY(
,¬
)=θ.
For example, we can resolve the two clauses
[Animal (F(x)) ∨ Loves(G(x), x)] and [¬Loves(u, v) ∨ ¬Kills(u, v)]
by eliminating the complementary literals Loves(G(x), x) and ¬Loves(u,
v), with unifier
θ={u/G(x), v/x}, to produce the resolve nt clause
[Animal (F(x)) ∨ ¬Kills(G(x), x)]
This rule is called the binary resolution rule because it resolves exactly
two literals.
Steps for Resolution:
1. Conversion of facts into first -order logic.
2. Convert First -Order logic statements into CNF.
3. Negate t he statement which needs to prove (proof by contradiction).
4. Draw resolution graph (unification).
8.5.3 Example Proof :
We will consider an example in which we will apply resolution.
a. John likes all kind of food.
b. Apple and vegetable are food
c. Anything anyone eats and not killed is food.
d. Anil eats peanuts and still alive
e. Harry eats everything that Anil eats.
Prove by resolution that:
f. John likes peanuts.
Step -1: Conversion of Facts into First -Order Logic
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154 Inference in First-Order Logic logic.

Step -2: Conversion of First -Order logic into CNF
In First order logic resolution, it is required to convert the First -Order
logic into CNF as CNF form makes easier for resolution proofs.
i. Eliminate all implication (→) and r ewrite
a. ∀x ¬ food(x) V likes(John, x)
b. food(Apple) Λ food(vegetables)
c. ∀x ∀y ¬ [eats(x, y) Λ ¬ killed(x)] V food(y)
d. eats (Anil, Peanuts) Λ alive(Anil)
e. ∀x ¬ eats(Anil, x) V eats(Harry, x)
f. ∀x¬ [¬ killed(x) ] V alive(x)
g. ∀x ¬ alive(x) V ¬ killed(x)
h. likes(John, Pe anuts)
ii. Move negation(¬) inwards and rewrite
a. ∀x ¬ food(x) V likes(John, x)
b. food(Apple) Λ food(vegetables)
c. ∀x ∀y ¬ eats(x, y) V killed(x) V food(y)
d. eats (Anil, Peanuts) Λ alive(Anil)
e. ∀x ¬ eats(Anil, x) V eats(Harry, x)
f. ∀x ¬killed(x) ] V alive(x)
g. ∀x ¬ ali ve(x) V ¬ killed(x)
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155 Artificial Intelligence iii. Rename variables or standardize variables
a. ∀x ¬ food(x) V likes(John, x)
b. food(Apple) Λ food(vegetables)
c. ∀y ∀z ¬ eats(y, z) V killed(y) V food(z)
d. eats (Anil, Peanuts) Λ alive(Anil)
e. ∀w¬ eats(Anil, w) V eats(Harry, w)
f. ∀g ¬killed(g) ] V alive(g)
g. ∀k ¬ alive(k) V ¬ killed(k)
h. likes(John, Peanuts)
iv. Eliminate existential instantiation quantifier by elimination.
In this step, we will eliminate existential quantifier ∃, and this process is
known as Skolemization. But in this example problem since there is no
existential quantifier so all the statements will remain same in this step.
v. Drop Universal quantifiers
In this step we will drop all universal quantifier since all the statements are
not implicitly quantified so we don't need it.
a. ¬ food(x) V likes(John, x)
b. food(Apple)
c. food(vegetables)
d. ¬ eats(y, z) V killed(y) V food(z)
e. eats (Anil, Peanuts)
f. alive(Anil)
g. ¬ eats(Anil, w) V eats(Harry, w)
h. killed(g) V alive(g)
i. ¬ alive(k) V ¬ killed(k)
j. likes(John, Peanuts)
vi. Distribute c onjunction ∧ over disjunction V :
This step will not make any change in this problem.
Step -3: Negate the statement to be proved :
In this step, we will apply negation to the conclusion statement, which will
be written as
¬likes(John, Peanuts)
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156 Inference in First-Order Logic Step -4: Draw Re solution graph :
Now in this step, we will solve the problem by resolution tree using
substitution. For the above problem, it will be given as follows:


