Logic Learning in Hopfield Networks

Transcription

1 Logc Learnng n Hopfeld Networks Saratha Sathasvam School of Mathematcal Scences, Unversty of Scence Malaysa Penang, Malaysa E-mal: saratha@cs.usm.my Wan Ahmad Tauddn Wan Abdullah Department of Physcs, Unverst Malaya 563 Kuala Lumpur, Malaysa E-mal: wat@um.edu.my The research s partly fnanced by an FRGS grant from the Mnstry of Hgher Educaton, Malaysa. Abstract Synaptc weghts for neurons n logc programmng can be calculated ether by usng Hebban learnng or by Wan Abdullah s method. In other words, Hebban learnng for governng events correspondng to some respectve program clauses s equvalent wth learnng usng Wan Abdullah s method for the same respectve program clauses. In ths paper we wll evaluate expermentally the equvalence between these two types of learnng through computer smulatons. Keywords: Logc programmng, Hebban learnng, Wan Abdullah s method, Program clauses. Introducton Recurrent neural networks are essentally dynamcal systems that feed back sgnals to themselves. Popularzed by ohn Hopfeld, these models possess a rch class of dynamcs characterzed by the exstence of several stable states each wth ts own basn of attracton. The (Lttle-)Hopfeld neural network [Lttle (974), Hopfeld (98)] mnmzes a Lyapunov functon, also known as the energy functon due to obvous smlartes wth a physcal spn network. Thus, t s useful as a content addressable memory or an analog computer for solvng combnatoral-type optmzaton problems because t always evolves n the drecton that leads to lower network energy. Ths mples that f a combnatoral optmzaton problem can be formulated as mnmzng the network energy, then the network can be used to fnd optmal (or suboptmal) solutons by lettng the network evolve freely. Wan Abdullah (99,99) and Pnkas (99) ndependantly defned b-drectonal mappngs between propostonal logc formulas and energy functons of symmetrc neural networks. Both methods are applcable n fndng whether the solutons obtaned are models for a correspondng logc program. Subsequently Wan Abdullah (99, 993) has shown on see how Hebban learnng n an envronment wth some underlyng logcal rules governng events s equvalent to hardwrng the network wth these rules. In ths paper, we wll expermentally carry out computer smulatons to support ths. Ths paper s organzed as follows. In secton, we gve an outlne of dong logc programmng on a Hopfeld network and n secton 3, Hebban learnng of logcal clauses s descrbed. In secton 4, we descrbe the proposed approach for comparng connecton strengths obtaned by Wan Abdullah s method and Hebban learnng. Secton 5 contans dscussons regardng the results obtaned from computer smulatons. Fnally concludng remarks regardng ths work occupy the last secton.. Logc Programmng on a Hopfeld network In order to keep ths paper self-contaned we brefly revew the Hopfeld model (extensve treatments can be found elsewhere [Geszt (99), Haykn (994)]), and how logc programmng can be carred out on such archtecture. The Hopfeld model s a standard model for assocatve memory. The Hopfeld dynamcs s asynchronous, wth each neuron updatng ts state determnstcally. The system conssts of N formal neurons, each of whch can be descrbed by Isng varables S ( t), ( =,,... N). Neurons then are bpolar, S { -,}, obeyng the dynamcs S h ), where the sgn( 57

