CS621 : Artificial Intelligence Pushpak Bhattacharyya CSE Dept., IIT Bombay Lecture 19: Fuzzy Logic and Neural Net Based IR

The IR scenario Docs Information Need Index Terms doc query Ranking match IR system Maker’s view

Definition of IR Model An IR model is a quadrupul [D, Q, F, R(q i, d j )] 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 d j with q i

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 = k a  (k b   k c ) Terms are either present or absent. Thus, w ij  {0,1} Consider –q = k a  (k b   k c ) –vec(q dnf ) = (1,1,1)  (1,1,0)  (1,0,0) –vec(q cc ) = (1,1,0) is a conjunctive component

The Boolean Model q = k a  (k b   k c ) sim(q,d j ) = 1 if  vec(q cc ) | (vec(q cc )  vec(q dnf ))  (  k i, g i (vec(d j )) = g i (vec(q cc ))) 0 otherwise (1,1,1) (1,0,0) (1,1,0) KaKa KbKb KcKc

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)  (A  B,u) = max(  (A,u),  (B,u))  (A  B,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))) cc1 cc3 cc2 KaKb 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: 10 9 operations/sec –Housefly brain: operations/sec

Brain facts & figures Basic building block of nervous system: nerve cell (neuron) ~ neurons in brain ~ 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 pathway

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 Expert Systems Robotics Search, RSN, LRN

Symbolic AI Connectionist AI is contrasted with Symbolic AI Symbolic 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 AI A 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 computers Pattern Recognition Numerical Processing Learning oriented Programming oriented Distributed & parallel processing Centralized & serial processing Content 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 W n-1 w1w1 X n-1 x1x1 Threshold = θ

θ 1 y Step function / Threshold function y = 1 for Σw i x i >=θ =0 otherwise ΣwixiΣwixi

Features of Perceptron Input output behavior is discontinuous and the derivative does not exist at Σw i x i = θ Σw i x i - 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 inputs X1 x2 y The parameter values (weights & thresholds) need to be found. y w1w1 w2w2 x1x1 x2x2 θ

Computing parameter values w1 * 0 + w2 * 0 = 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=1 w1 = 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 = 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 (2 n^2 ) K 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.