{"title": "Encoding Labeled Graphs by Labeling RAAM", "book": "Advances in Neural Information Processing Systems", "page_first": 1125, "page_last": 1132, "abstract": null, "full_text": "Encoding Labeled  Graphs  by  Labeling \n\nRAAM \n\nAlessandro  Sperduti* \n\nDepartment of Computer Science \n\nPisa University \n\nCorso Italia 40,  56125 Pisa, Italy \n\nAbstract \n\nIn  this  paper  we  propose  an  extension  to  the  RAAM  by  Pollack. \nThis  extension,  the  Labeling  RAAM  (LRAAM),  can  encode  la(cid:173)\nbeled  graphs  with  cycles  by  representing pointers explicitly.  Data \nencoded  in  an  LRAAM  can  be  accessed  by  pointer  as  well  as  by \ncontent.  Direct  access  by  content  can  be  achieved  by  transform(cid:173)\ning  the  encoder  network  of  the  LRAAM  into  an  analog  Hopfield \nnetwork  with  hidden  units.  Different  access  procedures  can  be \ndefined  depending  on  the  access  key.  Sufficient  conditions  on  the \nasymptotical stability of the associated Hopfield network are briefly \nintroduced. \n\n1 \n\nINTRODUCTION \n\nIn  the  last  few  years,  several  researchers  have  tried  to  demonstrate  how  symbolic \nstructures such  as  lists,  trees,  and  stacks can be represented and manipulated in  a \nconnectionist  system,  while  still preserving all the computational characteristics of \nconnectionism (and extending them to the symbolic representations)  (Hinton, 1990; \nPlate,  1991; Pollack,  1990;  Smolensky,  1990;  Touretzky,  1990).  The goal is  to high(cid:173)\nlight  the potential of the connectionist approach in handling domains of structured \ntasks.  The common background of their ideas is  an attempt to achieve distal access \nand  consequently compositionality.  The RAAM  model,  proposed  by  Pollack  (Pol(cid:173)\nlack,  1990), is  one example of how  a  neural network can discover compact recursive \n\n\"Work partially done while  at the International Computer  Science Institute, Berkeley. \n\n1125 \n\n\f1126 \n\nSperduti \n\nOutput Layer \n\nHidden Layer \n\nInput Layer \n\nLabel \n\nFigure 1:  The network for  a  general LRAAM.  The first  layer of the network imple(cid:173)\nments an encoder;  the second layer,  the corresponding decoder. \n\ndistributed  representations of trees with  a  fixed  branching factor. \n\nThis  paper  presents  an  extension  of the RAAM,  the  Labeling  RAAM  (LRAAM). \nAn  LRAAM  allows one  to store a  label for  each  component  of the structure to be \nrepresented,  so  as to generate reduced representations of labeled graphs.  Moreover, \ndata encoded in an LRAAM can be accessed not only by pointer but also by content. \nIn  Section  2  we  present  the  network and  we  discuss  some  technical  aspects of the \nmodel.  The  possibility  to  access  data by  content  is  presented in  Section  3.  Some \nstability  results  are  introduced in  Section  4,  and  the paper  is  closed  by  discussion \nand conclusions  in  Section 5. \n\n2  THE NETWORK \n\nThe  general  structure  of  the  network  for  an  LRAAM  is  shown  in  Figure  1.  The \nnetwork is trained by backpropagation to learn the identity function.  The idea is to \nobtain a  compressed representation (hidden layer activation) of a  node of a  labeled \ngraph  by  allocating  a  part  of  the  input  (output)  of the  network  to  represent  the \nlabel  (Nl  units)  and  the  remaining  part  to  represent  one  or  more  pointers.  This \nrepresentation is then used as pointer to the node.  To allow the recursive use of these \ncompressed  representations,  the  part  of  the  input  (output)  layer  which  represents \na  pointer  must  be of the same dimension  as  the hidden  layer  (N H  units) .  Thus,  a \ngeneral LRAAM  is  implemented  by a  NJ  - N H  - NJ  feed-forward  network,  where \nNJ  =  Nl + nN H,  and n  is  the number of pointer fields. \nLabeled  graphs  can  be easily  encoded  using  an  LRAAM.  