{"title": "The Statistical Mechanics of k-Satisfaction", "book": "Advances in Neural Information Processing Systems", "page_first": 439, "page_last": 446, "abstract": null, "full_text": "The Statistical Mechanics of \n\nk-Satisfaction \n\nScott Kirkpatrick* \n\nRacah Institute for  Physics  and \nCenter for  Neural  Computation \n\nHebrew  University \n\nJerusalem,  91904  Israel \n\nkirk@fiz.huji.ac .il \n\nGeza  Gyorgyi \n\nInstitute for  Theoretical  Physics \n\nEotvos  University \n\n1-1088 Puskin u.  5-7 \nBudapest,  Hungary \n\ngyorgyi@ludens.elte.hu, \n\nN aft ali Tishby \n\nand  Lidror Troyansky \n\nInstitute of Computer Science  and  Center for  Neural  Computation \n\nThe  Hebrew  University of Jerusalem \n\n91904  Jerusalem, Israel \n\n{tishby, lidrort }@cs.huji.ac.il \n\nAbstract \n\nThe  satisfiability of random  CNF  formulae  with  precisely  k  vari(cid:173)\nables per clause (\"k-SAT\") is a popular testbed for the performance \nof search  algorithms.  Formulae have  M  clauses  from  N  variables, \nrandomly  negated,  keeping  the  ratio  a  =  M / N  fixed .  For  k  =  2, \nthis  model  has  been  proven  to  have  a  sharp  threshold  at  a  = 1 \nbetween  formulae which  are almost aways satisfiable and formulae \nwhich  are  almost never  satisfiable  as  N  --jo  00 .  Computer  experi(cid:173)\nments for  k  = 2,  3,  4,  5  and  6,  (carried  out  in  collaboration  with \nB.  Selman  of ATT Bell  Labs). show  similar threshold  behavior for \neach  value  of k.  Finite-size  scaling,  a  theory  of the  critical  point \nphenomena used  in statistical physics, is shown  to characterize the \nsize  dependence  near  the  threshold.  Annealed  and  replica-based \nmean field  theories  give a  good account  of the results. \n\n\"Permanent  address:  IBM  TJ Watson  Research  Center,  Yorktown  Heights,  NY 10598 \nUSA.  (kirk@watson.ibm.com)  Portions  of  this  work  were  done  while  visiting  the  Salk \nInstitute,  with support from  the McDonnell-Pew  Foundation. \n\n439 \n\n\f440 \n\nKirkpatrick, Gyorgyi, Tishby, and Troyansky \n\n1  Large-scale computation without  a  length scale \n\nIt is  increasingly  possible  to  model  the  natural  world  on  a  computer.  Condensed \nmatter physics  has strategies  to manage the complexities of such  calculations,  usu(cid:173)\nally  depending  on  a  characteristic  length.  For  example,  molecules  or  atoms  with \nfinite  ranged interactions can be broken down into weakly interacting smaller parts. \nWe  may  also  use  symmetry  to  identify  natural  modes  of the  system  as  a  whole. \nEven  in  the most difficult case,  continuous phase transitions correlated  over  a  wide \nrange of scales,  the renormalization group provides a  way of collapsing the problem \ndown  to  its  \"relevant\"  parts  by  providing  a  generator  of behavior  on  all  scales  in \nterms of the critical point itself. \n\nBut length scales  are not much help  in  organizing another sort of large calculation. \nExamples include  large  rule-based  \"expert systems\"  that  model  the  particulars  of \ncomplex industrial processes.  Digital Equipment, for  example,  has used  a  network \nof three or more expert systems  (originally called  \"R1/XCON\") to check  computer \norders  for  completeness  and internal  consistency,  to  schedule  production  and ship(cid:173)\nping,  and  to  aid  a  salesman  to  anticipate  customers'  needs.  