2 Vol., No. 3 Modern Appled Scence feld,, and runnng over all neurons N, s the synaptc or connecto strength from neuron to h = S + neuron, and s the threshold of neuron. Restrctng the connectons to be symmetrc and zero-dagonal, or energy functon, =, =, allows one to wrte a Lyapunov E = SS S whch decreases monotoncally wth the dynamcs. The two-connecton model can be generalzed to nclude hgher order connectons. Ths modfes the feld nto h =... + k S Sk + S + k where.. denotes stll hgher orders, and an energy functon can be wrtten as follows: E =... k S S Sk SS S 3 k provded that k = for,, k dstnct, wth [ ] denotng permutatons n cyclc order, and = for any,, k [ k] k equal, and that smlar symmetry requrements are satsfed for hgher order connectons. The updatng rule mantans S ( t ) = sgn[ h ( t)] + (4) In logc programmng, a set of Horn clauses whch are logc clauses of the form A B, B,..., Bn where the arrow may be read f and the commas and, s gven and the am s to fnd the set(s) of nterpretaton (.e., truth values for the atoms n the clauses whch satsfy the clauses (whch yelds all the clauses true). In other words, we want to fnd models correspondng to the gven logc program. In prncple logc programmng can be seen as a problem n combnatoral optmzaton, whch may therefore be carred out on a Hopfeld neural network. Ths s done by usng the neurons to store the truth values of the atoms and wrtng a cost functon whch s mnmzed when all the clauses are satsfed. As an example, consder the followng logc program, A B, C.. D B. C. whose three clauses translate respectvely as A B C, D B and C. The underlyng task of the program s to look for nterpretatons of the atoms, n ths case A, B, C and D whch make up the model for the gven logc program. Ths can be seen as a combnatoral optmzaton problem where the nconsstency, E = ( S ) ( + S ) ( + S P A + ( S ) ( ) ( ) D + S B + S C (5) Where S A, etc. represent the truth values (true as ) of A, etc., s chosen as the cost functon to be mnmzed, as was done by Wan Abdullah. We can observe that the mnmum value for E P s, and has otherwse value proportonal to the number of unsatsfed clauses. The cost functon (5), when programmed onto a thrd order neural network yelds synaptc strengths as gven n Table. We address ths method of dong logc programmng n neural networks as Wan Abdullah s method. 3. Hebban Learnng of Logcal Clauses The Hebban learnng rule for a two-neuron synaptc connecton can be wrtten as B C ) λ S S = (6) where λ s a learnng rate. For connectons of other orders n, between n neurons {S, S,..., S m }, we can generalze ths to ( n)... m λnss... Sm = (7) 58

3 Ths gves the changes n synaptc strengths dependng on the actvtes of the neurons. In an envronment where selectve events occur, Hebban learnng wll reflect the occurrences of the events. So, f the frequency of the events s dctated by some underlyng logcal rule, logc should be entrenched n the synaptc weghts. Wan Abdullah (99, 993) has shown that Hebban learnng as above corresponds to hardwrng the neural network wth synaptc strengths obtaned usng Wan Abdullah s method, provded that the followng s true: λ n = (8) ( n )! We do not provde a detaled analyss regardng Hebban learnng of logcal clauses n ths paper, but nstead refer the nterested reader to Wan Abdullah s papers. 4. Comparng Connecton Strengths Obtaned By Hebban Learnng Wth Those By Wan Abdullah s Method In the prevous secton, we have elaborated how synaptc weghts for neurons can be equvalently calculated ether by usng Hebban learnng or by Wan Abdullah s method. Theoretcally, nformaton (synaptc strengths) produced by both methods are smlar. However, due to nterference effects and redundances, synaptc strengths could be dfferent [Sathasvam (6)], but the set of solutons for both cases should reman the same. Due to ths, we cannot use drect comparson of obtaned synaptc strengths. Instead, we carry out computer smulaton of artfcally generated logc programs and compare fnal states of the resultng neural networks. To obtan the logc-programmed Hopfeld network based on Wan Abdullah s method, the followng algorthm s carred out: ) Gven a logc program, translate all the clauses n the logc program nto basc Boolean algebrac form. ) Identfy a neuron to each ground neuron. ) Intalze all connectons strengths to zero. v) Derve a cost functon that s assocated wth the negaton of the conucton of all the clauses, such that ( + S ) x represents the logcal value of a neuron X, where S x s the neuron correspondng to logcal atom X. The value of S x s defned n such a way that t carres the values of f X s true and - f X s false. Negaton (X does not occur) s represented by ( S ) x ; a conuncton logcal connectve s represented by multplcaton whereas a dsuncton connectve s represented by addton. v) Obtan the values of connecton strengths by comparng the cost functon wth the energy. v)let the neural network programmed wth these connecton strengths evolve untl mnmum energy s reached. Check whether the soluton obtaned s a global soluton (the nterpretaton obtaned s a model for the gven logc program). We run the relaxaton for trals and combnatons of neurons so as to reduce statstcal error. The selected tolerance value s.. All these values are obtaned by try and error technque, where we tred several values as tolerance values, and selected the value whch gves better performance than other values. To compare the nformaton obtan n the synaptc strength, we make comparson between the stable states (states n whch no neuron changes ts value anymore) obtaned by Wan Abdullah s method wth stable states obtaned by Hebban learnng. The way we calculated the percentage of solutons reachng the global solutons s by comparng the energy for the stable states obtaned by usng Hebban learnng and Wan Abdullah s method. If the correspondng energy for both learnng s same, then we conclude that the stable states for both learnng are the same. Ths ndcates, the model (set of nterpretatons) obtaned for both learnng are smlar. In all ths, we assume that the global solutons for both networks are the same due to both methods consderng the same knowledge base (clauses). 5. Results and Dscusson Fgures - 6 llustrate the graphs for global mnma rato (rato= (Number of global solutons)/ (Number of solutons=number of runs)) and Hammng dstances from computer smulaton that we have carred out. From the graphs obtaned, we observed that the rato of global solutons s consstently for all the cases, although we ncreased the network complexty by ncreasng the number of neurons (NN) and number of lterals per clause (NC, NC, NC3). Due to we are gettng smlar results for all the trals, to avod graphs overlappng, we only presented the result obtaned for the number of neurons (NN) = 4. Besdes that, error bar for some of the cases could not be plotted because the sze of the pont s bgger than the error bar. Ths ndcates that the statstcal error for the correspondng pont s so small. So, we couldn t plot the error bar. Most of the neurons whch are not nvolved n the clauses generated wll be n the global states. The random generated program clause relaxed to the fnal states, whch seem also to be stable states, n less than fve runs. Furthermore, the 59