Each  node  of the  graph \nonly  needs  to  be  represented  as  a  record,  with  one  field  for  the  label  and  one \nfield  for  each  pointer  to  a  connected  node.  The  pointers  only  need  to  be  logical \npointers,  since  their  actual values  will  be  the  patterns  of hidden  activation  of the \nnetwork.  At  the beginning of learning,  their  values  are  set  at  random.  A  graph is \nrepresented by a list of these records, and this list constitutes the initial training set \nfor the LRAAM. During training the representations of the pointers are consistently \nupdated  according  to  the  hidden  activations.  Consequently,  the  training  set  is \ndynamic.  For example,  the network for  the graph shown in  Figure 2 can be trained \nas follows: \n\n\fEncoding Labeled Graphs by Labeling RAAM \n\n1127 \n\ninput \n\nhidden \n\noutput \n\n(Ll  dn2  dn4  dn5 )  - d~1  - (L\"  d\"  d\"  d\"  ) \n(L2  dn3  dn4  nil)  - d~2  - (L\"  d\"  d\"  nil\") \n(L3  dn6  nil nil)  - d~3  - (L\"  d\"  nil\" nil\") \n(L4  dn6  dn3  nil)  - d~4  - (L\"  d\"  d\"  nil\") \n(L5  dn4  dn6  nil)  - d~5  - (L\"  d\"  d\"  nil\") \n(L6  nil nil nil)  - d~6  - (L~ nil\" nil\" nil\") \n\nn3 \nn6 \nn6  n3 \nn4 \nn6 \n\nn5 \n\n1 \n\n2 \n3 \n4 \n5 \n\nn2 \n\nn4 \n\nn4 \n\nwhere  Li  and  dni  are  respectively  the label  and  the  pointer  (reduced  descriptor  to \nthe i-th node.  For the sake of simplicity,  the void  pointer is  represented by a  single \nsymbol,  nil,  but  each  instance  of it  must  be  considered  as  being  different.  This \nstatement will  be  made clear in  the next section. \n\nOnce the training is complete, the patterns of activation representing pointers can be \nused to retrieve information.  Thus, for example, if the activity of the hidden units of \nthe network is clamped to dn1 , the output of the network becomes (Ll ,dn2 ,dn4 ,dn5 ), \nenabling further  retrieval of information by decoding dn2 , or dn4 , or dn5 , and so on. \nNote  that  more labeled graphs can  be encoded in  the same LRAAM. \n\n2.1  THE VOID  POINTER PROBLEM \n\nIn the RAAM model there is a termination problem in the decoding of a compressed \nrepresentation:  due  to  approximation errors introduced  during decoding,  it  is  not \nclear when a decoded pattern is a terminal or a  nonterminal.  One solution is to test \nfor  \"binary-ness\", which  consists in checking whether all  the values of a  pattern are \nabove  1 - T  or below  T,  T  >  0,  T  \u00ab  1.  However,  a  nonterminal may  also  pass  the \ntest for  \"binary-ness\". \n\nOne  advantage of LRAAM  over  RAAM  is  the  possibility  to  solve  the  problem  by \nallocating one bit of the label for  each pointer to represent if the pointer is  void  or \nnot.  This  works  better than  fixing  a  particular  pattern  for  the  void  pointer,  such \nas  a  pattern  with  all  the  bits  to  1  or  0  or  -1  (if  symmetrical  sigmoids  are  used). \nSimulations  performed  with  symmetrical  sigmoids  showed  that  the  configurations \nwith  all  bits  equal  to  1  or  -1  were  also  used  by  non  void  pointers,  whereas  the \nconfiguration with  all  bits set  to zero considerably reduced the rate of convergence. \nU sing  a  part  of  the  label  to  solve  the  problem  is  particularly  efficient,  since  the \npointer  fields  are  free  to  take  on  any  configuration  when  they  are  void,  and  this \nincreases  the  freedom  of the  system.  To  facilitate  learning,  the  output  activation \nof the void  pointers in  one epoch  is  used  as  an input  activation  in  the next  epoch. \nExperimentation showed fast  convergence to different fixed  points for  different  void \n\nFigure 2:  An  example of a  labeled  graph. \n\n\f1128 \n\nSperduti \n\npointers.  