Hence, the negation of the conclusion has been proved as a complete
contradi ction with the given set of statements.
Explanation of Resolution graph:
a. In the first step of resolution graph, ¬likes(John, Peanuts) , and
likes(John, x) get resolved(cancelled) by substitution of {Peanuts/x},
and we are left with ¬ food(Peanuts).
b. In the second step of the resolution graph, ¬ food(Peanuts) , and
food(z) get resolved (cancelled) by substitution of { Peanuts/z}, and
we are left with ¬ eats(y, Peanuts) V killed(y).
c. In the third step of the resolution graph, ¬ eats(y, Peanuts) and eats
(Anil, Peanuts) get resolved by substitution {Anil/y}, and we are left
with Killed(Anil) .
d. In the fourth step of the resolution graph, Killed(Anil) and ¬ killed(k)
get resolve by substitution {Anil/k}, and we are left with ¬
alive(Anil) .
e. In the last step of the r esolution graph ¬ alive(Anil) and alive(Anil)
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157 Artificial Intelligence
8.5.4 Equality :
None of the inference methods described so far handle an assertion of the
form x = y. Three distinct approaches can be taken. The first approach is
to write down sentences about th e equality relation in the knowledge base.
Equality is reflexive, symmetric, and transitive and we can substitute
equals for equals in any predicate or function. So, we need three basic
axioms, and then one for each predicate and function:
∀x x=x
∀ x, y x=y ⇒ y =x
∀ x, y, z x=y ∧ y =z ⇒ x=z
∀ x, y x=y ⇒ (P1(x) ⇔ P1(y))
∀ x, y x=y ⇒ (P2(x) ⇔ P2(y))
...
∀ w, x, y, z w =y ∧ x=z ⇒ (F1(w, x)= F1(y, z))
∀ w, x, y, z w =y ∧ x=z ⇒ (F2(w, x)= F2(y, z))
...
Given these sentences, a standard inference procedure such as resolution
can perform tasks requiring equality reasoning, such as solving
mathematical equations. However, these axioms will generate a lot of
conclusions, most of them will not be helpful to a proof. So there has been
a search for more efficient ways of handling equality. One alternative is to
add inference rules rather than axioms.
The simplest rule, demodulation, takes a unit clause x=y and some clause
α that contains the term x, and yields a new clause formed by substituting
y for x within α. It works if the term within α unifies with x; it need not be
exactly equal to x.
Note that demodulation is directional; given x = y, the x always gets
replaced with y, never vice versa.
Example:
Given,
Father (Father (x)) = PaternalGrandfather (x)
Birthdate(Father (Father (Bella)), 1926)
we can conclude by demodulation
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158 Inference in First-Order Logic 1. Demodulation: For any terms x, y, and z, w here z appears
somewhere in literal
and where UNIFY(x, z) = θ,

Where, SUBST is the usual substitution of a binding list, and SUB(x, y,m)
means to replace x with y everywhere that x occurs within m.
The rule can also be extended to handle non -unit clauses in which an
equality literal appears .
3. Paramodulation: For any terms x, y, and z, where z appears
somewhere in literal
,
and where UNIFY(x, z) = θ,

For example, from
P(F(x,B), x) ∨ Q(x) and F(A, y)= y ∨ R(y)
we have θ =UNIFY(F(A, y), F(x,B))= {x/A, y/B}, and we can conclude by
paramodulation the sentence
P(B,A) ∨ Q(A) ∨ R(B)
Paramodulation yields a complete inference procedure for first -order logic
with equality.
3. Equational unification:
A third approach handles equality reasoning entirely within an extended
unification algo rithm. That is, terms are unifiable if they are provably
equal under some substitution, where “provably” allows for equality
reasoning. For example, the terms 1 + 2 and 2 + 1 normally are not
unifiable, but a unification algorithm that knows that x + y=y + x could
unify them with the empty substitution. Equational unification of this kind
can be done with efficient algorithms designed for the particular axioms
using commutativity, associativity, and so on rather than through explicit
inference with those ax ioms.
8.5.5 Resolution Strategies :
We know that repeated applications of the resolution inference rule will
eventually find a proof if one exists. In this subsection, we examine
strategies that help find proofs efficiently.
1. Unit preference:
This strat egy prefers to do resolutions where one of the sentences is a munotes.in

