{"title": "Analyzing the Energy Landscapes of Distributed Winner-Take-All Networks", "book": "Advances in Neural Information Processing Systems", "page_first": 626, "page_last": 633, "abstract": null, "full_text": "626 \n\nANALYZING THE ENERGY LANDSCAPES \n\nOF  DISTRIBUTED \n\nWINNER-TAKE-ALL  NETWORKS \n\nDavid S.  Touretzky \n\nSchool of Computer Science \nCarnegie Mellon University \n\nPittsburgh, P A 15213 \n\nABSTRACT \n\nDCPS  (the  Distributed  Connectionist  Production System)  is  a  neural \nnetwork  with  complex  dynamical  properties.  Visualizing  the  energy \nlandscapes of some of its component modules leads to a  better intuitive \nunderstanding  of the  model,  and  suggests  ways  in  which  its  dynamics \ncan be controlled in order to improve performance on difficult  cases. \n\nINTRODUCTION \n\nCompetition through mutual inhibition appears in a wide variety of network designs. \nThis paper discusses  a  system with unusually complex competitive dynamics.  The \nsystem  is  DCPS,  the  Distributed  Connectionist  Production  System of Touretzky \nand  Hinton  (1988).  DCPS  is  a  Boltzmann  machine  composed  of five  modules, \ntwo of which,  labeled  \"Rule Space\"  and  \"Bind  Space,\"  are winner-take-all (WTA) \nnetworks.  These modules interact via their effects on two attentional mod ules called \nclause spaces.  Clause spaces are another type of competitive architecture based on \nmutual  inhibition,  but  they  do  not  produce  WTA  behavior.  Both  clause  spaces \nprovide evidential  input  to  both  WTA  nets,  but  since  connections  are  symmetric \nthey  also  receive  top-down  \"guidance\"  from  the  WTA  nets.  Thus,  unlike  most \nother  competitive  architectures,  in  DCPS  the  external  input  to  a  WTA  net  does \nnot  remain constant as  its state evolves.  Rather,  the  present  output  of the  WTA \nnet helps to determine which evidence will become visible in the clause spaces in the \nfuture.  This dynamic  attentional mechanism allows  rule  and  bind  spaces  to work \ntogether even though they are  not directly connected. \n\nDCPS  actually uses  a  distributed  version of winner-take-all networks  whose  oper(cid:173)\nating characteristics differ  slightly from  the non-distributed  version.  Analyzing the \nenergy  landscapes  of DWTA networks  has  led  to a  better  intuitive understanding \nof their dynamics.  For a  complete discussion  of the role  of DWTA nets  in  DCPS, \nand  the  ways  in  which  insights  gained  from  visualization  led  to improvements  in \nthe system's stochastic search behavior, see  [Touretzky,  1989]. \n\n\fEnergy Landscapes of Distributed Winner-Take-All Networks \n\n627 \n\nDISTRIBUTED  WINNER-TAKE-ALL NETWORKS \n\nIn classical WTA nets [Feldman &  Ballard,  1982],  a unit's output value is a continu(cid:173)\nous quantity that reflects its activation level.  In this paper we  analyze  a stochastic, \ndistributed  version of winner-take-all dynamics using  Boltzmann machines,  whose \nunits have only binary outputs [Hinton &  Sejnowski,  1986].  The amount of eviden(cid:173)\ntial input  to these  units determines  its energy  gap  [Hopfield,  1982],  which  in turn \ndetermines  its  probability of being active.  The  network's  degree  of confidence  in \na  hypothesis  is  thus reflected  in  the  amount of time the  unit  spends in  the active \nstate.  A  good  instantaneous approximation to strength of support can be obtained \nby representing each  hypothesis  with a  clique  of k  independent units  looking  at  a \ncommon evidence  pool.  The number of active units in a clique reflects the strength \nof that hypothesis.  DCPS uses cliques of size 40.  