CS621: Artificial Intelligence

Pushpak BhattacharyyaCSE Dept., IIT Bombay

Lecture 37–Prolog- cut and backtracking; start of Fuzzy Logic

A Typical Prolog program

Compute_length ([],0).Compute_length ([Head|Tail], Length):-

Compute_length (Tail,Tail_length),Length is Tail_length+1.

High level explanation:The length of a list is 1 plus the length of the tail of the list, obtained by removing the first element of the list.

This is a declarative description of the computation.

Prolog Program Flow, BackTracking and Cut

Controlling the program flow

Prolog’s computation

Depth First Search Pursues a goal till the end

Conditional AND; falsity of any goal prevents satisfaction of further clauses.

Conditional OR; satisfaction of any goal prevents further clauses being evaluated.

Control flow (top level)

Giveng:- a, b, c. (1)g:- d, e, f (2)

If prolog cannot satisfy (1), control will automatically fall through to (2).

Control Flow within a rule

Taking (1),g:- a, b, c.

If a succeeds, prolog will try to satisfy b, succeding which c will be tried.

For ANDed clauses, control flows forward till the ‘.’, iff the current clause is true.

For ORed clauses, control flows forward till the ‘.’, iff the current clause evaluates to false.

What happens on failure

REDO the immediately preceding goal.

Fundamental Principle of prolog programming

Always place the more general rule AFTER a specific rule.

CUT Cut tells the system that

IF YOU HAVE COME THIS FAR

DO NOT BACKTRACK

EVEN IF YOU FAIL SUBSEQUENTLY.

‘CUT’ WRITTEN AS ‘!’ ALWAYS SUCCEEDS.

For example: belongs-to predicate

(1) belongs-to(E,[E|T]).(2) belongs-to(E,[E1|T]) :-

belongs-to(E,T).What about(3) belongs-to(E,[E|T]) :- !.(4) belongs-to(E,[E1|T]) :-

belongs-to(E,T). (4) will not be executed!!!

Fail

This predicate always fails. Cut and Fail combination is used to

produce negation. Since the LHS of the neck cannot

contain any operator, A ~B is implemented as

B :- A, !, Fail.

Predicate Calculus

Introduction through an example (Zohar Manna, 1974): Problem: A, B and C belong to the

Himalayan club. Every member in the club is either a mountain climber or a skier or both. A likes whatever B dislikes and dislikes whatever B likes. A likes rain and snow. No mountain climber likes rain. Every skier likes snow. Is there a member who is a mountain climber and not a skier?

Given knowledge has: Facts Rules

A wrong prolog program!1. member(a).2. member(b).3. member(c).4. mc(X);sk(X) :- member(X) /* X is a mountain climber or

skier or both if X is a member; operators NOT allowed in the head of a horn clause; hence wrong*/

5. like(X, snow) :- sk(X). /*all skiers like snow*/6. \+like(X, rain) :- mc(X). /*no mountain climber likes rain; \

+ is the not operator; negation by failure; wrong clause*/7. \+like(a, X) :- like(b,X). /* a dislikes whatever b likes*/8. like(a, X) :- \+like(b,X). /* a dislikes whatever b likes*/9. like(a,rain).10. like(a,snow).?- member(X),mc(X),\+sk(X).

How to represent in Prolog: Ram dislikes whatever Shyam likes and likes whatever Shyam dislikes Prolog uses Horn clauses, and so no

operator, including negation, is allowed on the Head of any rule

However, Prolog's machinery of backtracking (which tries to prove the goal in a different path), ! (Cut} operator which cuts-out backtracking) and Fail (which always returns false) can be used for solving the above problem.

Solution like(ram,X) :- like(shyam,X), !, Fail. like(ram,X). Justification: Suppose it is asserted in the

database that like(shyam,apple)

Now on query ?like(ram, apple)

Solution (contd.) like(shyam,apple) will get satisfied,

control will move forward, ! will get executed, Fail will get executed, returning No for like(ram,apple).

The system will NOT backtrack, since ! was executed before, and NOT try to resatisfy the next assertion like(ram, X).

Solution (contd.) Suppose, now, it is asserted that

like(shyam, apple) :- Fail. /*i.e., Shyam does not like apple

Now on query ?like(ram, apple) like(shyam,apple) will fail.

The system WILL backtrack and resatisfy the next assertion like(ram, X), returning like(ram, apple) as TRUE, binding X to apple.