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159 Artificial Intelligence single literal (also known as a unit clause). The idea behind the strategy is
that we are trying to produce an empty clause, so it might be a good idea
to prefer inferences that produce shorter clauses. Resolving a unit sentence
(such as P) with any other sentence (such as ¬P ∨¬Q∨R) always yields a
clause (in this case, ¬Q ∨ R) that is shorter than the other clause. This
strategy leads to a dramatic speedup, making it feasible to prove theorems
that could not be handled without the preference. Unit resolution is a
restricted form of resolution in which every resolution step must involve a
unit clause. Unit resolution is incomplete in general, but complete for
Horn clauses.
2. Set of support:
Preferences that try certain resolutions first are helpful, but in general it is
more effective to try to eliminate some potential resolutions altogether.
For example, we can insist that every resolution step involve at least one
element of a special set of c lauses called the set of support. The resolvent
is then added into the set of support. If the set of support is small relative
to the whole knowledge base, the search space will be reduced. The set -of-
support strategy has the additional advantage of genera ting goal -directed
proof trees that are often easy for humans to understand.
3. Input resolution:
In this strategy, every resolution combines one of the input sentences
(from the KB or the query) with some other sentence. The space of proof
trees of this shape is smaller than the space of all proof graphs. Linear
resolution is complete.
4. Subsumption:
The subsumption method eliminates all sentences that are subsumed by
(that is, more specific than) an existing sentence in the KB. For example,
if P(x) is in the KB, then there is no sense in adding P(A) and even less
sense in adding P(A) ∨ Q(B). Subsumption helps keep the KB small and
thus helps keep the search space small.
8.5.6 Practical Uses Of Resolution Theorem Provers :
1. Theorem provers can be applied to the problems involved in the
synthesis and verification of both hardware and softw are. Thus,
theorem -proving research is carried out in the fields of hardware
design, programming languages, and software engineering not just in
AI.
2. In the case of hardware, the axioms describe the interactions between
signals and circuit elements. Logical reasoners designed specially for
verification have been able to verify entire CPUs, including their
timing properties.
3. The AURA theorem prover has been applied to design circuits that
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160 Inference in First-Order Logic 4. In the case of software, re asoning about programs is quite similar to
reasoning about actions, axioms describe the preconditions and effects
of each statement.
5. Similar techniques are now being applied to software verification by
systems such as the SPIN model checker. For example, t he Remote
Agent spacecraft control program was verified before and after flight.
6. The RSA public key encryption algorithm and the Boyer –Moore
string -matching algorithm have been verified this way.
8.6 SUMMARY  In this chapter we analyzed logical inference in first-order logic.
 Inference rules (universal instantiation and existential instantiation)
can be used to propositionalize the inference problem.
 The use of unification to identify appropriate substitutions for
variables eliminates the instantiation step in first -order proofs, making
the process more efficient.
 A lifted version of Modus Ponens uses unification to provide a natural
and powerful inference rule, generalized Modus Ponens.
 The forward -chaining and backward chaining algorithms can be used
for inference.
 The generalized resolution inference rule provides a complete proof
system for first -order logic, using knowledge bases in conjunctive
normal form.
 Several strategies exist for reducing the search space of a resolution
system without compromisin g completeness. One of the most
important issues is dealing with equality; we studied how
demodulation and paramodulation can be used.
 Efficient resolution -based theorem provers have been designed to
prove interesting mathematical theorems and to verify an d synthesize
software and hardware.
8.7 UNIT END QUESTIONS 1. What is Unification? Explain in brief about Unification.
2. Explain Conjunctive Normal Form (CNF) in First -Order logic.
3. Give the outline of simple forward chaining algorithm.
4. Explain in detail backwar d chaining algorithm.
5. Give comparison between forward chaining and backward chaining.
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161 Artificial Intelligence 7. Explain various resolution strategies in detail.
8. From “Horses are animals,” it follows that “The head of a hors e is the
head of an animal.” Demonstrate that this inference is valid by
carrying out the following steps:
a. Translate the premise and the conclusion into the language of first -
order logic. Use three predicates: HeadOf (h, x) (meaning “h is the
head of x”), Horse(x), and Animal (x).
b. Negate the conclusion, and convert the premise and the negated
conclusion into conjunctive normal form.
c. Use resolution to show that the conclusion follows from the premise.
8.8 LIST OF REFERENCES 1. A First Course in Artificial In telligence , First Edition, Deepak
Khemani, Tata McGraw Hill Publisher
2. Artificial Intelligence: A Modern Approach, Third Edition, Stuart
Russel and Peter Norvig, Pearson Publisher
8.9 BIBLIOGRAPHY  Ayorinde, I. and Akinkunmi, B. (2013). Application of Fir st-Order
Logic in Knowledge Based Systems. African Journal of Computing &
ICT
 https://www.javatpoint.com/ai -resolution -in-first-order -logic
 https://www.javatpoint.com/forward -chaining -and-backward -chaining -
in-ai

