{"title": "Reasoning about Time and Knowledge in Neural Symbolic Learning Systems", "book": "Advances in Neural Information Processing Systems", "page_first": 921, "page_last": 928, "abstract": "", "full_text": "Reasoning about Time and Knowledge \nNeural-Symbolic Learning Systems \n\n. In \n\nArtur S. d' Avila Garcez\" and Luis C. Lamb A \n\n\"Dept. of Computing, City University London \nLondon, EC1V OHB, UK (aag@soi.city.ac.uk) \n\nADept. of Computing Theory, PPGC-II-UFRGS \n\nPorto Alegre, RS 91501-970, Brazil (lamb@inf.ufrgs.br) \n\nAbstract \n\nWe show that temporal logic and combinations of temporal logics \nand modal logics of knowledge can be effectively represented in ar(cid:173)\ntificial neural networks. We present a Translation Algorithm from \ntemporal rules to neural networks, and show that the networks \ncompute a fixed-point semantics of the rules. We also apply the \ntranslation to the muddy children puzzle, which has been used as a \ntestbed for distributed multi-agent systems. We provide a complete \nsolution to the puzzle with the use of simple neural networks, capa(cid:173)\nble of reasoning about time and of knowledge acquisition through \ninductive learning. \n\n1 \n\nIntroduction \n\nHybrid neural-symbolic \nsystems concern the use of problem-specific symbolic \nknowledge within the neurocomputing paradigm (d'Avila Garcez et al., 2002a). \nTypically, translation algorithms from a symbolic to a connectionist representation \nand vice-versa are employed to provide either (i) a neural implementation of a logic, \n(ii) a logical characterisation of a neural system, or (iii) a hybrid learning system \nthat brings together features from connectionism and symbolic artificial intelligence \n(Holldobler, 1993). \n\nUntil recently, neural-symbolic systems were not able to fully represent, reason and \nlearn expressive languages other than propositional and fragments of first-order \nlogic (Cloete & Zurada, 2000). However, in (d'Avila Garcez et al., 2002b; d'Avila \nGarcez et al., 2002c; d'Avila Garcez et al., 2003), a new approach to knowledge \nrepresentation and reasoning in neural-symbolic systems based on neural networks \nensembles has been introduced. This new approach shows that modal logics can be \neffectively represented in artificial neural networks. \n\nIn this paper, following the approach introduced in (d'Avila Garcez et al., 2002b; \nd'Avila Garcez et al., 2002c; d'Avila Garcez et al., 2003), we move one step further \nand show that temporal logics can be effectively represented in artificial neural \n\no Artur Garcez is partly supported by the Nuffield Foundation. Luis Lamb is partly \n\nsupported by CNPq. The authors would like to thank the referees for their comments. \n\n\fnetworks. This is done by providing a translation algorithm from temporal logic \ntheories to the initial architecture of a neural network. A theorem then shows \nthat the translation is correct by proving that the network computes a fixed-point \nsemantics of its corresponding temporal theory (van Emden & Kowalski, 1976) . The \nresult is a new learning system capable of reasoning about knowledge and time. We \nhave validated the Connectionist Temporal Logic (CTL) proposed here by applying \nit to a distributed time and knowledge representation problem known as the muddy \nchildren puzzle (Fagin et al., 1995). \n\nCTL provides a combined (multi-modal) connectionist system of knowledge and \ntime, which allows the modelling of evolving situations such as changing environ(cid:173)\nments or possible worlds. Although a number of multi-modal systems - e.g., com(cid:173)\nbining knowledge and time (Halpern & Vardi, 1986; Halpern et al., 2003) and com(cid:173)\nbining beliefs, desires and intentions (Rao & Georgeff, 1998) - have been proposed \nfor distributed knowledge representation, little attention has been paid to the inte(cid:173)\ngration of a learning component for knowledge acquisition. This work contributes \nto bridge this gap by allowing the knowledge representation to be integrated in a \nneural learning system. Purely from t he point of view of knowledge representation \nin neural-symbolic systems, this work contributes to the long term aim of