{"title": "Constructing Proofs in Symmetric Networks", "book": "Advances in Neural Information Processing Systems", "page_first": 217, "page_last": 224, "abstract": null, "full_text": "Constructing Proofs in Symmetric Networks \n\nGadi Pinkas \nComputer Science Department \nWashington University \nCampus Box 1045 \nSt. Louis,  MO  63130 \n\nAbstract \n\nThis paper considers the problem of expressing predicate calculus in con(cid:173)\nnectionist networks that are based  on energy minimization.  Given a  first(cid:173)\norder-logic  knowledge  base and a  bound  k,  a  symmetric  network is  con(cid:173)\nstructed  (like a  Boltzman  machine  or a  Hopfield  network)  that searches \nfor  a  proof for  a  given  query.  If a  resolution-based  proof of length  no \nlonger  than k  exists,  then the global  minima of the energy function  that \nis  associated  with  the  network represent such  proofs.  The  network  that \nis  generated  is  of size  cubic  in  the bound  k  and  linear in the  knowledge \nsize.  There are no restrictions on the type of logic  formulas  that can be \nrepresented.  The network  is  inherently fault  tolerant and  can cope with \ninconsistency and nonmonotonicity. \n\n1 \n\nIntroduction \n\nThe ability to reason from acquired knowledge is undoubtedly one of the basic and \nmost  important components  of human  intelligence.  Among  the  major  tools  for \nreasoning  in  the area of AI  are deductive  proof techniques.  However,  traditional \nmethods are  plagued  by intractability,  inability to learn and  adjust,  as  well  as  by \ninability  to cope  with  noise  and  inconsistency.  A connectionist  approach may  be \nthe  missing  link:  fine  grain,  massively  parallel architecture may give  us  real-time \napproximation; networks are potentially trainable and adjustable; and they may be \nmade tolerant to noise as a  result of their collective computation. \nMost  connectionist  reasoning  systems  that  implement  parts  of  first-order  logic \n(see for  examples:  (Holldobler 90],  [Shastri et a1.  90])  use  the spreading activation \nparadigm  and  usually  trade expressiveness  with  time efficiency.  In  contrast,  this \n217 \n\n\f218 \n\nPinkas \n\npaper uses  the energy minimization paradigm (like [Derthick 88],  [Ballard 86]  and \n[Pinkas 91c]),  representing an intractable problem,  but trading time with correct(cid:173)\nness;  i.e.,  as  more time is  given,  the probability of converging to a  correct answer \nincreases. \nSymmetric  connectionist  networks  used  for  constraint  satisfaction  are  the \ntarget  platform  [Hopfield 84b],  [Hinton, Sejnowski 86],  (peterson, Hartman 89], \n[Smolensky 86].  They are characterized by a quadratic energy function that should \nbe minimized.  Some of the models in the family may be seen as performing a search \nfor  a  global minimum of their energy function.  The  task is  therefore to represent \nlogic deduction that is bound by a finite proof length as energy minimization (with(cid:173)\nout a  bound  on  the  proof length,  the  problem  is  undecidable).  When  a  query  is \nclamped,  the network should search for  a  proof that supports the query.  If a  proof \nto the  query exists,  then every global  minimum of the  energy function  associated \nwith the network represents a proof.  If no proof exists,  the global minima represent \nthe lack of a  proof. \nThe paper elaborates the propositional case;  however, due to space limitations,  the \nfirst-order (FOL) case is only sketched.  For more details and full treatment of FOL \nsee [Pinkas 91j]. \n\n2  Representing proofs of propositional logic \n\nI'll start by assuming that the knowledge base is propositional. \nThe proof area: \nA  proof is  a  list  of clauses  ending  with  the query such  that every  clause  used  is \neither an original clause,  a  copy (or weakening)  of a  clause that appears earlier in \nthe  proof,  or  a  result  of a  resolution  step  of the  two clauses  that  appeared just \nearlier.  