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162
UNIT V
9
PLANNING
Unit Structure
9.0 Definition of Classical Planning
9.0.1 Planning assumptions
9.1 Algorithms for planning as State Space Search
9.1.1 Forward State Space
9.1.2 Backward State Space
9.2 Planning Graphs
9.2.1 Algorithm to build a plan grap h
9.3 Other Classical Planning Approaches
9.4 Analysis of planning approaches
9.5 Time, schedules and resources
9.5.1 Representing Temporal and resource constraints
9.5.2 Aggregation
9.5.3 Solving Scheduling Problems
9.6 Hierarchical Planning
9.6.1 Hierar chical Task Network (HTN)
9.6.2 Partial Order Planning (POP)
9.6.3 POP one level planner
9.7 Planning and acting in Nondeterministic Domains
9.7.1 Some strategies are listed below
9.8 Multiagent planning
9.9 Summary
9.10 Unit End Questions
9.11 References
9.0 DEFINITION OF CLASSICAL PLANNING 1. How to reach goal from initial state is nothing but planning.
2. In order to achieve goal a sequence of actions is needed such kind of
behaviour is called as planning.
3. During the execution after planning the age nt will need to do
prediction of the future so that it can move accordingly.
4. In simple words we can call planning as decision making system that
tries to achieve its goal.
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163 Artificial Intelligence 9.0.1 Planning assumpti ons:
GOAL, observations, actions, deterministic nature, events etc.
9.1 ALGORITHMS FOR PLANNING AS STATE SPACE SEARCH 1. Information is needed to predict something; all these information is
given by state.
2. This information is used to apply actions and decision ca n be made if
the state is a goal state.
3. There are 3 state space for a problem:
a. Initial state: The first state from which the action is applied and
begins
b. Goal state: Here the objective is satisfied
c. Solution: Target is achieved here
4. There are 2 types of st ate space search.
a. Forward state space
b. Backward state space.
9.1.1 Forward State Space :
1. In this type of planning the agent will start from initial state and will go
till the final i.e., target state.
2. The aim is to find the final path or full path which will give the target.
3. In the below example if you will notice then according to the forward
state space, the initial state is Train is at Mumbai, now when the train
moves it is getting two states after the action, one is train is in
Rajasthan and another state is the train is at Delhi.
4. Imagine the goal is to reach Delhi then we can say that the target
reached from initial till goal with the help of Forward state space.

9.1.2 Backward State Space :
1. Backward state space planning is opposite to forward state sp ace
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164 Planning 2. In this state space planning the state will start from the goal state the it
will go towards the initial state.
3. In the below example if the target is Delhi, then the first state from
which it will start is Train is at Delhi then it will move towards the
initial state i.e., Train is at Mumbai.

9.2 PLANNING GRAPHS 1. This a technique in which plan is represented graphically to make
work easy and to get the better an accurate planning.
2. All planning methods that is done can be rechecked or can be
converted into planning graphs and checked.
3. There are many algorithms available which can be utilized to create
this plan graph. One of the algorithms is called as GRAPHPLAN
algorithm which is used in creation of notion and actions of the graph.
4. There is also parallelism which is supported by plan graph it is used to
represent many states and how the particular state runs independently
is represented
5. In the below graph you will come across 3 categories
a. Action level nodes: The actions that can be executed in that
particular period
b. State level nodes: The state that can be true in that particular
period
c. Edge: The pre -condition which is applied in it and the effect
6. In the below example S 0 is the initial state
7. Every state will hav e its associated actions that can be true in that
particular period.
8. If any conflict occurs anywhere then that can be represented using
mutual exclusion links.
9. The level A 0 contains all the actions that can happed after the state S 0
10. The small box that is present above the edges represents facts.
11. Fact is nothing but it says that the literals that is getting used will not
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165 Artificial Intelligence

