CS621 : Artificial Intelligence

Pushpak BhattacharyyaCSE Dept., IIT Bombay

Lecture 19: Fuzzy Logic and Neural Net Based IR

The IR scenarioDocs

Information Need

Index Terms

doc

query

Rankingmatch

IR system Maker’s view

Definition of IR Model

An IR model is a quadrupul

[D, Q, F, R(qi, dj)]

Where,

D: documents

Q: Queries

F: Framework for modeling document, query and their relationships

R(.,.): Ranking function returning a real no. expressing the relevance of dj with qi

The Boolean Model

• Simple model based on set theory• Only AND, OR and NOT are used• Queries specified as boolean expressions

– precise semantics– neat formalism

– q = ka (kb kc)

• Terms are either present or absent. Thus, wij {0,1}

• Consider

– q = ka (kb kc)

– vec(qdnf) = (1,1,1) (1,1,0) (1,0,0)

– vec(qcc) = (1,1,0) is a conjunctive component

The Boolean Model

• q = ka (kb kc)

• sim(q,dj) = 1 if vec(qcc) | (vec(qcc) vec(qdnf)) (ki, gi(vec(dj)) = gi(vec(qcc))) 0 otherwise

(1,1,1)(1,0,0)

(1,1,0)

Ka Kb

Kc

Fuzzy Set Model

• Queries and docs represented by sets of index terms: matching is approximate from the start

• This vagueness can be modeled using a fuzzy framework, as follows:– with each term is associated a fuzzy set– each doc has a degree of membership in this fuzzy set

• This interpretation provides the foundation for many models for IR based on fuzzy theory

• In here, we discuss the model proposed by Ogawa, Morita, and Kobayashi (1991)

Fuzzy Set Theory

• Definition– A fuzzy subset A of U is characterized by a membership function

(A,u) : U [0,1] which associates with each element u of U a number (u) in the interval [0,1]

• Definition– Let A and B be two fuzzy subsets of U. Also, let ¬A be the

complement of A. Then,• (¬A,u) = 1 - (A,u) • (AB,u) = max((A,u), (B,u))• (AB,u) = min((A,u), (B,u))

Fuzzy Information Retrieval

• Fuzzy sets are modeled based on a thesaurus • This thesaurus is built as follows:

– Let vec(c) be a term-term correlation matrix– Let c(i,l) be a normalized correlation factor for (ki,kl):

c(i,l) = n(i,l) ni + nl - n(i,l)

– ni: number of docs which contain ki– nl: number of docs which contain kl– n(i,l): number of docs which contain both ki and kl

• We now have the notion of proximity among index terms.

Fuzzy Information Retrieval

• The correlation factor c(i,l) can be used to define fuzzy set membership for a document dj as follows: (i,j) = 1 - (1 - c(i,l)) ki dj

(i,j) : membership of doc dj in fuzzy subset associated with ki

• The above expression computes an algebraic sum over all terms in the doc dj

• A doc dj belongs to the fuzzy set for ki, if its own terms are associated with ki

Fuzzy Information Retrieval

• (i,j) = 1 - (1 - c(i,l)) ki dj

(i,j) : membership of doc dj in fuzzy subset associated with ki

• If doc dj contains a term kl which is closely related to ki, we have– c(i,l) ~ 1 (i,j) ~ 1– index ki is a good fuzzy index for doc

Fuzzy IR: An Example

• q = ka (kb kc)• vec(qdnf) = (1,1,1) + (1,1,0) + (1,0,0) = vec(cc1) + vec(cc2) + vec(cc3) (q,dj) = (cc1+cc2+cc3,j) = 1 - (1 - (a,j) (b,j) (c,j)) *

(1 - (a,j) (b,j) (1-(c,j))) * (1 - (a,j) (1-(b,j)) (1-(c,j)))

cc1cc3

cc2

Ka Kb

Kc

Fuzzy Information Retrieval

• Fuzzy IR models have been discussed mainly in the literature associated with fuzzy theory

• Experiments with standard test collections are not available

• Difficult to compare at this time

Basic of Neural Network

The human brain

Seat of consciousness and cognition

Perhaps the most complex information processing machine in nature

Historically, considered as a monolithic information processing machine

Beginner’s Brain Map

Forebrain (Cerebral Cortex): Language, maths, sensation, movement, cognition, emotion

Cerebellum: Motor Control

Midbrain: Information Routing; involuntary controls

Hindbrain: Control of breathing, heartbeat, blood circulation

Spinal cord: Reflexes, information highways between body & brain

Brain : a computational machine?