Prolog’s way of making and breaking a list

Problem: to remove duplicates from a list

rem_dup([],[]).rem_dup([H|T],L) :- member(H,T), !, rem_dup(T,L).rem_dup([H|T],[H|L1]) :- rem_dup(T,L1).

Note: The cut ! in the second clause needed, since after succeeding at member(H,T), the 3rd clause should not be tried even if rem_dup(T,L) fails, which prolog will otherwise do.

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M.Tech Seminar Topics: 2009

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1. Cross Lingual Information Retrieval The goal is to study the techniques of information retrieval when the language of the query is different from the language of the web page.

After covering basics of information retrieval (crawling, indexing, ranking etc.), the student is expected to study the complexities of cross lingual search (see http://clef-campaign.org).

Considerable work on this has happened at IIT Bombay which leads the national project on CLIR (http://www.clia.iitb.ac.in/clia-beta-ext; the site may not be up always).

2. Semantic Role Labeling (UNL) Sentences have inherent structure in terms of agent, object, instrument, time, place and such relationships between words.

When extracted, such information finds usage in a large number of applications like machine translation, entailment, question answering, information extraction etc.

We will study semantic roles in the form of Universal Networking Language (UNL; http://www.undl.org) and the ongoing work on UNL extraction spanning over last several years (visit http://www.cse.iitb.ac.in/~pb, click on publications and see publications on “Semantics”).

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3. Empirical Methods in NLP-ML This is a foundational topic proposing to investigate mathematical and statistical principles and methods cutting across different areas of NLP and Machine Learning.

Examples of these are Spectral Analysis, Expectation Maximization, Graphical Models, Lexical Semantic Association Techniques and so on. Additional topics of study will be NLP-suitable probability distributions, e.g., Dirichlet distribution. Specific NLP and ML applications will always be kept in focus.

Browse proceedings of EMNLP conference (http://www.isca-students.org/emnlp_2009) and CoNLL conference (http://www.cnts.ua.ac.be/conll/).

4. Text Entailment This studies the methodologies involved in deciding if a piece of text (H) is inferrable from another (e.g., “France beat Brazil in the FIFA World Cup semi final” entails “Brazil bowed out of the World Cup”).

Grounded on predicate calculus, there has been considerable work recently on using machine learning techniques for entailment (http://pascallin.ecs.soton.ac.uk/Challenges/RTE/). A number of students have worked with me on this topic and the use of UNL in entailment.

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5. Foundation of Artificial Intelligence We will study work by Minsky, Newel and Simon, McCarthy, Penrose, Dreyfus, Hofstadter, Marr, Chomsky and so on, including Indian thoughts on intelligence and consciousness.

6. Machine Translation, Principles and Paradigms Starting with Interligua Based Machine Translation using UNL, we have made inroads in Statistical Machine Translation (http://www.cse.iitb.ac.in/~pb/papers/acl09-smt.pdf).

This topic will study many different approaches to machine translation which is a key problem (along with CLIR) for a multilingual country like India.

We are members of large national projects on English-to-Indian-Language and Indian-language-to-Indian-Language machine translation.

7. Shallow Parsing for Indian Languages This is ongoing critical work attempting to build common platforms for morphology, POS tag and chunk processing for Indian Languages- especially Hindi and Marathi.

Talk to seniors Mugdha Bapat (mbapat@cse) and Harshada Gune (harshada@cse).

Fuzzy Logic

Uncertainty StudiesUncertainty Study

Probability Based Information Theory based

Fuzzy Logic Based

Probabilistic Reasoning

Markov Processes& GraphicalModels

EntropyCentricAlgos

Qualitative Reasoning

Bayesian Belief Network

Outlook(0)

Temp(T)

Humidity

(H)

Windy(W)

Decision

to play(D)

Sunny High High F N

Sunny High High T N

Cloudy High High F Y

Rain Med High F Y

Rain Cold Low N Y

Rain Cold Low T N

Cloudy Cold Low T YTo-play-or-not-to-play-tennis data vs. Climatic-Condition from Ross Quinlan’s paper on ID3 (1986), C4.5 (1993)

Weather

(0)

Temp(T)

Humidity

(H)

Windy(W)

Decision

(D)

Sunny Med High F N

Sunny Cold Low F Y

Rain Med Low F Y

Sunny Med Low T Y

Cloudy Med High T Y

Cloudy High Low F Y

Rain High High T N

Outlook

RainCloudySunny

WindyYesHumidity

High Low

YesNo

T F

YesNo

Rule Base

R1: If outlook is sunny and if humidity is high then Decision is No.