For  this  reason,  we  claimed  that  each  occurrence  of the  void  pointer  is \ndifferent,  and  that the nil symbol can  be considered as  a  \"don't  care\"  symbol. \n\n2.2  REPRESENTATION  OF THE TRAINING  SET \n\nAn  important  question  about  the  way  a  graph  is  represented  in  the  training  set \nis  which  aspects  of  the  representation  itself  can  make  the  encoding  task  harder \nor  easier.  In  (Sperduti,  1993a)  we  made  a  theoretical  analysis  on  the  constraints \nimposed  by  the  representation  on  the  set  of  weights  of  the  LRAAM,  under  the \nhypotheses  of  perfect  learning  (zero  total  error  after  learning)  and  linear  output \nunits.  Our findings  were: \n\ni)  pointers to nodes belonging to the same cycle of length k and represented in \nthe same pointer field  p,  must be eigenvectors of the matrix (W(p))k, where \nW(p)  is  the  connection  matrix  between  the  hidden  layer  and  the  output \nunits representing the pointer field  p; \n\nii)  confluent pointers,  i.e.,  pointers to the same node  represented in  the same \npointer  field  p  (of different  nodes),  contribute to  reducing  the  rank of  the \nmatrix  W(p),  the actual rank is  however  dependent  on  the constraints im(cid:173)\nposed  by  the label field  and the other pointer fields. \n\nWe  have observed  that  different  representations  of the  same structure can  lead  to \nvery  different  learning  performances.  However,  representations  with  roughly  the \nsame number of non  void  pointers for  each pointer field,  with cycles  represented in \ndifferent  pointer fields  and with  confluent  pointers seem  to be more effective. \n\n3  ACCESS  BY CONTENT \n\nRetrieval of coded information is performed in RAAM through the pointers.  All the \nterminals and nonterminals can be retrieved recursively by the pointers to the whole \ntree  encoded  in  a  RAAM.  If direct  access  to  a  component  of  the  tree  is  required, \nthe pointer to the component  must  be stored and used on demand. \n\nData encoded  in  an  LRAAM  can  also  be  accessed  directly  by  content.  In fact,  an \nLRAAM  network  can  be  transformed  into  an  analog  Hopfield  network  with  one \nhidden  layer and  asymmetric connection  matrix by feeding  back its output into its \ninput units.  1  Because each pattern is  structured in different  fields,  different  access \n\n1 Experimental results  have shown  that there is  a  high correlation  between elements of \nW(h)  (the set of weights from the input to the hidden layer) and the corresponding elements \nin W(o)T  (the set of weights  from the hidden to the output layer).  This is  particularly true \nfor  weights  corresponding  to  units  of the  label  field.  Such  result  is  not  a  total  surprise, \nsince in the case of a  static training set, the error function  of a  linear encoder network has \nbeen  proven  to  have  a  unique  global  minimum  corresponding  to the  projection  onto  the \nsubspace generated by the first  principal vectors of a  covariance matrix associated  with the \ntraining set (Baldi &  Hornik,  1989).  This implies that the weights  matrices are transposes \nof each other unless there is  an invertible transformation between them (see also  (Bourlard \n&  Kamp,  1988)) . \n\n\fEncoding Labeled Graphs by Labeling RAAM \n\n1129 \n\nn2=.-r..~= \n\nn5 \n\n=100=00=-==.=1.1 \n\nn9 \n\nnlO \n\n~IQl R\",.~.=. \n/n15  \\ \n\n101 \u2022\u2022 101.1. 1  O~.=lctJ~.=I.1 \n\nnl4 \n\nFigure 3:  The labeled graph encoded in  a  16-3-16 LRAAM  (5450 epochs),  and the \nlabeled tree encoded  in  a  18-6-18 LRAAM  (1719 epochs). \n\nprocedures can  be defined  on  the Hopfield  network according to the type of access \nkey.  An  access  procedure is  defined  by: \n\n1.  choosing one or more fields  in the input layer according to the access key(s); \n2.  clamping the output of such units to the the access  key(s); \n3.  setting randomly  the output of the remaining units in  the network; \n4.  letting the remaining units  of the network to  relax into a  stable state. \n\nA  validation  test  of the reached stable state can  be performed by: \n\n1.  