Information processing: brains vs computers brains better at perception / cognition slower at numerical calculations parallel and distributed Processing associative memory

Brain : a computational machine? (contd.)

• Evolutionarily, brain has developed algorithms most suitable for survival

• Algorithms unknown: the search is on• Brain astonishing in the amount of information it

processes

– Typical computers: 109 operations/sec– Housefly brain: 1011 operations/sec

Brain facts & figures

• Basic building block of nervous system: nerve cell (neuron)

• ~ 1012 neurons in brain

• ~ 1015 connections between them

• Connections made at “synapses”

• The speed: events on millisecond scale in neurons, nanosecond scale in silicon chips

Neuron - “classical”

• Dendrites– Receiving stations of neurons– Don't generate action potentials

• Cell body– Site at which information

received is integrated

• Axon– Generate and relay action

potential– Terminal

• Relays information to

next neuron in the pathwayhttp://www.educarer.com/images/brain-nerve-axon.jpg

Computation in Biological Neuron

• Incoming signals from synapses are summed up at the soma

• , the biological “inner product”• On crossing a threshold, the cell “fires” generating an

action potential in the axon hillock region

Synaptic inputs: Artist’s conception

The biological neuron

Pyramidal neuron, from the amygdala (Rupshi et al. 2005)

A CA1 pyramidal neuron (Mel et al. 2004)

A perspective of AI Artificial Intelligence - Knowledge based computing Disciplines which form the core of AI - inner circle Fields which draw from these disciplines - outer circle.

Planning

CV

NLP

ExpertSystems

Robotics

Search, RSN,LRN

Symbolic AI

Connectionist AI is contrasted with Symbolic AISymbolic AI - Physical Symbol System Hypothesis

Every intelligent system can be constructed by storing and processing symbols and nothing more is necessary.

Symbolic AI has a bearing on models of computation such as

Turing Machine Von Neumann Machine Lambda calculus

Turing Machine & Von Neumann Machine

Challenges to Symbolic AI

Motivation for challenging Symbolic AIA large number of computations and

information process tasks that living beings are comfortable with, are not performed well by computers!

The Differences

Brain computation in living beings TM computation in computersPattern Recognition Numerical ProcessingLearning oriented Programming orientedDistributed & parallel processing Centralized & serial processingContent addressable Location addressable

Perceptron

The Perceptron Model

A perceptron is a computing element with input lines having associated weights and the cell having a threshold value. The perceptron model is motivated by the biological neuron.

Output = y

wnWn-1

w1

Xn-1

x1

Threshold = θ

θ

1y

Step function / Threshold functiony = 1 for Σwixi >=θ =0 otherwise

Σwixi

Features of Perceptron

• Input output behavior is discontinuous and the derivative does not exist at Σwixi = θ

• Σwixi - θ is the net input denoted as net

• Referred to as a linear threshold element - linearity because of x appearing with power 1

• y= f(net): Relation between y and net is non-linear

Computation of Boolean functions

AND of 2 inputsX1 x2 y0 0 00 1 01 0 01 1 1The parameter values (weights & thresholds) need to be found.

y

w1 w2

x1 x2

θ

Computing parameter values

w1 * 0 + w2 * 0 <= θ θ >= 0; since y=0

w1 * 0 + w2 * 1 <= θ w2 <= θ; since y=0

w1 * 1 + w2 * 0 <= θ w1 <= θ; since y=0

w1 * 1 + w2 *1 > θ w1 + w2 > θ; since y=1w1 = w2 = = 0.5

satisfy these inequalities and find parameters to be used for computing AND function.

Other Boolean functions

• OR can be computed using values of w1 = w2 = 1 and = 0.5

• XOR function gives rise to the following inequalities:

w1 * 0 + w2 * 0 <= θ θ >= 0

w1 * 0 + w2 * 1 > θ w2 > θ

w1 * 1 + w2 * 0 > θ w1 > θ

w1 * 1 + w2 *1 <= θ w1 + w2 <= θ

No set of parameter values satisfy these inequalities.

Threshold functions

n # Boolean functions (2^2^n) #Threshold Functions (2n^2)

1 4 42 16 143 256 1284 64K 1008

• Functions computable by perceptrons - threshold functions

• #TF becomes negligibly small for larger values of #BF.

• For n=2, all functions except XOR and XNOR are computable.