The proof emerges as an activation pattern on special unit structures called \nthe  proof area,  and  is  represented  in  reverse  to the common  practice (the query \nappears first).  For example:  given a  knowledge base of the following clauses: \n1)  A \n2)  ..,Av B vC \n3)  ..,Bv D \n4)  ..,CV D \nwe would like to prove the query D, by generating the following list of clauses: \n\n1)  D \n2)  A \n3)  ..,Av D \n4)  ..,CV D \n5) -.AVCv D \n6)  -.Bv D \n7)  ..,Av B vC \n\n(obtained by resolution of clauses 2 and 3 by canceling A). \n(original clause no.  1). \n(obtained by resolution of clauses 4 and 5 by canceling C). \n(original clause no.  4). \n(obtained by resolution of clauses 6 and 7 by canceling B). \n(original clause no.  3). \n(original clause no.  2). \n\nEach clause in the proof is  either an original clause,  a copy of a clause from earlier \nin the proof, or a  resolution step. \nThe matrix C  in figure  1,  functions as a  clause list.  This list represents an ordered \nset of clauses  that form  the proof.  The query clauses  are clamped  onto this  area \n\n\fand activate hard constraints that force  the rest of the units of the matrix to form \na valid proof (if it exists). \n\nConstructing Proofs in Symmetric Networks \n\n219 \n\nQuery: \n\nD \n\nA \n\n.,AvD \n\n-CvD \n\n.,AvCvD \n\n-JJvD \n\n.,AvBvC \n\n1 \nA  0 \nB \n\nC \n\nD \n\nn \n\n3 \n\n@ \n\nr - -\n\n1  0 \n\n4 \n\ne \n\nIN  /2  C \n2  0  ~ \n4  0  ~  @ \n\nS  0  0@ \n6  0 \n7  0  0  G> \n\n3  0  0 \n\n~ \n\nG> \n!0~ \n\\G>J \n\nRES KB  O'Y \n\nR \n\nk \n\n123  \n0 \n\n0 \n\n0 \n\nk \n\n0 \n\n0 \n\n0 \n\n0 \n\n0 \n\n0 \n0 \n\nk  -\n\n2 \n\n3 \n\n4 \n\nk \n\n1 \n\n2 \n\n3 \n\n4 \n\nk \n\n1 \n\n2 \n\nk \n\n0 \n\n0 \n\n0 \n\np \n\n1 \n2  0 \n3 \n\n4 \n\nS \n\n6 \n\n7 \n\nt \n\n0 \n\n0 \n\n0 \n\nl\"igure 1:  The proof area for a propositional case \n\nK \n\nD \n\nVariable binding is performed by dynamic allocation of instances using a technique \nsimilar  to [Anand an et a!.  891  and  [Barnden 91].  In this  technique, if two symbols \nneed  to be bound together,  an instance is allocated from a pool of general purpose \ninstances,  and  is  connected  to both symbols.  An instance can be connected  to a \nliteral  in  a  clause,  to  a  predicate  type,  to a  constant,  to a  function  or  to  a  slot \nof another  instance  (for  example,  a  constant  that  is  bound  to the first  slot  of a \npredicate). \nThe  clauses  that  participate  in  the  proof are  represented  using  a  3-dimensional \nmatrix (C.\",;)  and  a  2-dimensional  matrix  (P\";)  as  illustrated  in  figure  1.  The \nrows  of C  represent  clauses  of the  proof,  while  the  rows  of P  represent  atomic \n\n\f220 \n\nPinkas \n\npropositions.  The columns of both matrices represent the pool of instances used for \nbinding propositions to clauses. \nA  clause  is  a  list  of negative and  positive  instances  that  represent  literals.  The \ninstance  thus  behaves  as  a  two-way  pointer  that  binds  composite  structures  like \nclauses  with  their constituents (the atomic  propositions).  A  row  i  in  the  matrix \nC  represents a  clause which is composed  of pairs of instances.  If the unit C+,i,i  is \nset, then the matrix represents a  positive literal in clause i.  If P A,i  is also set, then \nC+,',j represents a  positive literal of clause i that is bound to the atomic proposition \nA.  Similarly C-\"J represents a  negative literal. \nThe first  row of matrix C  in the figure  is the query clause D.  It contains only one \npositive literal that is  bound to atomic  proposition D  via instance 4.  For another \nexample consider the third row of the C  which represents a  clause of two literals:  a \npositive one that is bound to D  via instance 4,  and a  negative one bound  to A  via \ninstance 1 (it is the clause ..,A V D, generated as a  result of a  resolution step). \nParticipation  in  the  proof:  The  vector  IN  represents  whether  clauses  in  C \nparticipate in the proof.  In our example,  all  the clauses are in the proof;  however, \nin the general case some of the rows of C  may be meaningless.  