repre(cid:173)\nsenting expressive and computationally well-behaved symbolic formalisms in neural \nnetworks. \n\nThe remainder of this paper is organised as follows. We start , in Section 2, by \ndescribing the muddy children puzzle, and use it to exemplify the main features \nof CTL. In Section 3, we formally introduce CTL's Translation Algorithm, which \nmaps knowledge and time theories into artificial neural networks, and prove that \nthe t ranslation is correct. In Section 4, we conclude and discuss directions for future \nwork. \n\n2 Connectionist Reasoning about Time and Knowledge \n\nTemporal logic and its combination with other modalities such as knowledge and \nbelief operators have been the subject of intense investigation (Fagin et al., 1995). In \nthis section, we use the muddy children puzzle, a testbed for distributed knowledge \nrepresentation formalisms, to exemplify how knowledge and t ime can be expressed \nin a connectionist setting. We start by stating the puzzle (Fagin et al., 1995; Huth \n& Ryan, 2000). \n\nThere is a number n of (truthful and intelligent) children playing in a garden. A \ncertain number of children k (k :S n) has mud on their faces . Each child can see if \nthe other are muddy, but not themselves. Now, consider the following situation: A \ncaret aker announces that at least one child is muddy (k 2': 1) and asks does any of \nyou know if you have mud on your own face? To help understanding the puzzle, let \nus consider the cases in which k = 1, k = 2 and k = 3. If k = 1 (only one child is \nmuddy), the muddy child answers yes at the first instance since she cannot see any \nother muddy child. All the other children answer no at the first instance. If k = 2, \nsuppose children 1 and 2 are muddy. At the first instance, all children can only \nanswer no. This allows 1 to reason as follows: if 2 had said yes the first time, she \nwould have been the only muddy child. Since 2 said no , she must be seeing someone \nelse muddy; and since I cannot see anyone else muddy apart from 2, I myself must \nbe muddy! Child 2 can reason analogously, and also answers yes the second time \nround. If k = 3, suppose children 1, 2 and 3 are muddy. Every children can only \nanswer no the first two times round. Again, this allows 1 to reason as follows: if \n2 or 3 had said yes the second time, they would have been the only two muddy \nchildren. Thus, there must be a third person with mud. Since I can only see 2 and \n\n\f3 with mud, this third person must be me! Children 2 and 3 can reason analogously \nto conclude as well that yes, they are muddy. \n\nThe above cases clearly illustrate the need to distinguish between an agent's indi(cid:173)\nvidual knowledge and common knowledge about the world in a particular situation. \nFor example, when k = 2, after everybody says no at the first round, it becomes \ncommon knowledge that at least two children are muddy. Similarly, when k = 3, \nafter everybody says no twice, it becomes common knowledge that at least three \nchildren are muddy, and so on. In other words, when it is common knowledge that \nthere are at least k -1 muddy children; after the announcement that nobody knows \nif they are muddy or not , then it becomes common knowledge that there are at \nleast k muddy children, for if there were k - 1 muddy children all of them would \nknow that they had mud in their faces. I \n\nIn what follows, a modality K j is used to represent the knowledge of an agent j. In \naddition, the term Pi is used to denote that proposition P is true for agent i. For \nexample, KjPi means that agent j knows that P is true for agent i. We use Pi to \nsay that child i is muddy, and qk to say that at least k children are muddy (k :s; n). \nLet us consider the case in which three children are playing in the garden (n = 3). \nRule ri below states that when child 1 knows that at least one child is muddy and \nthat neither child 2 nor child 3 are muddy then child 1 knows that she herself is \nmuddy. Similarly, rule r~ states that if child 1 knows that there are at least two \nmuddy children and she knows that child 2 is not muddy then she must also be able \nto know that she herself is muddy, and so on. The rules for children 2 and 3 are \ninterpreted analogously. \n\nri: K Iql!