Units  in  rival cliques compete via \ninhibitory connections \n\nIf all  units  in  a  clique  have  identical  receptive  fields,  the  result  is  an  \"ensemble\" \nBoltzmann  machine  [Derthick  &  Tebelskis,  1988].  In  DCPS  the  units  have  only \nmoderately sized, but highly overlapped, receptive fields,  so the amount of evidence \nindividual units perceive is distributed binomially.  Small excitatory weights between \nsibling units help make up for  variations in external evidence.  They also make states \nwhere all  the units in  a  single clique  are active be powerful attractors. \n\nEnergy  tours in a  DWTA take one  of four  basic  shapes.  Examples  may be seen  in \nFigure  1a.  Let  e  be  the amount of external  evidence  available to each  unit,  0  the \nunit's  threshold,  k  the  clique  size,  and  W,  the excitatory  weight  between siblings. \nThe four  shapes are: \n\nEager vee:  the evidence is  above threshold  (e  >  0).  The system is  eager  to \nturn  units  on;  energy  decreases  as  the  number  of active  units  goes  up.  We \nhave a  broad, deep energy well, which the system will naturally fall  into given \nthe chance. \n\nReluctant  vee:  the  evidence  is  below  threshold,  but  a  little  bit  of sibling \ninfluence  (fewer  than k/2  siblings)  is  enough  to  make  up  the  difference  and \nput the system over the energy barrier.  We  have e < 0 < e +w,(k-1)/2.  The \nsystem is initially reluctant to turn units on because that causes the energy to \ngo up,  but once over the hump it willingly turns on more units.  With all units \nin  the  clique  active,  the  system is  in  an  energy  well  whose  energy  is  below \nzero. \n\nDimpled peak:  with higher  thresholds the total energy of the network may \nremain above  zero even when all units are on.  This happens  when more than \nhalf of the  siblings  must  be  active  to  boost  each  unit  above  threshold,  i.e., \ne + w,(k - 1)  >  0  >  e + w,(k - 1)/2.  The  system can still  be  trapped  in \nthe  small energy  well  that  remains,  but only  at  low  temperatures.  The well \nis  hard  to  reach  since  the  system  must first  cross  a  large  energy  barrier  by \ntraveling far  uphill in energy space.  Even if it does  visit  the well,  the system \nmay easily  bounce out of it again if the well is  shallow. \n\n\f628 \n\nTouretzky \n\nSmooth peak:  when ()  > e +  w.(k - 1),  units  will be below threshold  even \nwith  full  sibling  support.  In  this  case  there  is  no  energy  well,  only  a  peak. \nThe system wants to turn all units off. \n\nVISUALIZING  ENERGY LANDSCAPES \n\nLet's examine the energy landscape of one WTA space when there is ample evidence \nin the clause spaces for  the  winning  hypothesis.  We select  three hypotheses,  A,  B, \nand C,  with disjoint evidence populations.  Let hypothesis B  be the best supported \none with evidence 100, and let A have evidence 40  and C  have evidence 5.  We will \nsimplify the situation slightly by assuming that all units in a clique perceive exactly \nthe same  evidence.  In  the left  half of Figure  1 b  we  show  the  energy curves for  A, \nB,  and  C, using  a  value of 69  for  the unit  thresholds.1  Each curve is  generated  by \nstarting with all units turned off;  units for a particular hypothesis are turned on one \nat a  time until all  40  are on;  then they are turned off again one  at a  time,  making \nthe curve symmetric.  Since the evidence for  hypothesis A  is  a  bit below threshold, \nits curve is of the \"reluctant vee\"  type.  The evidence for  hypothesis B is  well above \nthreshold, so its curve is  an  \"eager  vee.\"  Hypothesis C  has almost no evidence;  its \n\"dimpled  peak\"  shape  is  due  almost  entirely  to  sibling support.  (Sibling  weights \nhave a  value of +2;  rival weights a  value of -2.) \n\nNote  that  the energy  well  for  B  is  considerably  deeper  than  for  A.  This  means  at \nmoderate  temperature  the  model  can  pop  out  of A's  energy  well,  but  it  is  more \nlikely to remain in B's well.  The well for  B is  also somewhat broader than the well \nfor  A,  making it easier for  the B attractor to capture the model;  its attract or region \nspans a  larger portion of state space. \n\nThe  energy  tours  for  hypotheses  A,  B,  and  C  correspond  to  traversing  three  or(cid:173)\nthogonal edges extending from a corner of a 40 x  40  x 40  cube.  A  point at location \n(x, y, z)  in  this  cube  corresponds  to  x  A  units,  y  B  units,  and  z  C  units  being \nactive.  During  the  stochastic  search,  A  and  B  units  will  be  flickering  on  and  off \nsimultaneously,  so the model  will  also visit  internal points of the cube not covered \nin the energy tour diagram.  To see  these points we  will  use  two additional graphic \nrepresentations of energy  landscapes.  First,  note  that  hypothesis  C  gets  so  little \nsupport  that  we  safely  can ignore  it and  concentrate on  A  and  B.  This allows  us \nto  focus  on just the front  face  of the state space  cube.  In  Figure  2a,  the  number \nof active A  units runs from  zero to forty along the vertical axis,  and the number of \nactive B units runs from zero to forty along the horizontal axis.  The arrows at each \npoint on the graph  show legal state transitions at zero temperature.  For example, \nat  the  point  where  there  are are  38  active B  units and  3  active  A  units  there  are \ntwo  arrows,  pointing down  and  to the right.  This  means  there  are  two states the \nmodel could  enter next:  it could  either turn off one  of the active A  units,  or turn \non  one  more B  unit,  respectively.  At nonzero temperatures other state transitions \n\n1 All the weights and thresholds used in this paper are actual DCPS values taken from [Touretzky \n\n&  Hinton, 1988]. \n\n\fEnergy Landscapes of Distributed Winner-Take-All Networks \n\n629 \n\nare possible,  corresponding  to uphill  moves  in energy space,  but these  two remain \nthe most  probable. \n\nThe  points in  the  upper  left  and  lower  right  corners  of Figure  2a  are  marked  by \n\"Y\"  shapes.  These  represent  point  attractors  at  the  bottoms of energy  wells;  the \nmodel will not move out of these states unless the temperature is  greater than zero. \nOther points in state space are said to be within the region of a  particular attractor \nif all legal transition sequences (at T  = 0)  from  those points lead  eventually to the \nattractor.  The attractor regions  of A  and  B  are  outlined  in  the figure.  Note that \nthe  B  attractor  covers  more  area  than  A,  as  predicted  by  its  greater  breadth  in \nthe  energy  tour  diagram.  Note  also  that  there  is  a  small  ridge  between  the  two \nattractor regions.  From starting points on the ridge the model can end up in either \nfinal  state. \n\nFigure 2b shows the depths of the two attractors.  The energy well for  B  is  substan(cid:173)\ntially deeper than the well for A.  Starting at the point in the lower left corner where \nthere are zero A units and zero B units active, the energy falls off immediately when \nmoving in the B  direction (right), but rises initially in  the A direction  (left)  before \ndropping  into  a  modest  energy  well  when  most  of the  A  units  are  on.  