This very  detailed  set \nof tasks  in  1979  required  2  programmers and  250  rules  to  deal  with  100  parts.  In \nthe ten years  described  by Barker (1989), it grew  100X, employing 60  programmers \nand nearly 20,000 rules  to deal with 30,000 part numbers.  100X in ten years is  only \nmoderate growth, and it would be valuable to understand how technical, social, and \nbusiness  factors  have constrained  it. \n\nMany important commercial and scientific problems without length scales are ready \nfor  attack  by  computer modelling or  automatic classification,  and  lie  within  a  few \ndecades  of XCON's size.  Retail industries  routinely  track  105  - 106  distinct  items \nkept  in stock.  Banks, credit  card  companies, and specialized  information providers \nare  building models  of what  108  Americans  have  bought  and  might  want  to  buy \nIn  biology,  human  metabolism  is  currently  described  in  terms  of  >  1000 \nnext. \nsubstances  coupled  through>  10,000  reactions,  and  the  data is  doubling  yearly. \nSimilarly, amino acid  sequences  are known  for> 60,000 proteins. \n\nA deeper  understanding of the computational cost of these problems of order  106 \u00b12 \nis  needed  to  see  which  are  practical  and  how  they  can  be  simplified.  We  study \nan  idealization of XC ON-style resolution search,  and find  obvious collective  effects \nwhich  may be at the heart  of its computational complexity. \n\n2  Threshold Phenomena and  Random  k-SAT \n\nProperties  of  randomly  generated  combinatorial  structures  often  exhibit  sharp \nthreshold phenomena analogous to the phase transitions studied in condensed  mat(cid:173)\nter physics.  Recently, thresholds have been observed in randomly generated Boolean \nformulae.  Mitchell  et  al.  (1992)  consider  the  k-satisfiability  problem  (k-SAT).  An \ninstance  of k-SAT  is  a  Boolean  formula  in  conjunctive  normal  form  (CNF),  i.e., \na  conjunction  (logical  AND)  of disjunctions  or  clauses  (logical  ORs),  where  each \ndisjunction contains exactly k literals.  A literal is  a  Boolean variable or, with equal \nprobability, its  negation.  The task  is  to determine  whether  there  is  an assignment \nto the variables such  that all clauses evaluate to true.  Here,  we  will use N  to denote \nthe number of variables  and  M  for  the number of clauses  in  a  formula. \n\n\fThe Statistical Mechanics of k-Satisfaction \n\n441 \n\nFor  randomly generated  2-SAT  instances,  it  has  been  shown  analytically that  for \nlarge  N,  when  the  ratio  a:  =  M / N  is  less  than  1  the  instances  are  almost  all \nsatisfiable,  whereas  for  ratios  larger  than  1,  almost  all  instances  are  unsatisfiable \n(Chvatal and  Reed  1992;  Goerdt  1992).  For  k  ~ 3,  a  rigorous  analysis has  proven \nto  be  elusive.  Experimental evidence,  however, strongly suggests  a  threshold  with \na:  ~ 4.3 for  3SAT (Mitchell  et  al.  1992; Crawford  and Auton  1993; Larrabee  1993). \n\nOne of the main reasons for studying randomly generated 3CNF formulae is for  their \nuse  in the empirical evaluation of combinatorial search  algorithms.  3CNF formulae \nare good candidates for the evaluation of such algorithms because determining their \nsatisfiability is  an  NP-complete problem.  This also holds for  larger values of k.  