\\KI\"\"'P2!\\KI\"\"'P3 ---+KIPI \nrj: K Iq2!\\K I\"\"'P3 ---+KIPI \n\nd: K Iq2!\\KI\"\"'P2 ---+KIPI \nrl: K Iq3 ---+KIPI \nTable 1: Snapshot rules for agent ( child) 1 \n\nEach set of snapshot rules r~ (1 :s; I :s; n; mE N+) can be implemented in a single \nhidden layer neural network Ni as follows. For each rule, a hidden neuron is created. \nEach rule antecedent (e.g., KIql in ri) is associated with an input neuron. The rule \nconsequent (KIPI) is associated with an output neuron. Finally, the input neurons \nare connected to the output neuron through the hidden neuron associated with \nthe rule (ri). In addition, weights and biases need to be set up to implement the \nmeaning of the rule. When a neuron is activated (i.e. has activation above a given \nthreshold), we say that its associated concept (e.g., KIql) is true. Conversely, when \na neuron is not activated, we say that its associated concept is false. As a result , \neach input vector of Ni can be associated with an interpretation (an assignment of \ntruth-values) to the set of rules . Weights and biases must be such that the output \nneuron is activated if and only if the interpretation associated with the input vector \nsatisfies the rule antecedent. In the case of rule ri, the output neuron associated \nwith KIPI must be activated (true) if the input neuron associated with KIql, the \ninput neuron associated with K I\"\"'P2, and the input neuron associated with K I\"\"'P3 \nare all activated (true). \n\nThe Connectionist Inductive Learning and Logic Programming (C-ILP) System \n(d'Avila Garcez et al., 2002a; d'Avila Garcez & Zaverucha, 1999) makes use of the \nabove kind of translation. C-ILP is a massively parallel computational model based \non an artificial neural network that integrates inductive learning from examples and \nbackground knowledge with deductive learning through logic programming. Follow-\n\nINotice that this reasoning process can only start once it is common knowledge that \n\nat least one child is muddy, as announced by the caretaker. \n\n\fing (Holldobler & Kalinke, 1994) (see also (Holldobler et al. , 1999)) , a Translation \nAlgorithm maps any logic program P into a single hidden layer neural network N \nsuch t hat N computes the least fixed point of P . This provides a massively parallel \nmodel for computing the stable model semantics of P (Lloyd, 1987) . In addition, \nN can b e t rained wit h examples using, e.g., Backpropagation, and using P as back(cid:173)\nground knowledge (Pazzani & Kibler, 1992) . The knowledge acquired by training \ncan then be extracted (d'Avila Garcez et al. , 2001) , closing the learning cycle (as \nin (Towell & Shavlik, 1994)). \n\nFor each agent (child) , a C-ILP network can be created. Each network can be \nseen as representing a (learnable) possible world containing information about the \nknowledge held by an agent in a distributed system . Figure 1 shows the implemen(cid:173)\ntation of rules ri to d. In addition, it contains output neurons PI 2 and Kql , Kq2 \nand Kq3 , all represented as facts. 3 This is highlighted in grey in Figure 1. Neurons \nthat appear on both the input and output layers of a C-ILP network (e.g., Kqd \nare recurrently connected using weight one, as depicted in Figure 1. This allows the \nnetwork to iterate the computation of truth-values when chains occur in the set of \nrules. For example, if a ---+ b and b ---+ C are rules of the theory, neuron b will appear \non both the input and output layers of the network, and if a is activated then c will \nbe activated through the activation of b. \n\nFigure 1: The implementation of rules {ri, ... , rn. \n\nIf child 1 is muddy, output neuron PI must be activat ed. Since, child 2 and 3 can \nsee child 1, they will know that PI is muddy. This can be represented as PI ---+ K 2PI \nand PI ---+ K 3PI , and analogously for P2 and P3 . This means that the activation of \noutput neurons KI 'P2 and K I'P3 in Figure 1 depends on the activation of neurons \nthat are not in this network (NI ), but in N2 and N 3 . We need, therefore, to model \nhow the networks in the ensemble interact with each other. \n\nFigure 2 illustrat es