When IN.  is on,  it \nmeans that the clause i  is in the proof and must be proved as well.  Every clause that \nparticipates in the proof is either a  result of a  resolution step (RES. is set),  a copy \nof a  some clause (CPYi  is set),  or it is an original clause from the knowledge base \n(K B. is set).  The second  clause of C  in  figure  1  for  example is an original clause \nof the knowledge  base.  If a  clause j  is  copied,  it must be in  the proof itself and \ntherefore I Nj is set.  Similarly, if clause i is a result of a resolution step, then the two \nresolved clauses must also be in the proof (I Ni+l,i  and I Ni+2,i) and therefore must \nbe  themselves  resolvents,  copies  or originals.  This  chain  of constraints  continues \nuntil all constraints are satisfied and a  valid  proof is generated. \nPosting a  query:  The user posts a  query clamping its clauses onto the first  rows \nof C  and  setting the appropriate IN  units.  This indicates  that the query clauses \nparticipate in the proof and  should  be proved  by either a  resolution  step,  a  copy \nstep or by an original clause.  Figure 1 represents the complete proof for  the query \nD .  We start by allocating an instance (4)  for  D  in  the P  matrix,  and clamping a \npositive literal D  in the first  row of C  (C+,l ,4);  the rest of the first  row's units are \nclamped  zero.  The unit  INl  is  biased  (to have the value  of one),  indicating that \nthe query is in the proof;  this cause a  chain of constraints to be activated that are \nsatisfied  only  by a  valid  proof.  If no  proof exists,  the I Nl  unit  will  become zero; \ni.e.,  the global minima is  obtained by setting I Nl  to zero despite the bias. \nRepresenting  resolutions  steps:  The  vector  RES is  a  structure of units  that \nindicates which are the clauses in C  that are obtained by a resolution step.  If RES, \nis  set,  then  the  ith row  is  obtained  by  resolving  row  i + 1  of C  with  row  i + 2. \nThus,  the  unit  RESl  in  figure  1  indicates  that  the clause  D  of the first  row  of \nC  is  a  resolvent of the second  and  the third  rows  of C  representing ..,A V  D  and \nA  respectfully.  Two literals  cancel each other if they have  opposite signs and  are \nrepresented by the same instance.  In figure  1,  literal A  of the third  row of C  and \nliteral ..,A  of the second  row  cancel  each other,  generating  the clause of the first \nrow. \n\nThe rows of matrix R  represent literals canceled  by resolution steps.  If row i of \n\n\fConstructing Proofs  in Symmetric Networks \n\n221 \n\nC  is the result of a  resolution step, there must be one and only one instance j  such \nthat both clause i + 1 and clause i + 2 include it with opposite signs.  For example \n(figure  1):  clause  D  in the first  row of C  is  the  result  of resolving clause  A  with \nclause..,A V D  which are in the second and third rows of C  respectfully.  Instance 1, \nrepresenting atomic proposition A,  is the one that is canceled;  RI,I  is set therefore, \nindicating that clause 1 is obtained by a  resolution step that cancels the literals of \ninstance  1. \nCopied and original clauses:  The matrix D  indicates which clauses are copied \nto other clauses in  the proof area.  Setting Di,i  means that clause i  is  obtained by \ncopying (or weakening) clause j  into clause i (the example does not use copy steps). \nThe matrix K  indicates which original knowledge-base clauses participate in the \nproof.  The unit Ki,J  indicates that a clause i in the proof area is an original clause, \nand  the syntax of the j-th clause  in  the  knowledge  base must  be imposed  on the \nunits of clause i.  In figure  1 for  example,  clause 2  in the proof (the second  row in \nC),  assumes the identity of clause  number  1 in the knowledge base  and  therefore \nK l ,2  is set. \n\n3  Constraints \n\nWe are now ready to specify the constraints that must be satisfied  by the units so \nthat a  proof is found.  