For \nk =  1 or  2,  the satisfiability problem can  be  solved efficiently  (Aspvall  et  al.  1979) . \nDespite the worst-case  complexity, simple heuristic  methods can usually determine \nthe  satisfiability  of random  formulae.  However,  computationally challenging  test \ninstances are found  by generating formulae at or near the threshold  (Mitchell  et  al. \n1992).  Cheeseman (1991) has made a similar observation of increased computational \ncost for  heuristic  search  at a  boundary between  two  distinct phases  or behaviors of \na  combinatorial model. \n\nWe  will  provide  a  precise  characterization  of  the  N -dependence  of  the  threshold \nphenomena for  k-SAT with k ranging from 2 to 6.  We will employ finite size scaling, \na  method  from  statistical  physics  in  which  direct  observation  of the  width  of the \nthreshold , or  \"critical region\"  of a transition is used  to characterize  the  \"universal\" \nbehavior  of quantities  across  the  entire  critical  region,  extending  the  analysis  to \ncombinatorial  problems  in  which  N  characterizes  the  size  of the  model  observed. \nFor discussion  of the applicability of finite-size  scaling to systems without  a metric, \nsee  Kirkpatrick  and  Selman (1993). \n\nill \n~ \n\n~ \n\n'\" ~ \n... ill \n\n\u00a7 \n\n~ \n0 \n\ng \n',j \n~ \n~ \n\n1. \n\nO . B \n\n0 . 6 \n\n0 . 4 \n\n0 . 2 \n\n0 \n\n0 \n\ni\n\n.' \n\n,': \n\nif  1(/ \nII! \nif \n!i \ni! \n!i  11/ \n~i \nii \n\n'1,1 \n\nJ \n\n1.0 \n\nThr \u2022\u2022 ho~d.  rOr  2SAT.  3SAT ,  4SAT,  5SAT ,  and  6SAT \n\n/'<> \n\n.'\" \n/ \n:' \n.... \nI :  \n! / \nIi' \n// \n// \nf/ \n}' \n\n:1 \n\n..... ; \n.. ' \",' \n\n20 \n\nJ.. \n\n30 \nM I N \n\n40 \n\nso \n\n60 \n\nFig. 1:  Fraction of unsatisfiable formulae for  2-,  3- 4-,  5- and 6-SAT. \n\n\f442 \n\nKirkpatrick, Gyorgyi, Tishby, and Troyansky \n\n3  Experimental data \n\nWe  have  generated  extensive  data  on  the  satisfiability  of randomly  generated  k(cid:173)\nCNF  formulae  with  k  ranging from  2  to  6.  Fig.  1  shows  the  fraction  of random \nk-SAT  formulae  that  is  unsatisfiable  as  a  function  of the  ratio,  a.  For  example, \nthe left-most curve in  Fig.  1 shows  the fraction  of formulae that is  unsatisfiable for \nrandom 2CNF formulae with 50  variables over  a  range of values  of a. \n\nEach  data  point  was  generated  using  10000  randomly  generated  formulae,  giving \n1 %  accuracy.  We  used  a  highly  optimized  implementation  of the  Davis-Putnam \nprocedure  (Crawford and Auton  1993).  The procedure  works  best on formulae with \nsmaller k .  Data was  obtained for  k  =  2 on  samples  with  N  ~ 500,  for  k  =  3 with \nN  ~ 100,  and for  k  =  5  with  N  ~ 40,  all at comparable computing cost. \n\nFig.  1 (for  N  ranging from  10  to 50)  shows  a  threshold  for  each  value of k.  Except \nfor  the case  k  =  2,  the curves cross  at a  single point and sharpen up with increasing \nN.  For  k  =  2,  the intersections  between  the curves for  the largest  values  of N  seem \nto be converging to a single point as well,  although the curves for smaller N  deviate. \nThe  point where  50%  of the formulae are  unsatisfiable  is  thought  to  be  where  the \ncomputationally hardest problems are found  (Mitchell  et  al.  1992;  Cheeseman  et  al. \n1991).  