9.2.1 Algorithm to build a plan graph :
1. Start at S0
2. Initialize the variable
3. Find the all -possible actions ( A)that can be applied in that particular
state, by incrementing it A++
4. Check if there is any mutex present anywhere.
5. Move ahead with next State by incrementing it S++
6. If mutex is present then compute that as well
7. If solution found then return success and exit.
8. If no solution found or possible then return failure
9.3 OTHER CLASSICAL PLANNING APPROACHES 1. Classical planning as Boolean satisfiability
 Boolean satisfiability is also called as propositional satisfiability
problem
 It will check and let you know whe ther the plan is satisfying or not
satisfying.
 As the name is Boolean it uses two values i.e. true or false so by using
this approach if it returns true then we can say that the plan is
satisfying otherwise not.
 It is also called as SAT.
2. Planning as constr aint satisfaction
 In this mathematical query are applied for a set of objects and its state
should agree with the constraints.
 Constraint Satisfaction Problem (CSP)
 It is a planning where it ignores the fact of partially ordered plan
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166 Planning  It searches through space not through state of the plan.
3. Planning as first order logical deduction: situation calculus
 The language called as PDDL (Planning Domain Definition
Language) is used to balance and tackle by operating the high
complex algorithm.
 The stating state is called as situation then action is applied on this
situation. After applying particular set of action result is achieved.
Even the result is regarded as situation.
 Fluent is the one which occurs when the relation is di fferent from the
situation
 On the basis of precondition, the action is executed.
9.4 ANALYSIS OF PLANNING APPROACHES  Planning = Search + Logic
 From the above formula we can say that planning is a combination of
searching and logic.
 Planner is a program that he lps us to find the solution.
 GRAPHPLAN can be used as it will save or it will help in finding the
difficult interactions which are due to mostly by mutex
 Even SATPLAN deals with the same kind of mutex relations.
 Subgoal is another task which is used to rea ch goal.
9.5 TIME, SCHEDULES AND RESOURCES  Planning is done first, scheduling is done later
 With the help of planning the different types of actions that can be
applied is decided again with the help of these set of actions that was
decided during planning, ou t of this a suitable action is chosen and that
action is scheduled to get the result.
 According the need of action resources are provided to it.
9.5.1 Representing Temporal and resource constraints :
 Job Shop Scheduling (JSS) or Job Sop Problem is used to d o
operational research and get the optimized results out of it.
 The main role of Job Shop Scheduling is to assign a job to the
particular resource in that particular period of time.
 It has a particular section each section has its particular task
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167 Artificial Intelligence  When it comes to Job Shop Scheduling machine can do only one
work at a time.
 It follows non -pre-emptive approach i.e., if the job starts with one task
then it has to complete fully, it can’t stop in between.
 In the below example J1, J2 and J3 re presents job and at the left end
Machines are present with a number line starting at zero and going till
7.
 So, from the diagrammatic representation we can say that it takes
around 7 for all the jobs to finish.
 It should be noted that the values by this is not always optimal in
nature it may vary on the basis of machine, resource, kind of job etc.

9.5.2 Aggregation :
 Aggregation is one in which the variable represented in the following
manner like Man(3) instead of Man(l1), Man(l2) and Man(l3).
 The main a im of aggregation is nothing but groupism i.e., it may
happen that there is a need of whole object to be used together in such
scenarios we may use aggregation instead of calling each object
individually.
9.5.3 Solving Scheduling Problems :
 In order to lowe r the time duration of planning phase we must start
the plan and actions at the earliest so that it can be considered on the
basis of ordering of the constraints and move on.
 With the help of directed graph, the orders of your plan and actions
can be analy sed.
 For efficient use of the above graph, CPM which stands for Critical
Path Method can be applied to get the possible outcome for where to
start and end.