The constraints are specified  as  well formed  logic formulas. \nFor example the formula (A V B) \"C imposes a  constraint over the units (A,B,C) \nsuch that the only possible valid assignments to those units are (011), (101), (111). \nA  general  method  to implement  an  arbitrary  logical  constraint  on  connectionist \nnetworks is shown in [Pinkas 90b].  Most of the constraints specified in this section \nare hard constraints; i.e.,  must be satisfied for a valid proof to emerge.  Towards the \nend of this section, some soft constraints are presented. \n\nIn-proof constraints:  If a  clause  participates  in  the  proof,  it  must  be either  a \nresult of a resolution step, a copy step or an original clause.  In logic,  the constraints \nmay be expressed as:  Vi  : INi- RESi V CP'Yi V  K Bi.  The three units (per clause \ni) consist  a  winner  takes all subnetwork (WTA). This means that only one of the \nthree units is actually set.  The WTA constraints may be expressed as: \nRESi-..,CP'Yi \"  ..,K Bi \nCP'Yi--,RESi \"  ..,K Bi \nK Bi--,RESi \"  ..,C P'Yi \nThe WTA property may be enforced by inhibitory connections between every pair \nof the three units. \nCopy constraints:  If CPYi  is set then clause i  must be a  copy of another clause \nj  in the proof.  This can be expressed as Vi  : C P'Yi- V . (Di,i  \"  I Ni ).  The rows of \nDare WTAs allowing-i to be a copy of only one j. In addition, if clause j  is copied \nor weakened into clause i  then every unit set in clause j  must also be set in clause \ni.  This may be specified as:  Vi,j,l : Di,,- \u00abC+,.,'  +- C+\",') \"  (C_,.,'  +- C_\",,\u00bb. \nResolution  constraints:  If a  clause  i  is  a  result  of resolving  the  two  clauses \ni + 1 and  i + 2,  then there must be one and only one instance (j) that is  canceled \n(represented by Ra,i)' and C.  is obtained by copying both the instances of CHI and \nCH2, without the instance j. These constraints may be expressed  as: \n\n\f222 \n\nPinkas \n\nVi  : RE Si- Vi Ri,i \nat least one instance is canceled \nVi,j,j',j' \u00a2  j: Ri,i--'Ri,i' \nonly one instance is canceled (WT.t \nVi, j  : ~,i-(C+,i+l,i \"  C-,i+2,i) V (C-,i+1J \"C+,i+2,j) cancel literals with opposite signs. \nVi  : RESi-INi+l \"INi+2 \nthe two resolvents are also in proof \nVi  : RE Si-( C+,i,i +-+( C+,i+l,i  V C+,i+2,i) \"  \"'Ri,i \ncopy positive literals \nVi: RESi-(C-,iJ+-+(C-,i+1J V C-,i+2J) \"  -'~,i \ncopy negative literals \n\nClause-instance  constraints:  The  sign  of an  instance  in  a  clause  should  be \nunique;  therefore, any instance pair in the matrix Cis WTA: Vi, j  : C+,i,i--,C-,iJ' \nThe columns of matrix P  are WTAs since an instance is allowed to represent only \none atomic  prop06ition:  VA, i, B  :F  A  :  PA,i-\",PB,i.  The  rows of P  may  be also \nWTAs:  VA,i,j:f; i: PA,i-\"'PA,j (this constraint is not imposed in the FOL case). \nKnowledge base constraints:  If a  clause i  is an original knowledge base clause, \nthen there must be a  clause j  (out of the m  original clauses)  whose syntax is forced \nupon  the  units of the i-th row of matrix C.  This constraint can be expressed  as: \nVi  :  K Bi- Vj Ki,i'  The rows of K  are  WTA networks so that only one original \nclause is forced  on the units of clause i:  Vi, j, j' :F  j  : K',i--,Ki,i\" \nThe only hard constraints that are left are those that force the syntax of a particular \nclause from the knowledge base.  Assume for example that Ki,4  is set, meaning that \nclause i in C  must have the syntax of the fourth clause in the knowledge base of our \nexample (..,CV D).  Instances j  and j' must be allocated to the atomic propositions \nC  and  D  respectfully,  and  must appear  also  in clause i  as  the literals C-,iJ  and \nC+,i,i\"  The following constraints capture the syntax of (..,CV D): \nVi  : Ki,4- V . (C_ ,iJ \"  PC,i) \nVi: K i ,4-V; (D+,i,i \"Pc,i) \nFOL extension: \nIn  first-order  predicate  logic  (FOL)  instead  of atomic  propositions  we  must  deal \nwith predicates (see  [pinkas 91j]  for details).  As in the propositional case,  a  literal \nin a  clause is  represented by a  positive or negative instance;  however,  the instance \nmust be allocated now to a  predicate name and may have slots to be filled  by other \ninstances (representing functions and constants).  