The 50%  point lies consistently to the right of the scale-invariant point  (the \npoint where  the  curves  cross  each  other),  and shifts with N. \n\nThere  is  a  simple  explanation  for  the  rapid  shift  of  the  thresholds  to  the  right \nwith  increasing  k .  The  probability  that  a  given  clause  is  satisfied  by  a  random \ninput  configuration  is  (2k  - 1)/2k  =  (1  - 2- k )  _  'k.  If we  treat  the  clauses  as \nindependent,  the  probability that  all clauses  are  satisfied  is  ,~ =  ,kN .  We  define \nconfigurations,2N 'k  . 5  =  1 + alog2(,k) = 1- a/aann, and the  vanishing of the \nthe entropy, 5, per in~ut as  l/N times the log2  of the expected number of satisfying \n\nentropy  gives  an  estimate  of the  threshold,  identical  to  the  upper  bound  derived \nby  several  workers  (see  Franco  (1983)  and  citations  in  Chvatal  (1992)):  aann  = \n-(log2(1 - 2- k))-1  ~ (ln2)2k.  This is  called  an  annealed estimate for  C\u00a5c,  because \nit  ignores  the  interactions  between  clauses,  just  as  annealed  theories  of materials \n(see  Mezard  1986) average over many details of the disorder.  We have marked aann \nwith  an  arrow for  each  k  in  the figures,  and tabulate it in Table  1. \n\n4  Results of Finite-Size Scaling Analysis \n\nFrom  Fig.  1,  it  is  clear  that  the  threshold  \"sharpens  up\"  for  larger  values  of N. \nBoth  the  threshold  shift  and  the  increasing  slope  in  the  curves  of  Fig.  1  can  be \naccounted for  by finite size scaling.  (See Stauffer and Aharony (1992) or Kirkpatrick \nand  Swendsen  (1985).)  We  plot  the  fraction  of samples  unsatisfied  against  the \ndimensionless rescaled  variable, \n\ny  =  Nl/V(a - c\u00a5c)/ac  . \n\nValues  for  a c  and  1I  must  be  derived  from  the  experimental  data.  First  a c  is \ndetermined  as  the  crossing  point  of the  curves  for  large  N  in  Fig.  1.  Then  1I  is \ndetermined  to make the slopes match up  through  the critical  region.  In  Fig. 2  (for \nk =  3)  we  find  that these  two  parameters capture  both the  threshold  shift and the \nsteepening of the  curves,  using a c  = 4.17 and  1I  = 1.5.  We  see  that  F,  the fraction \n\n\fThe Statistical Mechanics of k-Satisfaction \n\n443 \n\nscakMf CFOuover functton,  III SAT modele \n\n_>SAT.,. \n\n\",.12  \u2022 \n\"=20  \u2022 \nN=24  a \nN=tO \nIl \nN. 50  a. \nN .. 100  .... . \n\n\u2022 .. .fi' \n\ni. \n\n01 \n\n01 \n\nJ \ni \na \n\nf '0 \nOf I \n\n02 \n\nFig.  2:  Rescaled  3-SAT data using a c  = 4.17,  lJ = 1.5. \nFig.  3:  Rescaled  data for  2-,  3-,  4-,  5-,  and 6-SAT approach  annealed limit. \n\n-2 \n\n-\\ \n\n2 \nY \n\n3 \n\nof unsatisfiable formulae, is  given by F(N, a) =  I(y)  , where  the invariant function, \nI,  is  that graphed  in  Fig.  2. \nA  description  of the  50%  threshold  shift  follows  immediately.  If we  define  y'  by \nI(y') =  0.5,  then  a50  =  a c(1 + y' N- 1/ V ) .  From  Fig.  2  we  find  that  a50  ~ 4.17 + \n3.1N- 2 / 3 .  Crawford and Auton (1993) fit  their data on the 50% point as a function \nof N  by  arbitrarily  assuming  that  the  leading  correction  will  be  O(I/N) .  They \nobtain a50 =  4.24 + 6/ N.  However,  the two expressions differ  by  only a few  percent \nas  N  ranges from  10  to 00. \n\nWe  also  obtained  good  results  in  rescaling  the  data for  the  other  values  of k.  In \nTable  1  we  give  the  critical  parameters  obtained  from  this  analysis.  