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168 Planning 9.6 HIERARCHICAL PLANNING

 In Hierarchical planning the importance is given to goal by ignoring
all oth er actions that is also possible along with it.
 It is also known as plan decomposition
 The main plus point of hierarchical planning is that it’s giving more
important to the final target -based goal and ignoring the other things.
 Initially at the abstract l evel a plan is taken and applied by considering
it.
 Then a particular solution on the basis of abstract consideration is
applied and achieved and moved ahead by taking it as an input.
 Also, the abstract level can be single or multiple levels based on the
need the selection and the process is done.
 Hierarchical planning involves to methods such as HTN also called as
Hierarchical Task Network and POP called as Partial Order Planning.
 We will look at each types below:
9.6.1 Hierarchical Task Network (HTN) :
 In the higher -level a less selection of actions is done and level by
level it gets lessen at the lower level.
 HTN is regarded as an act of selecting the goal using a particular
actions, to do this a proper selection of action is done at the top level
this is called as ACT, then implementation of this act is followed in
the lower level as well until the goal is achieved.
 When it comes to HTN, the first plan is regarded as very big level as
many planning and actions has to be considered at this phase.
 As soon as the action is chosen and applied in that particular level, the
decomposition of the level takes place into a partial one and this
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169 Artificial Intelligence 9.6.2 Partial Order Planning (POP) :
 In this the planning and action is done at the part ial level i.e.; we will
go little deeper in our scenario and try to achieve the goal.
 POP doesn’t follows the action sequences, if you have 2 action then
any of the one will be chosen and executed the main aim is to reach
the goal.
 To understand POP let’s take an example of colouring a flower which
has a stem and petals
 So, imagine we coloured the petal first, then we coloured the stem.
 At the end the goal is achieved called as colouring.

9.6.3 POP one level planner :
 As the name is suggesting one level it does the same when it comes to
application i.e., it will plan level by level.
 Let’s take an example and understand this, so if you want to go travel
to a particular destination then the first thing that you will do is
buying a ticket so to do that you nee d to go to ticket counter stand in
queue then wait for your turn to come then fill the ticket slip, submit
id proof and at the end you will get the ticket. Such kind of planning
and achieving the goal is not hing but one level planner.
9.7 PLANNING AND ACTI NG IN NONDETERMINISTIC DOMAINS  Nondeterministic domains are those in which nothing is known prior. munotes.in

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170 Planning  In this plan has to be done with very less or no details
 Because of unknown details uncertainty occurs
 As the environment is less available or not available at all in this case
perception is the best useful way to handle the situation, so that if the
agent faces any sudden situation or need to do something in this case
it can use the perception that was done earlier to apply particular
action in that situation .
 The good solution to these kinds of problems is nothing but strategies.
 Knowledge is the key for every situation.
9.7.1 Some strategies are listed below :
 Sensor less planning
 In this there will be partial observability or no observability at all
 The agen t as to use belief state which means that the agent needs to
believe that something is present and move accordingly.
 It is also called as conformant planning
 It is not done on the basis of any perception.
 The only moto of this strategy is to reach goal
 Contingent Planning :
 It is also called as conditional planning
 On the basis of condition, the agent makes the plan, applies the action
and executes it to reach the goal.
 The plan made on the basis of environment which can be partially
observable or fully obse rvable is appropriate.
 The variable which is used in contingent should be existential
quantifier in nature.
 In this it can also use belief state if needed but this belief state must
satisfy the condition, it may use formula for the same.
 Online replanning :
 In this particular technique a continuous checking is done which is
nothing but the monitoring.
 Replanning is done by execution of it and a new plan is created.
 It is not always necessary that the whole plan has to be replanned,
sometimes replanning may b e done in particular part as well
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171 Artificial Intelligence  In the below figure P is nothing but the plan that has been applied
using this plan the agent will move ahead and side by side it will also
observe the whole execution and once it gets to know that i n the
particular state say E the execution or the action that has been applied
is not according to the need to reach the goal it will change the plan
which is nothing but replanning or we can also call it as repairing.
Now after replanning again the new ac tions are applied and executed
to reach the goal in a very efficient manner.
 3 monitoring are supported
1. Action monitoring: here the agent will check whether the actions are
according to the precondition.
2. Plan monitoring: In this before running the action the agent will verify
if it is according to the plan and whether it will give success.
3. Goal monitoring: Betterment of the goal is checked here by checking
it.

9.8 MULTIAGENT PLANNING  In multiagent each agent has its own goal
 A agent’s main role is to pla n, act and sense
 When there are multiagent in the environment each agent has its own
goal and planning which leads to problem.
Categories:
 Multieffector planning: An agent having multiple effector associated
with it is called as multieffector planning age nt. To understand this,
consider the following example: A person can sing and dance at the
same time
 If these effectors are divided then we will get multibody.