4 Vol., No. 3 Modern Appled Scence network never gets stuck n any suboptmal solutons. Ths ndcates good solutons (global states) can be found n lnear tme or less wth less complexty. Snce all the solutons we obtaned are global soluton, so the dstance between the stable states and the attractors are zero. Supportng ths, we obtaned zero values for Hammng dstance. Ths ndcates the stable states for both learnng are the same. Therefore they are no dfferent n the energy value. So, models for both learnng are proved to be smlar. Although the way of calculatng synaptc weghts are dfferent, snce the calculatons revolve around the same knowledge base (clauses), the set of nterpretatons wll be smlar. Ths mples that, Hebban learnng could extract the underlyng logcal rules n a gven set of events and provde good solutons as well as Wan Abdullah s method. The computer smulaton results support ths hypothess. 6. Concluson In ths paper, we had evaluated expermentally the logcal equvalent between these two types of learnng (Wan Abdullah s method and Hebban learnng) for the same respectve clauses (same underlyng logcal rules) usng computer smulaton. The results support Wan Abdullah s earler proposed theory. References Geszt, T. (99). Physcal Models of Neural Networks. Sngapore: World Scentfc Publcaton. Haykn, S. (994). Neural Network: A Comprehensve Foundaton. New York: Macmllan. Hopfeld,.. (98). Neural Networks and Physcal Systems wth Emergent Collectve Computatonal abltes. Proc. Natl. Acad. Sc. USA, 79, Lttle, W. A. (974). The exstence of persstent states n the bran. Math. Bosc., 9, -. Pnkas, G. (99). Energy mnmzaton and the satsfablty of propostonal calculus. Neural Computaton, 3, 8-9. Sathasvam, S. (6). Logc Mnng n Neural Networks. PhD Thess. Unversty of Malaya, Malaysa. Wan Abdullah, W. A. T. (99). Neural Network logc. In O. Benhar et al. (Eds.), Neural Networks: From Bology to Hgh Energy Physcs. Psa: ETS Edtrce. pp Wan Abdullah, W. A. T. (99). Logc programmng on a neural network. Int.. Intellgent Sys., 7, Wan Abdullah, W. A. T. (993). The logc of neural networks. Phys. Lett.,76A, -6. 6

5 Table. Synaptc strengths for A B, C. & D B & C usng Wan Abdullah s method Synaptc Clause Strengths A B, C. D B. C. Total /6 /6 [ABC] [ABD] [ACD] [BCD] [AB] [AC] /8 /8 /8 /8 [AD] [BC] -/8 -/8 /4 ¼ [BD] [CD] [A] [B] [C] /8 /8 -/8 -¼ -3/8 -/8 / 3/8 /4 ¼ [D] Global Mnma For NC.5 Rato NC/NN Fgure. Global Mnma Rato for NC 6

6 Vol., No. 3 Modern Appled Scence Global Mnma For NC.5 Rato NC/NN Fgure. Global Mnma Rato for NC Global Mnma For NC3.5 Rato NC3/NN Fgure 3. Global Mnma Rato for NC3 Hammng Dstance For NC Rato NC/NN Fgure 4. Hammng Dstance for NC 6

7 Hammng Dstance For NC Rato NC/NN Fgure 5. Hammng Dstance for NC Hammng Dstance For NC3 Rato NC3/NN Fgure 6. Hammng Dstance for NC3 63