The  error \nbars  are  subjective,  and  show  the  range  of each  parameter  over  which  the  best \nfits  were  obtained.  Note  that  v  appears  to  be  tending  to  1,  and  aann  becomes \nan increasingly good  approximation to a c  as  k  increases.  The success  of finite-size \nscaling  with  different  powers,  v,  is  strong  evidence  for  criticality,  i.e.,  diverging \ncorrelations,  even  in  the absence  of any length. \n\nFinally,  we  found  that  all  the crossovers  were  similar in  shape.  In  fact,  combining \nthe various  rescaled  curves  in  figure  3 shows  that  the  curves  for  k ~ 3 all coincide \nin the vicinity of the 50% point, and tend to a limiting form, which can  be obtained \nby  extending  the annealed  arguments of the previous  section.  If we  define \n\nthen  the probability that a  formula remains unsatisfied  for  all 2N  configurations is \n\nThe curve for  k = 2 is similar in form,  but shifted to the right from the other ones. \n\n\f444 \n\nKirkpatrick, Gyorgyi, Tishby, and Troyansky \n\nk \n2 \n3 \n4 \n5 \n6 \n\n0'2 \nO'ann \n2.41 \n1.38 \n5.19 \n4.25 \n10.74  9.58 \n21.83  20.6 \n44.01 \n42.8 \n\nO'c \n1.0 \n\n0\" \n2.25 \n4.17\u00b1.03  0.74 \n9.75\u00b1.05  0.67 \n20.9\u00b1.1  0.71 \n43.2\u00b1.2  0.69 \n\nV \n2.6\u00b1.2 \n1.5\u00b1.1 \n1.25\u00b1.05 \n1.1\u00b1.O5 \n1.05\u00b1.05 \n\nTable  1:  Critical parameters for  random k-SAT. \n\n5  Outline  of Statistical  Mechanics  Analysis \n\nSpace  permits only  a  sketch  of our  analysis  of this  model.  Since  the  N  inputs  are \nbinary, we  may represent  them as  a  vector,  X, of Ising spins: \n\nEach  random formula, F, can  be  written  as  a  sum of its  M  clauses,  Cj, \n\nX={xi=\u00b1l} \n\ni=l, ... N. \n\nwhere \n\nM \n\nF  =  LCj, \n\nj=1 \n\nk \n\nCj  = II (1  - Jj 1X)/2. \n\n1=1 \n\nwhere  the  vector,  Jj,l,  has  only  one  non-zero  element,  \u00b11,  at  the  input  which \nit  selects.  F  evaluates  to  the  number  of  clauses  left  unsatisfied  by  a  particular \nconfiguration.  It is  natural to take  the  value  of F  to  be  the energy.  The partition \nfunction, \n\nz =  tr{x.}e.6.r =  tr{x.} II e.6Cj , \n\nj \n\nwhere  f3  is  the  inverse  of a  fictitious  temperature,  factors  into  contributions  from \neach  clause.  The  \"annealed\"  approximation  mentioned  above  consists  simply  of \ntaking the trace over each subproduct individually, neglecting their interactions.  In \nthis construction, we  expect  both energy and entropy, S,  to be extensive quantities, \nthat is,  proportional to  N.  Fig.  4 shows that this is  indeed  the case for  S( a).  The \nlines  in  Fig.  4  are the annealed  predictions  S( a, k)  =  1 - 0'/ aann.  Expressions for \nthe  energy  can  also  be  obtained  from  the  annealed  theory,  and  used  to  compare \nthe specific  heat  observed  in numerical experiments with  the simple limit in  which \nthe  clauses  do  not  interact.  This  gives  evidence  supporting  the  identification  of \nthe  unsatisfied  phase  as  a  spin  glass.  Finally,  a  plausible  phase  diagram  for  the \nspin  glass-like  \"unsatisfied\"  phase  is  obtained  by  solving  for  S(T)  = 0  at  finite \ntemperatures. \n\nTo  perform  the  averaging  over  the  random  clauses  correctly  requires  introducing \nreplicas  (see  Mezard  1986),  which  are  identical  copies  of the  random formula,  and \ndefining  q,  the  overlap  between  the  expectation  values  of  the  spins  in  any  two \nreplicas, as the new order parameter.  