unfreezing the clamped units in  the input layer; \n2.  if the stable state is no longer stable the result of the procedure is considered \n\nwrong and  another  run is  performed; \n\n3.  otherwise the stable state is  considered  a  success. \n\nThis  validation  test,  however,  sometimes  can  fail  to  detect  an  erroneous  retrieval \n(error)  because of the existence of spurious stable states that share the same known \ninformation with  the desired one. \n\nThe  results obtained  by  the  access  procedures on  an LRAAM  codifying  the graph \nand  on  an  LRAAM  codifying  the  tree  shown  in  Figure  3  are  reported  in  Table \n1.  For  each  procedure  100  trials  were  performed. \nThe  \"mean\"  column  in  the \ntable  reports  the  mean  number of iterations employed  by  the Bopfield  network  to \nconverge.  The access  procedure by outgoing pointers was applied only for  the tree. \nIt can  be  seen  from  Table  1  that  the  performances  of the  access  procedures  were \nhigh  for  the  graph  (no  errors  and  no  wrong  retrievals),  but  not  so  good  for  the \ntree, in particular for  the access by label procedure, due to spurious memories.  It is \ninteresting to note that the access by label procedure is  very efficient  for  the leaves \nof the tree.  This feature can be used to build a  system with  two identical networks, \none accessed  by  pointer and  the other  by  content.  The search for  a  label  proceeds \nsimultaneously into the two networks.  The network accessed by pointer will be very \nfast  to  respond  when  the label is  located on  a  node at lower levels  of the tree,  and \nthe  network  accessed  by  content  will  be  able  to  respond  correctly  and  very  fast  \"2 \nwhen  the label is  located on a  node at  higher levels of the tree. \n\n2 Assuming an  analog implementation of the Hopfield  network. \n\n\f1130 \n\nSperduti \n\nkey(s) \n\n(d1 , d 2) \n(d3,d4) \n(d5, d6) \n(d7 , ds) \n(d9, d lO ) \n\nd~~) \n\n(d12 ,d13 ) \nld14, d 15 ) \n(*)  one  pointer \n\n49% \n10% \n40% \n78% \n9% \n14% \n14% \n28% \n\nGRAPH:  Access  by  Label \n\nsuccess  wrong \n100% \n100% \n100% \n100% \n100% \n100% \n100% \n\nio \ni1 \ni2 \ni3 \ni4 \n15 \ni6 \nTREE:  Access  by Children Pointers \n\nerror  mean \n7.35 \n0% \n36.05 \n0% \n6.04 \n0% \n0% \n3.99 \n23.12 \n0% \n18.12 \n0% \n29.26 \n0% \n\n0% \n0% \n0% \n0% \n0% \n0% \n0% \n\n51% \n90% \n60% \n22% \n91% \n86% \n86% \n72% \n\n0% \n0% \n0% \n0% \n0% \n0% \n0% \n0% \n\n6.29 \n8.55 \n12.48 \n6.57 \n6.22 \n14.01 \n7.87 \n6.07 \n\nTREE:  Access  by Label \nsuccess  wrong \n100% \n6% \n53% \n0% \n0% \n0% \n51% \n58% \n43% \n0% \n0% \n0% \n0% \n71% \n0% \n0% \n\nerror  mean \n16.48 \n0% \n14.57 \n0% \n16.92 \n0% \n18.07 \n0% \n32.64 \n3% \n16.03 \n0% \n27.50 \n0% \n27.10 \n0% \n62.45 \n0% \n80% \n14.75 \n19.11 \n0% \n10.83 \n0% \n19.12 \n0% \n23.87 \n0% \n12.09 \n0% \n13.11 \n0% \n\n0% \n94% \n47% \n100% \n97% \n100% \n49% \n42% \n57% \n20% \n100% \n100% \n100% \n29% \n100% \n100% \n\nkey \nio \nit \ni2 \ni3 \ni4 \n15 \n16 \ni7 \nis \ni9 \n11O \nIII \nlt2 \nit3 \n114 \n115 \n\nTable 1:  Results obtained by the access  procedures. \n\n4  STABILITY RESULTS \n\nIn  the LRAAM  model two stability problems are encountered.  The first  one arises \nwhen considering the decoding of a  pointer along a  cycle of the encoded structures. \nSince the decoding process suffers, in  general, of approximation errors, it  may hap(cid:173)\npen  that  the  decoding  diverges  from  the  correct  representations  of  the  pointers \nbelonging to the cycle.  Thus, it is  fundamental  to discover  under  which conditions \nthe representations obtained for  the pointers are asymptotically stable with respect \nto  the  pointer  transformation.  