the interaction between three C-ILP networks in the muddy \nchildren puzzle. The arrows connecting the networks implement the fact that when \na child is muddy, the other children can see her. So if, e.g., neuron PI is activated \nin N I , neuron KPI must be activat ed in N2 and N3 . For the sake of clarity, the \nsnapshot rules r;\" shown in Figure 1 are omitted here, and this is indicat ed in Figure \n\n2Note Pl means 'child 1 is muddy' while KPl means 'child 1 knows she is muddy'. \n3 A fact is normally represented as a rule with no antecedents. C-ILP represents fact s by \nnot connecting the rule's hidden neuron to any input neuron (in the case of fully-connected \nnetworks, weights with initial value zero are used). \n\n\f2 by neurons highlighted in black. In addition, only positive information about the \nproblem is shown in Figure 2. Negative information such as -'PI, K-'PI, K-'P2 and \nK -'P3 would be implemented analogously. \n\nI \nI \nI \nI \n\n--------- - - -\n\nFigure 2: Interaction between agents in t he muddy children puzzle. \n\nFigure 2 illustrates well the idea behind this paper. By combining a number of \nsimple C-ILP networks, we are able to model individual and common knowledge. \nEach network represents a possible world or an agent's current set of beliefs (d' Avila \nGarcez et al. , 2002b). If we allow a number of ensembles like the one of Figure 2 to \nbe combined, we can represent the evolution in time of an agent's set of beliefs. This \nis exactly what is required for a complete solution of the muddy children puzzle, as \ndiscussed below. \n\nAs we have seen, the solution to the muddy children puzzle illustrated in Figures 1 \nand 2 considers only snapshots of knowledge evolution along time rounds without \nthe addition of a time variable (Ruth & Ryan, 2000). A complete solution, however, \nrequires the addition of a temporal variable to allow reasoning about t he knowledge \nacquired after each time round. The snapshot solution of Figures 1 and 2 should \nthen be seen as representing the knowledge held by the agents at an arbitrary time \nt. The knowledge held by the agents at time t + 1 would then be represented \nby anot her set of C-ILP networks, appropriately connected to the original set of \nnetworks. Let us consider again the case where k = 3. There are alternative ways \nof representing that , but one possible representation for child 1 would be as follows: \n\ntl : -,KIPI /\\ -,K 2P2 /\\ -,K 3P3 ---+ O K I Q2 \nt2 : -,KIPI /\\ -,K2P2 /\\ -,K3P3 ---+ O K I Q3 \n\nTable 2: Temporal rules for agent(child) 1 \n\nEach temporal rule is labelled by a time point ti in which the rule holds. In addition, \nif a rule labelled t i makes use of the n ext time temporal operator 0 then whatever \no qualifies refers to the next time ti+l in a linear time flow. As a result , the first \ntemporal rule above states that if, at tl, no child knows whether she is muddy or \nnot then, at t 2 , child 1 will know that at least two children are muddy. Similarly, \nthe second rule states that, at t2, if still no child knows whether she is muddy or \nnot then, at t3, child 1 will know that at least three children are muddy. As before, \nanalogous temporal rules exist for agents (children) 2 and 3. The temporal rules, \ntogether with the snapshot rules, provide a complete solution to the puzzle. This \nis depicted in Figure 3 and discussed below. 4 \n\nIn Figure 3, networks are replicated to represent an agent's knowledge evolution in \ntime. A network represents an agent 's knowledge today (or at tl), a network repre-\n\n41t is worth noting that each network remains a simple, single hidden layer neural \n\nnetwork that can be trained with the use of standard Backpropagation or other off-the(cid:173)\nshelf learning algorithm. \n\n\fTo Agents 2 and 3 (Kpl) at tl \n\n$ \"~;~~;'---:-\\ )if~~;;;3) ~\\o \n\nTo Agents 2 and 3 (Kp1) at t2 \n\n) J, .6~o:s;(t:).~~_~_ );::~AgrnU(Kp3) \n\n\u2022 CL)(). CLX I) \n\n1 at t1~. \n\n/ \n,.'\" \n'\" \\:. '\" ~// \n\n-, ~K, \n\n- ____ ~ \n\n;' \n\n' .! \n\n\" \n\n--\".. -- From Agent 2 (Kp2) \n\n\\~ , 1 at t2 \n\\\n.... \n\",,\"\", __ ~ __ ~ _ l /~/ \n\n'~K~' / \n~.