The results appear to be capable of accounting \n\n\fThe Statistical Mechanics of k-Satisfaction \n\n445 \n\nfor  the difference  between experiment and the annealed predictions at finite  k.  For \nexample,  an  uncontrolled  approximation  in  which  we  consider  just  two  replicas \ngives the values of a2 in Table 1,  and accounts rather closely for  the average overlap \nfound experimentally between pairs of lowest energy states, as shown in Fig.  5.  The \n2-replica theory  gives  q as  the solution of \n\na(k, q)  =  2k(1 + q)k-l(4k - 2k+l + (1  - ql)/ln\u00abl + q)(l - q)) \n\nfor  q  as  a  function  of a.  This  gives  the  lines  in  Fig 5.  We  defined  a2  (in  Table  1) \nas  the point of inflection,  or  the maximum in the slope of q(a). \n\nEntropy  tor  It- SAT. \n\nl  =  2.  3,  t .  S \n\no . \n\n0 ' \n\no . \n\no 1 \n\n' l i.frlk.ll'Sp\u00b7(cid:173)\n' n6k2  p'  \u2022 \n' n2 f.k2  p'_ \n\n' nlO, p'  D \n\u00b7 nH .p\u00b7  ....... \n\n\u2022 \n\n' n12U  p2 ' \n'n20 kf,  p '  ..-....t \n' nlOkS  p'  \u2022 \n' n20kS  p ' -\n\n,~~:-\n'qob.I2CM<A'  \u2022 \n'qob.l2_  0 \n'qobNek3'  x \n',,1211<2'  \u2022 \n'qob.t2Ok2'  \u2022 \n'qob1121c2'  \u2022 \n\n07 \n\n0.1 \n\n0.5 \n\n~ \n\n04 \n\n03 \n\n02 \n\n01 \n\n10 \n\n15 \n\nu t loO H/ N \n\n25 \n\n30 \n\n111M \n\n20 \n\n2S \n\nFig. 4:  Entropy  as function  of a  for  k  =  2,  3,  4,  and 5. \nFig.  5:  q  calculated  from  2-replica  theory  vs  experimental  ground  state  overlaps. \nArrows  pointing up  are  O'ann,  arrows  pointing down  are a2. \n\n6  Conclusions \n\nWe  have shown how finite size  scaling methods from statistical physics can  be used \nto model the  threshold  in  randomly generated  k-SAT problems.  Given the good fit \nof our  scaling analysis,  we  conjecture  that this  method can  also give  useful  models \nof phase  transitions in  other  combinatorial problems with  a  control parameter. \n\nSeveral  authors  have  attempted  to relate  NP-hardness  or  NP-completeness  to  the \ncharacteristics of phase transitions in models of disordered  systems.  Fu and Ander(cid:173)\nson  (see  Fu  1989)  have  proposed  spin  glasses  (magnets  with  2-spin  interactions  of \nrandom sign)  as having inherent exponential complexity.  Huberman and colleagues \n(see  Clearwater  1991)  were  first  to  focus  on  the  diverging  correlation  length  seen \nat  continuous  phase  transitions  as  the  root  of computational complexity.  In  fact, \nboth  effects  can  play  important  roles,  but  are  not  sufficient  and  may not  even  be \nnecessary. \n\nThere  are  NP-complete  problems  (e.g.  travelling salesman,  or  max-clique)  which \nlack  a  phase  boundary  at  which  \"hard  problems\"  cluster.  Percolation  thresholds \nare  phase  transitions,  yet  the  cost  of exploring  the  largest  cluster  never  exceeds \nN  steps,  Exponential  search  cost  in  k-SAT  comes  from  the  random  signs  of the \ninputs,  which  require  that the space  be searched  repeatedly.  Note that a  satisfying \n\n\f446 \n\nKirkpatrick, Gyorgyi, TIshby, and Troyansky \n\ninput  configuration  in  2-SAT  can  be  determined,  or  its  non-existence  proven,  in \npolynomial time,  because  it  can  be  reduced  to  a  percolation  problem on a  random \ndirected  graph  (Aspvall  1979).  