R2: If outlook is sunny and if humidity is low then Decision is Yes.

R3: If outlook is cloudy then Decision is Yes.

Fuzzy Logic

Fuzzy Logic tries to capture the human ability of reasoning with imprecise information

Models Human Reasoning Works with imprecise statements such as:

In a process control situation, “If the temperature is moderate and the pressure is high, then turn the knob slightly right”

The rules have “Linguistic Variables”, typically adjectives qualified by adverbs (adverbs are hedges).

Underlying Theory: Theory of Fuzzy Sets Intimate connection between logic and set

theory. Given any set ‘S’ and an element ‘e’, there is a

very natural predicate, μs(e) called as the belongingness predicate.

The predicate is such that, μs(e) = 1, iff e ∈ S

= 0, otherwise For example, S = {1, 2, 3, 4}, μs(1) = 1 and μs(5)

= 0 A predicate P(x) also defines a set naturally.

S = {x | P(x) is true}For example, even(x) defines S = {x | x is

even}

Fuzzy Set Theory (contd.)

Fuzzy set theory starts by questioning the fundamental assumptions of set theory viz., the belongingness predicate, μ, value is 0 or 1.

Instead in Fuzzy theory it is assumed that, μs(e) = [0, 1]

Fuzzy set theory is a generalization of classical set theory also called Crisp Set Theory.

In real life belongingness is a fuzzy concept.Example: Let, T = set of “tall” people

μT (Ram) = 1.0μT (Shyam) = 0.2

Shyam belongs to T with degree 0.2.

Linguistic Variables Fuzzy sets are named

by Linguistic Variables (typically adjectives).

Underlying the LV is a numerical quantityE.g. For ‘tall’ (LV), ‘height’ is numerical quantity.

Profile of a LV is the plot shown in the figure shown alongside.

μtall(h)

1 2 3 4 5 60

height h

1

0.4

4.5

Example Profiles

μrich(w)

wealth w

μpoor(w)

wealth w

Example Profiles

μA (x)

x

μA (x)

x

Profile representingmoderate (e.g. moderately rich)

Profile representingextreme

Concept of Hedge Hedge is an intensifier Example:

LV = tall, LV1 = very tall, LV2 = somewhat tall

‘very’ operation: μvery tall(x) = μ2

tall(x) ‘somewhat’ operation: μsomewhat tall(x) = √(μtall(x))

1

0h

μtall(h)

somewhat tall tall

very tall

Representation of Fuzzy setsLet U = {x1,x2,…..,xn}

|U| = n

The various sets composed of elements from U are presented as points on and inside the n-dimensional hypercube. The crisp sets are the corners of the hypercube.

(1,0)(0,0)

(0,1) (1,1)

x1

x2

x1

x2

(x1,x2)

A(0.3,0.4)

μA(x1)=0.3

μA(x2)=0.4

Φ

U={x1,x2}

A fuzzy set A is represented by a point in the n-dimensional space as the point {μA(x1), μA(x2),……μA(xn)}

Degree of fuzziness

The centre of the hypercube is the “most fuzzy” set. Fuzziness decreases as one nears the corners

Measure of fuzziness

Called the entropy of a fuzzy set

),(/),()( farthestSdnearestSdSE

Entropy

Fuzzy set Farthest corner

Nearest corner

(1,0)(0,0)

(0,1) (1,1)

x1

x2

d(A, nearest)

d(A, farthest)

(0.5,0.5)A

Definition

Distance between two fuzzy sets

|)()(|),(21

121 is

n

iis xxSSd

L1 - norm

Let C = fuzzy set represented by the centre point

d(c,nearest) = |0.5-1.0| + |0.5 – 0.0|

= 1

= d(C,farthest)

=> E(C) = 1

Definition

Cardinality of a fuzzy set

n

iis xsm

1

)()( [generalization of cardinality of classical sets]

Union, Intersection, complementation, subset hood

)(1)( xx ssc

Uxxxx ssss )](),(max[)(2121

Uxxxx ssss )](),(min[)(2121

Note on definition by extension and intension

S1 = {xi|xi mod 2 = 0 } – Intension

S2 = {0,2,4,6,8,10,………..} – extension

How to define subset hood?