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172 Planning Planning with multiple simultaneous actions :
 Multiactor: We will consider all multiagent, multib ody and
multieffector as one. So by doing this we are going to call all these as
multiactor.
 Here the actor is pointing to agents, models and body.
 In this case the action is not always individual, over here the joint
action is decided by all or many agent s.
 Given by joint action(a1, a2, ….an)
 Cooperation and Coordination : The problem that arise is, as we
say that joint decision has to be made then if there is any decision arises
through any of the agent then each agent should agree with it. Achieving
such agreement by every agent is what the main problem is. For this the
agents should cooperate and coordinate with each other.
 The one solution that is possible is convention.
 If convention is not present then the substitution to it is
communication between th e agents.
 For example, the person who wants someone to play with them can
call them and make team join in there team e.g.: “Join my team,
John!”
 With evolution convention can arise
 One of the good examples of cooperative process is fish school where
the gr oup of fish swim together in particular fashion.
9.9 SUMMARY  In this chapter how are what kind of planning has to be implemented
to achieve goal is getting explained in a very brief manner.
 In forward chaining a initial phase is considered and it wil l move
further till the final state whereas in backward chaining vice a versa is
applied.
 Graphical representation of plan is done using GRAPHPLAN
 Decomposition of plan is done to get the other solutions that is
possible in the graph such a way of finding the solution is called as
hierarchical plan approach.
 Planning in nondeterministic approach is the one in which the details
like environment or data are not known prior.
 Online planning is the one in which continuous monitoring is
emphasised and replanning is made again to fine the most apt
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173 Artificial Intelligence 9.10 UNIT END QUESTIONS  Explain replanning in detail.
 What do you mean by subgoals?
 What are nondeterministic domains? Explain planning adaption for
non-deterministic domains
 Explain general characteristics of uncertain environments.
 Write a short note on conditional planning or contingency planning
 Explain online replanning in detail.
 Write a short note on how planning is done with Multiple
simultaneous actions.
 Write a short note one sensorless planning or con formant planning

9.11 REFERENCES  Artificial Intelligence: A modern approach by Stuart Russel and peter
Norvig
Publisher: Pearson 3rd edition year is 2015
 A first course in artificial intelligence by Deepak Khemani, TMH
publisher frist edition 217
 Artificial intelligence: A ration approach by Rahul Deva, Shroff
publishers, 1st edition with the year as 2018
 Artificial Intelligence by Elaine Rich, Kevin Knight and Shivashankar
Nair 3rd edition 2018
 Artificial Intelligence and soft computing for beginne rs by Anandita
das Bhattacharjee


*****
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174
10
KNOWLEDGE REPRESENTATION
Unit Structure
10.0 Categories and Objects
10.1 Events
10.2 Mental events and mental objects
10.3 Reasoning systems for categories
10.3.1 Semantic networks
10.3.2.1.1 Description logics
10.4 Reasoning with default information
10.4.1 C ircumscription and default logic
10.5 Internet shopping world
10.5.1 Following links
10.6 Summary
10.7 Unit End Questions
10.8 References
10.0 CATEGORIES AND OBJECTS  Facts are nothing but objects.
 Pink colour pen falls under object property.
 In this way many pr operties of an object can be grouped together and
we can make them fall under categories.
 All dogs are mammals so these mammals can be called as category.
 With the help of inheritance, we can make it easy by using categories.
 We can show categories in two ways:
 One is using objects and predicates of FOL
Eg: Dog(x)
 Second way is by transforming the proposition into objects.
 Next important thing to consider is Subclass, subcategory.
 To understand subcategory let’s understand the following example:
Let’s take football, this football falls under category Ball, so now we can
say that football is subcategory of ball.
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175 Artificial Intelligence  One another thing is consider this, sentence1: if its cloudy then it will
rain, sentence 2: If it’s raining then use umbrella, from these 2
sentences we can say that if its cloudy then use umbrella. This kind of
creating new sentences or predicting something using the given data is
nothing but inheritance.
10.1 EVENTS  In the above topic we saw that the things were working on the basis of
situation, but in this topic the event is done on the basis of time.
 It uses fluent and object to proceed further.
 Fluent is something where the behaviour is changed according to the
time.
 We may represent fluent in this way:
On(dress, flower pic)
To show that the fluent is true in that particular time, The alphabet T is
used as follows:
T(On(dress, flower pic)
 In the above manner each predicate can be represented in there own
unique way on the basis of there facts, let’s see some example of the
same:
 When the fluent becomes true after some gap, we can call it as
restoration, so we will represent this as follows:
Restored(f,i)
Where f: fluent
i: interval
 When the fluent is true particular time, it is r epresented as follows:
T(f,t)
Where, T: true
f: fluent
t: time
 If it has to end then the word terminates is use to represent it, below is
an example:
Terminates(e,f,t)
Where: e: Event
f: fluent
t: time
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176 Knowledge Representation 10.2 M ENTAL EVENTS AND MENTAL OBJECTS  In this deducing of knowledge happens.
 In our life also many times it happens that we want to know about
something. We want to gain knowledge of something new. This new
knowledge about something can be g ained by asking a query from
someone.
 This kind of gaining knowledge from someone’s mind is called as
mental objects and the way by changing these objects is called as
mental events.
10.3 REASONING SYSTEMS OF CATEGORIES  In order to apply reasoning in the c ategories we need to represent it in
any of the one way from this;
 Semantic network
 Efficient algorithms
10.3.1 Semantic networks :
 It is also called as existential graphs.
 It is made up of nodes and edges.
 It uses propositional information to create graph so it is also called as
propositional network.
 It is 2d dimension
 The 2d is on the basis of knowledge.
 The graph is made up of nodes, links and labels.
 Objects are represented by nodes.
 Link represents the relationships.
 Nodes can be represented using oval or circle symbol.
 Directed link with “IS” or “has” etc representing the relationships
Eg: Dog is mammal
Animals are mammal