To accommodate such complexity \na  new  matrix (NEST) is added, and the role of matrix P  is revised. \nThe matrix P  must accommodate now  function  names,  predicate names and con(cid:173)\nstant names instead of just atomic propositions.  Each row of P  represents a name, \nand the columns represent instances that are allocated  to those  names.  The rows \nof P  that are associated  with predicates  and  functions  may contain several differ(cid:173)\nent instances of the same predicate or function,  thus,  they are not WTA anymore. \nIn order to represent compound terms and predicates,  instances may be bound to \nslots  of other  instances.  The  new  matrix (N ESn,i,p)  is  capable  of representing \nsuch bindings.  If N ESn,i,p is set, then instance i is bound to the p slot of instance \nj.  The columns  of NEST are  WTA,  allowing  only one instance  to  be bound  to \na  certain  slot  of another  instance.  When  a  clause  i  is  forced  to have  the syntax \nof some original clause I,  syntactic constraints are triggered  so that the literals of \nclause  i  become  instantiated  by  the relevant  predicates,  functions,  constants and \nvariables imposed  by clause I. \n\nthere exists a  negative literal that is bound to C; \nthere exists a  positive literal that is bound to D. \n\n\fConstructing Proofs  in Symmetric Networks \n\n223 \n\nUnification  is  implicitly  obtained  if two  predicates  are  representing  by  the  same \ninstance  while  still  satisfying  all  the  constraints  (imposed  by  the  syntax of the \ntwo clauses).  When a  resolution  step is  needed,  the network  tries  to allocate  the \nsame instance  to the two literals that need  to cancel  each  other.  If the syntactic \nconstraints on  the  literals  permit such  sharing  of an  instance,  then  the  attempt \nto share  the  instance  is  successful  and  a  unification  occurs  (occur  check  is  done \nimplicitly since  the matrix NEST allows the only finite  trees to be represented). \nMinimizing  the violation of soft  constraints:  Among  the valid  proofs some \nare preferable to others.  By means of soft constraints and optimization it is possible \nto encourage the network to search for  preferred  proofs.  Theorem-proving thus is \nviewed  as  a  constraint optimization  problem.  A  weight  may be assigned  to each \nof the constraints [Pinkas 91c)  and the network tries to minimize the weighted sum \nof the violated constraints, so that the set of the optimized solutions is  exactly the \nset  of the  preferred  proofs.  For  example,  preference  of proofs  with  most  general \nunification  is  obtained  by  assignment  of small  penalties  (negative  bias)  to  every \nbinding of a  function  to a  position of another instance (in  NEST).  Using similar \ntechniques, the network can be made to prefer shorter, more parsimonious or more \nreliable  proofs,  low-cost plans or even more specific arguments as in nonmonotonic \nreasonmg. \n\n4  Summary \n\nGiven  a  finite  set T  of m  clauses,  where  n  is  the  number  of different  predicates, \nfunctions  and constants,  and  given also  a  bound  k  over the proof length,  we  can \ngenerate  a  network  that  searches for  a  proof with  length  not  longer  then  k,  for \na  clamped  query Q.  If a  global  minimum is  found  then an answer  is  given as  to \nwhether  there exists such a  proof,  and the proof (with  MGU's)  may be  extracted \nfrom  the  state  of the  visible  units.  Among  the  possible  valid  proofs  the  system \nprefers  some  \"better\"  proofs  by  minimizing  the violation of soft constraints.  The \nconcept of \"better\"  proofs  may apply  to applications like  planning (minimize the \ncost), abduction (parsimony) and nonmonotonic  reasoning (specificity). \nIn the propositional case the generated  network is  of O(k2 + km + kn)  units and \nO( k 3 + km + kn) connections.  