Points in \nthe  interior  of the  diagram,  representing  a  combination of A  and  B  units  active, \nhave higher energies than points along the edges due to the inhibitory connections \nbetween units  in rival cliques. \n\nWe can see from  Figures  lb and 2 that the attractor for  A,  although narrower and \nshallower than the one for  B, is  still sizable.  This is  likely to mislead  the model,  so \nthat some  of the time  it will  get  trapped  in the  wrong energy  well.  The fact  that \nthere  is  an  attractor  for  A  at  all  is  due  largely  to sibling  support,  since  the  raw \nevidence for  A is less  than the rule  unit threshold. \n\nWe can eliminate the unwanted energy well for  A by choosing thresholds that exceed \nthe maximum sibling support of 2 x  39 = 78.  DCPS  uses  a  value of 119.  However, \nearly in the stochastic search the evidence visible  in  the clause  spaces will be lower \nthan at  the conclusion  of the  search;  high  thresholds  combined  with  low  evidence \nwould  make the  B  attractor  small  and  very  hard  to  find. \n(See  the  right  half of \nFigure  Ie,  and  Figure  3.)  Under  these  conditions  the  largest  attractor is  the  one \nwith all units turned off:  the null hypothesis.  ' \n\nDISCUSSION \n\nOur analysis of energy landscapes pulls us in two directions:  we need low thresholds \nso  the  correct  attractor  is  broad  and easy  to find,  but we  need  high  thresholds  to \neliminate unwanted at tractors associated with local energy minima.  Two solutions \nhave been investigated.  The first is  to start out with low thresholds and raise them \ngradually during the stochastic search.  This  \"pulls the rug out from under\"  poorly(cid:173)\nsupported  hypotheses  while giving the model  time to find  the desired  winner.  The \nsecond  solution  involves clipping  a  corner  from  the state space  hypercube  so  that \nthe model may never  have fewer  than 40  units active at a  time.  This prevents the \n\n\f630 \n\nTouretzky \n\nmodel from falling into the null attractor.  When it attempts to drop the number of \nactive units below 40  it is  kicked  away from the clipped  edge  by forcing  it to turn \non a  few  inactive units at  random. \n\nAlthough  DCPS  is  a  Boltzmann  machine  it  does  not  search  the  state  space  by \nsimulated  annealing  in  the  usual  sense.  True  annealing  implies  a  slow  reduction \nin  temperature  over  many  update cycles.  Stochastic  search  in  DCPS  takes  place \nat  a  single  temperature  that  has  been  empirically  determined  to  be  the  model's \napproximate  \"melting  point.\"  The  search  is  only  allowed  to  take  a  few  cycles; \ntypically it takes less than 10.  Therefore the shapes of energy wells and the dynamics \nof the search are particularly important, as they determine  how likely the model is \nto wander into particular attractor regions. \n\nThe work reported here suggests that stochastic search dynamics  may be improved \nby manipulating parameters other than just absolute temperature and cooling rate. \nThreshold  growing  and  corner  clipping  appear  useful  in  the  case  of DWTA  nets. \nAdditional details are available in [Touretzky,  1989]. \n\nAcknowledgments \n\nThis research was supported by the Office of Naval Research under contract N00014-\n86-K-0678,  and by National Science Foundation grant EET-8716324.  I thank Dean \nPomerleau, Roni Rosenfeld,  Paul Gleichauf, and Lokendra Shastri for  helpful com(cid:173)\nments, and  Geoff Hinton for  his collaboration in  the development of DCPS. \n\nReferences \n\n[1]  Derthick,  M.  A.,  &  Tebelskis,  J.  M.  (1988)  \"Ensemble\"  Boltzmann  machines \n\nhave  collective computational  properties  like  those  of Hopfield  and  Tank  neu(cid:173)\nrons.  In  D.  Z.  Anderson  (ed.),  Neural  Information  Processing  Systems.  