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177 Artificial Intelligence

10.5.2 Description logics:
 It is an advance version of semantic network.
 It represents the knowledge in a diagrammatic way.
10.4 REA SONING WITH DEFAULT INFORMATION  It is non monotonic in nature
 Non monotonic is nothing but additional information that can get
added in between and can lead to earlier conclusions.
 Changed notion on the truth is applied.
10.4.1 Circumscription and default logic :
 The meaning of circumscription is restriction of anything within some
limits.
 A model may substitute by another if it has any unwanted or abnormal
things inside it.
 The rule looks like this:
Dog(i): barks(i)/barks(i)

 Based on the above rule we can say that the Dog(i) if true if it barks(i) .
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178 Knowledge Representation  If any probability interferes in the rule then it may be replaced or
substituted.
10.5 INTERNET SHOPPING WORLD  Almost all the present shopping websites and applications are using
artifi cial intelligence.
 Using AI we can even judge which shopping website has the maximum
visitors and accordingly ranking can be done.
 Also, if a particular website has customers who visit frequently then a
personalized offers can be given.
 World wide web is t he environment where the agent.

10.5.1 Following links :
 On all the modules of the website the links are added between the
modules.
 This module will be able to interact and make it work with each other
easily
 These re artificial intelligence based.
 A cate gorization is also done under this.
 All product details, customer details are available and stored.
10.6 SUMMARY  In this chapter we saw that how the we categories the data.
 There are subcategories that can be applied. It uses fluent and objects.
 Knowledge gaining from someone else mind is explained in which
mental object comes into picture.
 Reasoning system of categories is of two types one is semantic
network and second one is efficient algorithms both helps us to
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179 Artificial Intelligence  Artificial Intelligence plays one of the important roles in internet
shopping world as almost everything from creating the account till the
product/service delivery requires intelligence.
10.7 UNIT END QUESTIONS 1. What do you mean by classical language?
2. What are categories with respect to Artificial Intelligence?
3. Write a short note on Mental Events
4. Write a short note on Events.
5. With the help of example explain in detail about semantic networks.
6. Explain briefly about internet shoppin g world
10.8 REFERENCES  Artificial Intelligence: A modern approach by Stuart Russel and peter
Norvig Publisher: Pearson 3rd edition year is 2015
 A first course in artificial intelligence by Deepak Khemani, TMH
publisher frist edition 217
 Artificial intelli gence: A ration approach by Rahul Deva, Shroff
publishers, 1st edition with the year as 2018
 Artificial Intelligence by Elaine Rich, Kevin Knight and Shivashankar
Nair 3rd edition 2018
 Artificial Intelligence and soft computing for beginners by Anandita
das Bhattacharjee

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