In  fact,  if  the  representations  are  asymptotically \nstable,  the errors introduced by the decoding function  are automatically corrected. \nThe following  theorem  can  be proven  (Sperduti,  1993b): \n\nTheorem  1  A  decoding  sequence \n\nl(i;+I) = F(p';)(l(iJ\u00bb), \n\nj  = 0, .. . ,L \n\nwith l(iL+d =  l(t o) ,  satisfying \n\nm L Ibikl  < 1, \n\nk=l \n\ni  =  1, ... ,m \n\n(1) \n\n(2) \n\nfor  some  index Pi'l'  q  = 0, ... , L,  is  asymptotically  stable,  where  btk  is  the  (i, k) th \nelement  of a  matrix B,  given  by \nB  = J(P\"I) (l( i'l) )J(P\"I-l ) (l( i'l_ J)) ... J(p'{J) (l( io) )J(p, L \\  l(iL\u00bb) ... J(P\"I+l ) (d (i'l+d). \n\nIn the statement of the theorem, F(p;) (l) =  F(D(p; )l+~;\u00bb) is the transformation \nof the  reduced  descriptor  (pointer)  d by  the  pointer  field  Pj,  and  J(pJ)(l)  is  its \n\n\fEncoding Labeled Graphs by Labeling RAAM \n\n1131 \n\nJacobian matrix.  As  a  corollary of this theorem we  have that if at  least  one pointer \nbelonging  to  the  cycle  has  saturated  components,  then  the  cycle  is  asymptotically \nstable  with  respect  to  the  decoding  process.  Moreover,  the  theorem  can  be  applied \nwith a few  modifications to the stability analysis of the fixed  points of the associated \nHopfield  network. \n\nThe  second  stability  problem  consists  into  the  discovering  of sufficient  conditions \nunder which the property of asymptotical stability of a fixed  point in one particular \nconstrained  version  of  the  associated  Hopfield  network,  i.e.,  an  access  procedure, \ncan  be  extended  to  related fixed  points of different  constrained  versions  of it,  i.e., \naccess  procedures  with  more  information  or  different  information.  The  result  of \nTheorem  1  was  used  to  derive  three  theorems  regarding  this  issue  (see  (Sperduti, \n1993b) ). \n\n5  DISCUSSION  AND  CONCLUSIONS \n\nThe LRAAM  model  can be seen from  various  perspectives.  It can be considered as \nan extension of the RAAM  model,  which  allows one  to encode  not  only  trees  with \ninformation on  the leaves,  but also labeled graphs with  cycles.  On  the other hand, \nit  can  be  seen  as  an  approximate  method  to  build  analog  Hopfield  networks  with \na  hidden  layer.  An  LRAAM  is  probably somewhere  in  between.  In  fact,  although \nit extends the representational capabilities of the RAAM  model,  it doesn't  possess \nthe  same  synthetic capabilities  as  the  RAAM,  since  it  explicitly  uses  the  concept \nof pointer.  Different  subsets  of units  are  thus  used  to  codify  labels  and  pointers. \nIn  the  RAAM  model,  using  the  same  set  of  units  to  codify  labels  and  reduced \nrepresentations is  a more natural way of integrating a previously developed reduced \ndescriptor  as  a  component  of  a  new  structure.  In  fact,  this  ability  was  Pollack's \noriginal rationale behind the RAAM model, since with this ability it is possible to fill \na  linguistic role with the reduced descriptor of a  complex sentence.  In the LRAAM \nmodel  the  same  target  can  be  reached,  but  less  naturally.  There  are two  possible \nsolutions.  One  is  to  store  the  pointer of some  complex  sentence  (or  structure,  in \ngeneral),  which  was  previously  developed,  in  the  label  of  a  new  structure.  The \nother  solution  would  be  to  have  a  particular  label  value  which  tells  us  that  the \ninformation we  are looking for can be retrieved using one conventional or particular \npointer among the current ones. \n\nAn  issue  strictly  correlated  with  this  is  that,  even  if  in  an  LRAAM  it  is  possible \nto encode  a  cycle,  what  we  get  from  the LRAAM is  not  an explicit  reduced  repre(cid:173)\nsentation  of the  cycle,  but  several  pointers  to  the  components  of the  cycle  forged \nin  such  a  way  that  the  information  on  the  cycle  is  only  represented  implicitly  in \neach of them.  However,  the ability  to synthesize  reduced  descriptors for  structures \nwith cycles is  what makes the difference between the LRAAM and the RAAM.  