~ I \n\n,. \n\n/// \n\n,\n\n-~~ / ~ att2 \n\nat t1 \nFrom Agent 3 (p3) \n\nFrom Agent 2 (p2) \nat t2 \n\n___ - - From Agent 3 (p3) at t1 \n\n'\",,' _____ ->~---. \n\n-\n\nFrom Agent 2 (p2) at t1 \n\nFigure 3: Knowledge evolution of agent (child) 1 from time tl to time h \n\nsents the same agent's knowledge tomorrow (t 2 ), and the appropriate connections \nbetween networks model the relations between today and tomorrow according to \nO. In the case of tl : ,KIPI 1\\ ,K2P2 1\\ ,K3P3 -+ OKl q2, for example, output \nneuron KIPI of the network that represents agent 1 at t l , output neuron K 2P2 of \nthe network that represents agent 2 at tl, and output neuron K 3P3 of the network \nthat represents agent 3 at tl need to be connected to output neuron K l q2 of the \nnetwork that represents agent 1 at t2 (the next time) such that K l q2 is activated \nif KIPI, K 2P2 and K 3P3 are not activated. In conclusion, in order to represent \ntime, in addition to knowledge, we need to use a two-dimensional C-ILP ensemble. \nIn one dimension we encode the knowledge interaction between agents at a given \ntime point, and in the other dimension we encode the agents' knowledge evolution \nthrough time. \n\n3 Temporal Translation Algorithm \n\nj \n\nIn this section, we present an algorithm to translate temporal rules of the form \nt : OKaLI' ... , OKbLk -+ OKcLk+I' where a, b, c ... are agents and 1 :s; t :s; n,5 \ninto (two-dimensional) C-ILP network ensembles. Let P represent a number q of \nground6 temporal rules . In such rules, we call Li (1 :s; i :s; k + 1) a literal, and \ncall KjLi (1 :s; \n:s; m) an annotated literal. Each Li can be either a positive \nliteral (p) or a negative literal ('p). Similarly, KjL i can be preceded by , . We \nuse Amin to denote the minimum activation for a neuron to be considered active \n(true), Amin E (0,1). We number the (annotated) literals7 of P from 1 to v such \nthat, when a C-ILP network N is created, the input and output layers of N are \nvectors of length v, where the i-th neuron represents the i-th (annotated) literal. \nFor convenience, we use a bipolar semi-linear activation function h(x) = l+e2- IlX -1, \nand inputs in {-I, I}. \n\nLet kz denote the number of (annotated) literals in the body of rule rl; f..L1, the \nnumber of rules in P with the same (annotated) literal as consequent , for each \nrule Tl; MAXrz (kl' f..L1), the greater element between kz and f..L1 for rule Tl; and \nMAX p (kl' ... , kq, f..LI, ... , f..Lq), the greatest element among all kl's and f..Lz'S of P. We \n\n5There may be n + 1 time points since, e.g., h : Kja, K k f3 -> OKj, means that if \n\nagent j knows a and agent k knows f3 at time tl then agent j knows / at time t2. \n\n6Variables such as ti are instantiated into the language's ground terms (tl, t2, t3 ... ). \n7We use ' (annotated) literals' to refer to any literal, annotated or not annotated ones. \n\n\f-----+ \n\n-----+ \n\nalso use k as a shorthand for (k1, ... , kq), and fJ, as a shorthand for (fJ,1, ... , fJ,q). \nFor example, for P = {r1 : b /\\ c /\\ ---,d ----+ a, r2 : e /\\ f \n----+ a, r3 : ----+ b}, k1 = 3, \nk2 = 2, k3 = 0, fJ,1 = 2, fJ,2 = 2, fJ,3 = 1, MAXr 1 (k1,fJ,1) = 3, MAXr2 (k2,fJ,2) = 2, \nM AXr 3 (k3, fJ,3) = 1 and M AXp( k , fJ, ) = 3. \nCTL Translation Algorithm: \n\n-----+ \n\n-----+ \n\n1. For each time point t in P do: For each agent j in P do: Create a C-ILP Neural \nNetwork Nj,t. \n2. Calculate W such that W 2': 2. . \n\nIn(l\u00b1,~in)-ln(l -Amin) \n\n; \n(3 MAXp(k , M ).(Amin-1)+Amin+1 \n\nI \n\n. \n\n. \n\nI \n\nI \n\nI \n\n2 \n\n' \n\n2 \n\nm \n\nk+1 \n\nto et = (1+Amin )(l-Md W \u00b7 \n\n3. For each rule in P of the form t : OK1L 1, ... , OKm- 1L k ----+ OKmL k+1,8 do: \n(a) Add a hidden neuron LO to N m,t+1 and set h(x) as the activation function \nof L O; (b) Connect each neuron OKjLi (1 ::; i ::; k) in Nj,t to LO. If L i is a \npositive (annotated) literal then set the connection weight to W; otherwise, set the \nconnection weight to -W Set the threshold eO of L O to eO = \n(1+ A min)(k l -1)W' \n' \n(c) Connect L O to KmLk+1 in N m,t+1 and set the connection weight to W. Set the \n(1+ A mi;)(l-Md W ; (d) Add a hidden neuron L e \nthreshold e;+l of KmLk+1 to e;+l = \nto Nm ,t and set h(x) as the activation