For predica;te logic there are O( k3 + km + kn) units \nand connections, and we need  to add O( Pm) connections and  hidden units, where \ni  is the complexity-level of the syntactic constraints [Pinkas 91j). \nThe results  improve an earlier approach [Ballard 86]:  There are no restrictions on \nthe  rules  allowed;  every  proof no  longer  than  the bound  is  allowed;  the  network \nis compact and the representation of bindings  (unifications)  is efficient;  nesting of \nfunctions and multiple uses of rules are allowed; only one relaxation phase is needed; \ninconsistency is allowed  in the knowledge base, and the query does not need  to be \nnegated and  pre-wired (it can be clamped during query time). \nThe architecture discussed  has a  natural  fault-tolerance  capability:  When  a  unit \nbecomes  faulty,  it simply cannot  assume  a  role  in the  proof,  and  other  units  are \nallocated instead. \nAcknowledgment:  I  wish  to  thank  Dana  Ballard,  Bill  Ball,  Rina  Dechter, \nPeter  Had dawy,  Dan  Kimura,  Stan  Kwasny,  Ron  Loui  and  Dave  Touretzky for \n\n\f224 \n\nPinkas \n\nhelpful conunents. \n\nReferences \n\n[Anand an et al.  89]  P. Anandan, S. Letovsky, E. Mjolsness,  \"Connectionist variable \nbinding  by optimization,\"  Proceedings  of the  11th  Cognitive  Science \nSociety 1989. \n\n[Ballard 86]  D.  H.  Ballard  \"Parallel Logical  Inference and  Energy Minimization,\" \nProceedings  of the  5th  National  Conference  on  Artificial Intelligence, \nPhiladelphia, pp.  203-208,  1986. \n\n[Bamden 91]  J .A. Barnden, \"Encoding complex symbolic data structures with some \nunusual connectionist  techniques,\"  in  J.A Barnden and J.B.  Pollack, \nAdvances  in  Connectionist and Neural  Computation  Theory  1,  High(cid:173)\nlevel connectionist models,  Ablex Publishing Corporation, 1991. \n\n[Derthick 88]  M.  Derthick \"Mundane reasoning by parallel constraint satisfaction,\" \nPhD thesis,  CMU-CS-88-182 Carnegie Mellon University, Sept. 1988 \n[Hinton, Sejnowski 86]  G.E Hinton and T.J. Sejnowski,  \"Learning and re-learning \nin  Boltzman  Machines,\"  in  J.  L.  McClelland  and  D.  E.  Rumelhart, \nParallel  Distributed  Processing:  Explorations  in  The  Microstructure \nof Cognition I,  pp.  282  - 317,  MIT Press,  1986. \n\n[Holldobler 90]  S.  Holldobler,  \"CHCL,  a  connectionist  inference  system for  Horn \n\nlogic based on connection method and using limited resources,\"  Inter(cid:173)\nnational Computer Science Institute TR-90-042,  1990. \n\n[Hopfield 84b]  J.  J.  Hopfield  \"Neurons  with  graded  response  have collective  com(cid:173)\n\nputational properties like those of two-state neurons,\"  Proceedings  of \nthe  National Academy of Sciences 81,  pp. 3088-3092,  1984. \n\n[Peterson,  Hartman 89]  C.  Peterson, E. Hartman,  \"Explorations of mean field  the(cid:173)\n\nory learning algorithm,\"  Neural Networks  t,  no.  6,  1989. \n\n[Pinkas 90b]  G. Pinkas,  \"Energy minimization and the satisfiability of propositional \n\ncalculus,\"  Neural Computation  9,  no.  2,  1991. \n\n[Pinkas 91c]  G.  Pinkas,  \"Propositional  Non-Monotonic  Reasoning  and  Inconsis(cid:173)\n\ntency in Synunetric Neural Networks,\"  Proceedings  of IlCAI, Sydney, \n1991. \n\n[Pinkas 91j]  G.  Pinkas,  \"First-order logic  proofs using connectionist constraint re(cid:173)\nlaxation,\"  technical report,  Department of Computer Science,  Wash(cid:173)\nington University,  WUCS-91-S4,  1991. \n\n[Shastri et al.  90]  L.  Shastri,  V.  Ajjanagadde,  \"From simple  associations  to  sys(cid:173)\n\ntematic  reasoning:  A  connectionist  representation of rules,  variables \nand  dynamic bindings,\"  technical report,  University of Pennsylvania, \nPhiladelphia, MS-CIS-90-0S,  1990. \n\n[Smolensky 86]  P.  Smolensky,  \"Information processing in dynamic systems:  Foun(cid:173)\ndations  of harmony  theory,\"  in  J.L.McClelland  and  D.E.Rumelhart, \nParallel  Distributed  Processing:  Explorations  in  The  Microstructure \nof Cognition I  , MIT Press,  1986. \n\n\f", "award": [], "sourceid": 473, "authors": [{"given_name": "Gadi", "family_name": "Pinkus", "institution": null}]}