The \nonly system that we  know of which is able to represent labeled graphs is  the DUAL \nsystem  proposed  by  Dyer  (Dyer,  1991).  It is  able  to  encode  small  labeled  graphs \nrepresenting  relationships  among  entities.  However,  the  DUAL  system  cannot  be \nconsidered  as  being  on  the  same  level  as  the  LRAAM,  since  it  devises  a  reduced \nrepresentation  of a  set  of functions  relating  the  components  of  the  graph  rather \nthan a  reduced  representation for  the graph.  Potentially also Holographic Reduced \nRepresentations  (Plate,  1991)  are able to encode cyclic  graphs. \n\n\f1132 \n\nSperduti \n\nThe  LRAAM  model  can  also  be  seen  as  an  extension  of  the  Hopfield  networks \nphilosophy.  A relevant aspect of the use of the Hopfield  network associated with an \nLRAAM,  is  that  the access  procedures defined  on  it  can  efficiently  exploit  subsets \nof the weights.  In fact,  their use corresponds to generating several smaller networks \nfrom  a  large  network,  one for  each  kind  of access  procedure,  each specialized  on  a \nparticular feature  of  the  stored  data.  Thus,  by  training  a  single  network,  we  get \nseveral useful  smaller networks. \n\nIn  conclusion  an  LRAAM  has  several  advantages over  a  standard RAAM.  Firstly, \nit  is  more  powerful,  since it  allows  to encode directed  graphs where  each  node  has \na  bounded  number of outgoing arcs.  Secondly,  an LRAAM  allows  direct  access  to \nthe components of the encoded structure not only  by  pointer,  but also by content. \nConcerning the applications where LRAAMs can  be exploited, we  believe  there are \nat  least  three  possibilities:  in  knowledge  representation,  by  encoding  Conceptual \nGraphs  (Sowa,  1984);  in  unification,  by  representing  terms  in  restricted  domains \n(Knight,  1989);  in  image  coding,  by storing  Quadtrees  (Samet,  1984); \n\nReferences \n\nP.  Baldi &  K. Hornik.  (1989)  Neural networks and principal component analysis:  Learning \nfrom  examples without local  minima.  Neural  Networks,  2:53-58. \n\nH.  Bourlard &  Y.  Kamp.  (1988)  Auto-association  by  multilayer  perceptrons and singular \nvalue decomposition.  Biological  Cybernetics,  59:291-294. \n\nM.  G. Dyer.  (1991)  Symbolic  NeuroEngineering for Natural  Language  Processing:  A  Multi(cid:173)\nlevel  Research  Approach.,  volume 1 of Advances  in Connectionist  and Neural  Computation \nTheory,  pages  32-86.  Ablex. \n\nG.  E.  Hinton.  (1990)  Mapping part-whole hierarchies into connectionist networks.  A rtifi(cid:173)\ncial  Intelligence,  46:47-75. \n\nK.  Knight. \n21:93-124. \n\n(1989)  Unification:  A  multidisciplinary  survey.  A CM  Computing  Surveys, \n\nT.  Plate.  (1991)  Holographic  reduced  representations.  Technical  Report  CRG-TR-91-1, \nDepartment of Computer Science,  University of Toronto. \n\nJ.  B.  Pollack.  (1990)  Recursive  distributed  representations.  Artificial  Intelligence,  46(1-\n2):77-106. \n\nH.  Samet.  (1984)  The quadtree and related hierarchical data structures.  A CM Computing \nSurveys,  16:187-260. \n\nP.  Smolensky.  (1990)  Tensor product variable  binding and the representation of symbolic \nstructures in  connectionist systems.  Artificial Intelligence,  46:159-216. \n\nJ.F.  Sowa.  (1984)  Conceptual  Structures:  Information  Processing  in  Mind  and  Machine. \nAddison-Wesley. \n\nA.  Sperduti.  (1993a)  Labeling RAAM.  TR 93-029,  ICSI,  Berkeley. \n\nA.  Sperduti.  (1993b)  On  some  stability  properties  of  the  LRAAM  model.  TR  93-031, \nICSI,  Berkeley. \n\nD.  S.  Touretzky.  (1990)  Boltzcons:  Dynamic symbol structures in a connectionist network. \nA rtificial  Intelligence,  46:5-46. \n\n\fPART XI \n\nADDENDA TO NIPS 5 \n\n\f\f", "award": [], "sourceid": 860, "authors": [{"given_name": "Alessandro", "family_name": "Sperduti", "institution": null}]}