function of L e ; (e) Connect neuron KmLk+1 \nin N m,t+1 to Le and set the connection weight to W; Set the threshold ei of Le to \nzero; (f) Connect L e to OKmLk+1 in Nm ,t and set the connection weight to W. \nSet the threshold et of K L \n4. For each rule in P of the form t : OK1L 1, ... , OKm-1Lk ----+ KmLk+1 ' do: \n(a) Add a hidden neuron L O to Nm, t and set h(x) as the activation function of \ni ::; k) in Nj ,t to L O . If L i is a \nL O; (b) Connect each neuron OKjLi (1 \npositive (annotated) literal then set the connection weight to W; otherwise, set the \nconnection weight to -W Set the threshold eO of LO to eO = \n(1+ A min)(k l -1)W' \n' \n(c) Connect LO to K mL k+1 in Nm ,t and set the connection weight to W . Set the \nthreshold ei+1 of K mL k+1 to e;+l = (1+ Ami;)(l- Md W; \n5. If N ought to be fully-connected, set all other connections to zero. \nIn the above algorithm it is worth noting that, whenever a rule consequent is pre(cid:173)\nceded by 0, a forward connection from t to t + 1 and a feedback connection from \nt + 1 to t need to be added to the ensemble. For example, if t : a ----+ Ob is a \nrule of P then not only must the activation of neuron a at t activate neuron b at \nt + 1, but the activation of neuron b at t + 1 must also activate neuron Ob at t . \nThis is implemented in steps 3(d) to 3(1) of the algorithm. The remainder of the \nalgorithm is concerned with the implementation of snapshot rules (as in Figure 1). \nThe values of Wand e come from C-ILP's Translation Algorithm (d'Avila Garcez \n& Zaverucha, 1999), and are chosen so that the behaviour of the network matches \nthat of the temporal rules, as the following theorem shows. \n\n::; \n\n2 \n\nI \n\nI \n\nTheorem 1 (Correctness of Translation Algorithm) For each set of ground tem(cid:173)\nporal rules P, there exists a neural network ensemble N such that N computes the \nfixed-point operator T p of P. \n\nProof. (sketch) This proof follows directly from the proof of the analogous theorem \nfor single C-ILP networks presented in (d 'Avila Garcez fj Zaverucha, 1999). This \nis so because C-ILP's definition for Wand e values makes hidden neurons L O and \nLe behave like and gates, while output neurons behave like or gates. D \n\n8Note that 0 is not required to precede every rule antecedent. In the network, neurons \n\nare labelled as OKILI or KILl to differentiate the two concepts. \n\n\f4 Conclusions \n\nIn his seminal paper (Valiant, 1984), Valiant argues for the need of rich logic-based \nknowledge representation mechanisms within learning systems. In this paper, we \nhave addressed such a need, yet complying with important principles of connec(cid:173)\ntionism such as massive parallelism. In particular, a very important feature of the \nsystem presented here (CTL) is the temporal dimension that can be combined with \nan epistemic dimension. This paper provides the first account of how to integrate \nsuch dimensions in a neural-symbolic learning system. The CTL framework opens \nup several interesting research avenues in the domain of neural-symbolic integra(cid:173)\ntion, allowing for the representation and learning of expressive formalisms. In this \npaper, we have illustrated this by providing a full solution to the muddy children \npuzzle, where agents reason about their knowledge at different time points. In the \nnear future, we plan to also apply the system to a large, real world case study. \n\nReferences \nC loete, 1., & Zurada, J. M. (Eds.). (2000) . Knowl edge-based neurocomputing. The MIT Press. \nd'Avila Garcez, A. S., Broda, K., & Gabbay, D. M. (2001). Symbolic knowledge extraction from trained \n\nneural networks: A sound approach. Artificial Intelligence , 125, 155- 207. \n\nd'Avila Garcez, A. S., Broda, K., & Gabbay, D. M. (2002a) . Neural-symbolic learning systems: Foun(cid:173)\n\ndations and applications. Perspectives in Neural Computing. Springer-Verlag. \n\nd'Avila Garcez, A. S ., Lamb, L. C., Broda, K. , & Gabbay, D. M . (2003). Distributed knowledge re p(cid:173)\n\nresentation in neural-symbolic learning systems: a case study. Accepted for Proceedings of 16th \nInternational FLAIRS Conference. St . 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