The  spin  glass  Hamiltonians  studied  by  Fu  and \nAnderson  have  a  form  close  to our  2-SAT formulae,  but  the  questions  studied  are \ndifferent.  Finding  an  input  configuration  which  falsifies  the  minimum number  of \nclauses  is  like  finding  the ground state in  a  spin glass  phase,  and is  NP-hard  when \na  > a c ,  even  for  k =  2.  Therefore,  if both  diverging  correlations  (diverging in size \nif no  lengths  are  defined)  and  random  sign  or  \"spin-glass\"  effects  are  present,  we \nexpect  a  local  search  like  Davis-Putnam  to  be  exponentially  difficult  on  average. \nBut these  characteristics  do  not imply NP-completeness. \n\n7  References \n\nAspvall,  B.,  Plass,  M.F.,  and Tarjan,  R.E.  (1979)  A linear-time  algorithm  for  testing  the \ntruth of certain quantified  Boolean formulae.  Inform.  Process.  Let., Vol.  8.,  1979, \n289-314. \n\nBarker,  V.  E.,  and  O'Connor,  D.  (1989).  Commun.  Assoc.  for  Computing  Machinery, \n\n32(3),  1989,  298-318. \n\nCheeseman,  P.,  Kanefsky,  B.,  and  Taylor,  W.M.  (1991).  Where  the really  hard  problems \n\nare.  Proceedings  IJCAI-91,  1991,  163-169. \n\nClearwater,  S.H.,  Huberman  B.A.,  Hogg,  T.  (1991)  Cooperative  Solution  of  Constraint \n\nSatisfaction  Problems.  Science,  Vol.  254,  1991,  1181-1183 \n\nCrawford,  J.M.  and  Auton  L.D.  (1993).  Experimental  Results  on  the  Crossover  Point in \n\nSatisfiability  Problems.  Proc.  of AAAI-99, 1993. \n\nChvatal,  V.  and  Reed,  B.  (1992)  Mick  Gets  Some:  The  Odds  are  on  his  Side.  Proc.  of \n\nSTOC,  1992,  620-627. \n\nFu,  Y.  (1989).  The  Uses  and  Abuses  of  Statistical  Mechanics  in  Computational  Com(cid:173)\n\nin  Lectures  in  the  Sciences  of Complexity,  ed.  D.  Stein,  pp.  815-826, \n\nplexity. \nAddison-Wesley,  1989. \n\nFranco,  J.  and  Paull,  M.  (1988).  Probabilistic  Analysis  of the  Davis-Putnam  Procedure \nfor solving  the Satisfiability  Problem.  Discrete Applied Math., Vol.  5,  77-87,  1983. \nGoerdt,  A.  (1992).  A  threshold  for  unsatisfiability.  Proc.  17th  Int.  Symp.  on  the  Math. \n\nFoundations of Compo  Sc.,  Prague,  Czechoslovakia,  1992. \n\nKirkpatrick,  S.  and  Swendsen,  R.H.  (1985).  Statistical  Mechanics  and  Disordered  Sys(cid:173)\n\ntems.  CA CM,  Vol.  28,  1985,  363-373. \n\nKirkpatrick,  S.,  and Selman,  B.  (1993),  submitted for  publication. \nLarrabee,  T.  and Tsuji,  Y.  (1993)  Evidence  for  a  Satisfiability  Threshold  for  Random \n3CNF Formulas,  Proc.  of the AAAI Spring Symposium on AI and NP-hard prob(cid:173)\nlems,  Palto Alto,  CA,  1993. \n\nMezard,  M.,  Parisi, G., Virasoro, M.A.  (1986).  Spin Glass Theory and Beyond, Singapore: \n\nWorld  Scientific,  1986. \n\nMitchell,  D.,  Selman,  B.,  and  Levesque,  H.J.  (1992)  Hard  and  Easy  Distributions  of SAT \n\nproblems.  Proc.  of AAAI-92, 1992,  456-465. \n\nStauffer,  D.  and  Aharony,  A.  (1992)  Introduction to  Percolation  Theory.  London:  Taylor \n\nand  Francis,  1992.  See  especially  Ch.  4. \n\n\f", "award": [], "sourceid": 737, "authors": [{"given_name": "Scott", "family_name": "Kirkpatrick", "institution": null}, {"given_name": "G\u00e9za", "family_name": "Gy\u00f6rgyi", "institution": null}, {"given_name": "Naftali", "family_name": "Tishby", "institution": null}, {"given_name": "Lidror", "family_name": "Troyansky", "institution": null}]}