New \nYork:  American Institute of Physics. \n\n[2]  Feldman,  J.  A.,  &  Ballard,  D.  H.  (1982)  Connectionist  models and  their prop(cid:173)\n\nerties.  Cognitive  Science 6:205-254. \n\n[3]  Hinton, G. E.,  &  Sejnowski, T. J.  (1986)  Learning and relearning in Boltzmann \nmachines.  In D.  E.  Rumelhart and J.  L.  McClelland  (eds.),  Parallel Distributed \nProcessing:  Explorations  in  the  Microstructure  of Cognition,  volume  1.  Cam(cid:173)\nbridge,  MA:  Bradford Books/The MIT Press. \n\n[4]  Hopfield,  J. J. (1982)  Neural  networks and physical systems with emergent col(cid:173)\n\nlective computational abilities. Proceedings  of the  National Academy of Sciences \nUSA,  79:2554-2558. \n\n[5]  Touretzky, D. S., &  Hinton, G. E. (1988)  A distributed connectionist product.ion \n\nsystem.  Cognitive  Science 12(3):423-466. \n\n[6]  Touretzky,  D.  S.  (1989)  Controlling  search  dynamics  by  manipulating  energy \nlandscapes.  Technical  report  CMU-CS-89-113,  School  of  Computer  Science, \nCarnegie Mellon  University,  Pittsburgh,  PA. \n\n\fEnergy Landscapes of Distributed Winner-Take-All Networks \n\n631 \n\n. \n\nAJ'\\ \n, \n! \n\\ \n! \n. \n\u00b7 \n\u00b7 \n. \n. \n\u00b7 \n. \n\u00b7 \n. \n. \n\u00b7 \n\u00b7 \n\n\\  ( \n~ ! \n\n~ \n\n! \n\nEvldlncl:  A&4O.  \"100.  C:5. \n\nEvldlncl:  A&4O.  \"100.  C:5. \n\n/ \\  \n\nI  \\ \n\n\\ , \n\\ \\ \n\nj \n\n: \n: \n\n! : \n! \n\n~ \n\n/\\ \n! \n\\ \n! \nl \n\\ \n\u00b7 \n. \n. \n\u00b7 \n. \n\u00b7 \n. \n\u00b7 \n. \n\u00b7 \n\u00b7 \n.  - .  \n\n\\  ! \n\n\":. \n\n0. \n\n: \n\\f \n\nllnIhold \u2022  69 \n\nllnIhold = 119 \n\nEvldlncl:  A&4O.  1060,  C:5. \n\nEvldlncl:  Aa4O.  1060.  C:5. \n\n\\  ! \n\n, . '= \n\nnr.hold = 69 \n\n\\( \n\nr\\ \n\\ \n! \n\\ \n! \n\\ \n! \n. \n\u00b7 \n. \n\u00b7 \n\u00b7 \n. \n\u00b7 \n. \n\nf\\\\ \n. \nf \n\n\\ \n. \n\nnr.hold = 119 \n\nI \n; \n\n, \n; \n\n1\\ \n.f \n\\ : \n\n1\\ \n1\\ \n!  \\ \n\u00b7 \n\u00b7 \n. \n. \n\u00b7 \n. \n\u00b7 \n. \n\u00b7 \n. \n\u00b7 \n. \n. \n\u00b7 \n\u00b7 \n. \n\u00b7 \n. \n. \n\u00b7 \n\u00b7 \n. \n\u00b7 \n. \n. \n\u00b7 \n\u00b7 \n. \n. \n\u00b7 \n\n!A'. \n: \n! \n! \n\n\\ !  \\ \n\u00b7 \n. \n. \n\u00b7 \n. \n\u00b7 \n. \n\u00b7 \n. \n\u00b7 \n\u00b7 \n. \n\u00b7 \n. \n\u00b7 \n. \n\u00b7 \n. \n. \n\u00b7 \n\nFigure  1:  (a)  four  basic  shapes  for  DWTA  energy  tours;  (b)  comparison  of low \nvs.  high  thresholds  in  energy  tours  where  there  is  a  high  degree  of evidence  for \nhypothesis  B;  (c)  corresponding tours with low  evidence for  B. \n\n\f'~~'eJms A~J~U~ ~u!puods~JJo~ ~q'l  (q)  !~m'l'eJ~dw~'l OJ~Z 'l'e  SUO!'l!su'eJ'l  ~'l'e'ls I'e~~1 \n('e) \n'q 1 ~m~!d JO  Jl'eq  U~l ~q'l  U!  S'e  '~~U~P!A~ q~!q pU'e  sPloqs~Jq'l MOl  :~ ~JI1~!d \n\n\"QIlI1 r \n\n\u20221Ik\u00bb. \n\n~':J. \n\nfb e\" \n\n.... \\~t'> \n\n\u2022\u2022\u2022 \n\n'69  = PT o4sa.J41 \n\n'001=8  'O~=~  :aouapT A3 \n\nA~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ \n1444444444444444444444444444444444~44444 \n1444444444444444444444444444444444444444 \n1444444444444444444444444444444444444444 \n1444444444444444444444444444444444444444 \n1444444444444444444444444444444444444444 \n144444444444444444444444444~444444444444 \n1444444~4444444444444444444~444444444444 \n1444444444444444444444444444444444444444 \n1444444444444444444444444444444444444444 \n1444444444444444444444444444444444444444 \n1444444444444444444444444444444444444444 \n'4444444444444444~4444444444444444444 \n44444444444444444444 \n14444444444444444 \n14444444444444444  4  44444444444444444444 \n14444444444444444 \n4444444444444444444 \n14444444444444444 \n14444444444444444 \n144444444444444444 \n14444444444444444444444444444444444 \n1444444444444444444444444444444444 \n144444444444444444444444444444444 \n14444444444444444444444444444444 \n1444444444444444444444444444444 \n144444444444444444444444444444 \n14444444444444444444444444444 \n1444444444444444444444444444 \n144444444444444444444444444 \n14444444444444444444444444 \n1444444444444444444444444 \n144444444444444444444444 \n14444444444444444444444 \n1444444444444444444444 \n144444444444444444444 \n14444444444444444444 \n1444444444444444444 \n144444444444444444 \n14444444444444444 \n1444444444444444 \n144444444444444 \n144 \n\n~ \n~~ \n~~~ \n~~~~ \n~~~~~ \n~~~~~~ \n~~~~~~~ \n~~~~~~~~ \n~~~~~~~~~ \n~~~~~~~~~~ \n~~~~~~~~~~~ \n~~~~~~~~~~~~ \n~~~~~~~~~~~~~ \n~~~~~~~~~~~~~~ \n~~~~~~~~~~~~~~~ \n~~~~~~~~~~~~~~~~ \n~~~~~~~~~~ ~~~~ \n~~~~~~~~~  ~~  ~~~~~ \n~~~~~ \n~~~~~ \n~~~~~~~~~~~~~~  ~~~~~~ \n~~~~~~~~~~~~~~~~~~~~~~ \n~~~~~~~~~~~~~~~~~~~~~~~ \n\n44444444444444444 \n4444444444444444 \n444444444444444 \n\n~~~~~~~~~~~~~~~~~~~~~~~ \n\n~~~~~~~~~~ \n\n~~~~~~~~~~~~ \n\nAlIZla,m0J, \n\n~f!9 \n\n\f-('I) \n~ \n~~ \n\n~ \n\nt-t:;.! \n~~ ~  ... \n~  .. \n('I)  == ~ ..... \n...  oq \n~  ::r \n;  ~ \n.....  ::r \nC-.  ... \no  ('I) \n~ -~o.. \n~  ~ \n~  I:T' o \nN \n...  ~ \n('I)  ~ \no  0.. \n~-('I)  0 \nS  == \n('I) \n~  < \n~  ..... \n~o.. \n~  ('I) \n...  ~ \n('I) \n_. \n..--\nO\"'~ \n~ ..... \n\"-\"~ \n\n(\") \n('I) \n\n\"'C:j \n\nfI) \n\n::r~ \n('I)  ~ \nI:T' \n(\") \n... ... \no \n('I) \n... \n('I) \n..... \n~oq \n\"t:I::r \no  ~ 5..;-\n..... -~  ...., \n\noq  0 \n('I) \n...., \n~  ~ \n('I) \n..... \n\"'oq \noq  ~ \n'< \n... \n~ .... \n~  ('I) \n\n... \n(\") \n~. \n(\")..-\n~~ \n\n~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ \n\n1444444444444444444444444444444444444444 \n1444444444444444444444444444444444444444 \n1444444444444444444444444444444444444444 \n1444444444444444444444444444444444444444 \n1444444444444444444444444444444444444444 \n1444444444444444444444444444444444444444 \n14444444444444444444~44444444444444 \n14444444444444444444444  444444444444444 \nt44444444444444444444 \n4  44444444444444 \nt44444444444444444444 \n44444444444444 \nt444444444444444444444444444444444444444 \nt44444444444~4444444~4444444444444444 \n4444444444444444 \nt4444444444444444444 \nt44444444444444444444  44444444444444444 \nt44444444444444444441 \n44444444444444 \nt44444444444444444444.  1144444444444444 \n1444444444444444444444444444444444444444 \nt444444444444444444444444444444444444444 \nt444444444444444444444444444444444444444 \n1444444444444444444444444444444444444444 \n1444444444444444444444444444444444444444 \nt444444444444444444444444444444444444444 \nt444444444444444444444444444444444444444 \nt444444444444444444444444444444444444444 \nt444444444444444444444444444444444444444 \n1444444444444444444444444444444444444444 \n1444444444444444444444444444444444444444 \nt444444444444444444444444444444444444444 \n14  4444444  444 \n4  4 \n,  444444444444444444444444444444444444444 \n,~  44444444444444444444444444444444444444 \n,~  4444444444444444444444444444444444444 \n,~~  444444444444444444444444444444444444 \n,~~~  44444444444444444444444444444444444 \n4444444444444444444444444444444444 \n, \n,  ~ \n444444444444444444444444444444444 \n~~  44444444444444444444444444444444 \nl \nl~~~~~~~  4444444444444444444444444444444 \n444444444444444444444444444444 \n\n,.., \n< .... \na. \nIII \n:J \n0 \nIII \n\nl> \nII \nA \n\n0 \u2022 \ntIl \nII m \n0 \u00b7 \n\n-i \n\n~ .., \n\nIII \nUl \n~ \n\n0 .... \n\n0.. \n\nII \n..... \n..... \n\n\\0 \u00b7 \n\ntTl. \n\n4 I \n\n'\\ \u2022 ., \n\n~ \u2022 \n\ntE.1 \n\n~ \nr \n\nfI) \n\n~ (D \n~ \n~ \nfI) &: \n0'\" \n~ \n~ \nj;l.. \n~ \n~ \n~ (D \n\n~ -z (D i \n\n~ \n\na-\n\n0) \n\nto c.o \n\n\f", "award": [], "sourceid": 160, "authors": [{"given_name": "David", "family_name": "Touretzky", "institution": null}]}