Methods%20for%20SAT-%20a%20Survey

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Methods for SAT- a Survey

• Robert Glaubius
• CSCE 976
• May 6, 2002

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SAT Outline

• Definitions
• Solving SAT
• Testing SAT
• Attempted Contribution
• Conclusions

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SAT The Propositional Satisfiability Problem

• Given ?, a predicate in CNF, e.g.,
• ? (a1 ? a2) ? (a2 ? a3) ? (a1 ? a2 ? a3)
• Question does a model of ? exist?

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SAT Definitions

• Variable ai
• Literal ? ai or ai
• Clause (?i ? ?j ? ... ? ?n)
• ? (a1 ? a2) ? (a2 ? a3) ? (a1 ? a2 ? a3)

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SAT Outline

• Definitions
• Solving SAT
• Testing SAT
• Attempted Contribution
• Conclusions

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SAT Solution methods

• Systematic search
• SATO Zhang, 1993
• Satz Li and Ambulagan, 1997
• Chaff Moskewicz, et al., 2001
• Stochastic search
• GSAT Selman et al., 1992
• walkSAT Selman et al., 1996

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SAT Systematic Search I

• Davis-Putnam procedure
• ? Satz, Sato, Chaff
• Splitting and unit propagation
• ? ai ? ? and ? ai ? ?

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SAT Systematic Search II

• Splitting and unit propagation
• ? a2 ? (a1 ? a2) ? (a2 ? a3) ? (a1 ? a2 ?
  a3)
• ? unit_propagate(?) ? (a1 ? a3)
• (a1 ? a2), (a2 ? a3) subsumed by a2
• (a1 ? a2 ? a3) ? (a1 ? a3)

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SAT Systematic Search III
Heuristics and Hacks

• SATO Splitting heuristics

• Satz More Splitting heuristics

• Chaff Squeaky clean implementation

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SAT Stochastic Search

• Selection Heuristics mechanisms for variable
  selection
• Random restart
• Restart
• Move
• Walk

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SAT Stochastic Search - GSAT

• procedure GSAT
• begin
• for i 1 to MAX-TRIES
• T a randomly generated truth assignment
• for j 1 to MAX-FLIPS
• if T satisfies ? then return T
• p variable s.t. flip(p) maximizes
  satisfied clauses
• T T after flip(p)
• end for
• end for
• return fail
• end

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SAT Stochastic Search - WalkSAT

• procedure WalkSAT
• begin
• T a randomly generated truth assignment
• for i 1 to MAX-TRIES
• if T satisfies ? then return T
• k random unsatisfied clause
• p variable in k s.t. flip(p) maximizes
  satisfied clauses
• if flip(p) does not unsatisfy any clause
  then T T after flip(p)
• else j a random number in 0,1
• if j gt 0.5 then flip(p)
• else T T after flip(q), where q is
  a random variable in k
• end for
• end

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SAT Outline

• Definitions
• Solving SAT
• Testing SAT
• Attempted Contribution
• Conclusions

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SAT Algorithm Verification

• Benchmark problems planning, verification
• International Competition and Symposium on
  Satisfiability Testing
• SAT2002 Competetion
• Random problems
• naïve random generation
• structured random problems, QCP Selman and
  Gomes, 1997
• QWH, Achlioptas et al., 2000

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SAT Random Generation - Naïve

• Generate n clauses with 1..l variables per clause
  drawn from a pool of k variables.
• Does not guarantee satisfiability
• Requires filtering of instances by slower,
  systematic means

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Digression I Latin Square
Latin Square a configuration of n symbols in n
columns and n rows s.t. each symbol occurs
exactly once in each row and each column.
Rhetorical Question If we had a table that was
only partially filled, can we make it a Latin
Square?
This turns out to be an NP-complete problem
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SAT Random Generation QCP

• Quasigroup completion problem
• Fill some percentage of squares in an n?n matrix
  and try to complete the quasigroup (latin
  square).
• Does offer a structured random problem
• Doesnt guarantee a solution

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SAT Random Generation QWH I

• Quasigroup with Holes
• Start with a randomly generated completed latin
  square, then poke random holes in it.
• Does offer a structured problem
• Does have at least one solution

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SAT Random Generation QWH II

• Improvement balanced QWH
• Instead of random holes, specify a set number for
  each row and column

But how do we make a random latin square?
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Digression II Random Latin Squares
Jacobsen and Matthews (1996) identify a set
of moves that translate an order n Latin Square
into a new Latin square, and prove that these
moves connect the space of all order n Latin
Squares.
Iterated random application of these moves allows
us to randomly select new squares.
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Digression III Encoding the Square I

• Exactly one symbol per cell
• (aij?bij?...?nij)?(aij?bij)?(aij?cij)?...?(mi
  j?nij)
• Exactly one occurrence of s in row r
• (sr1?sr2?...?srn)?(sr1?sr2)?(sr1?sr3)?...?(sr
  n-1?srn)
• Exactly one occurrence of s in column c
• (s1c?s2c?...?snc)?(s1c?s2c)?(s1c?s3c)?...?(sn
  -1c?snc)

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Digression III Encoding the Square II

• Conjoin the appropriate positive literal for each
  preassigned cell
• How much space does this cost us?
• n3 variables
• 1.5n4 - 1 .5n3 3n2 clauses

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SAT Outline

• Definitions
• Solving SAT
• Testing SAT
• Attempted Contribution
• Conclusions

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SAT Return to Stochastic Search

• Premise WalkSAT uses a random start
• What if we can derive us a heuristic that
• can give us a better start frequently?
• Systematic MoMs heuristic Most occurrences in
  clauses of minimum length.
• Another rhetorical question
• Can we adapt this to stochastic search?

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SAT ANP heuristic I

• Given
• ? (a1 ? a2) ? (a2 ? a3) ? (a1 ? a2 ? a3)
• What is the P(ai ? ?)?
• The only way to evaluate this is by finding all
  models.
• Is P(ai ? ?) approximable?

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SAT ANP heuristic II

• Given k (a2 ? a3), we define P(ai ? kj).
• P(a2 True k ) 0.67, since a2 is true in 2
  models of k
• P(a2 True k ) 0.33, since a2 is true in 1
  models of k
• Approximate P(ai ? ?) ? ?j P(ai ? kj )
• Normalize P(ai ? ?) P(ai ? ?) 1

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SAT ANP heuristic III

• (a1 ? a2) ? (a2 ? a3) ? (a1 ? a2 ? a3)
• P(a1 True ?) ? 0.73, P(a1 False ?) ?
  0.27
• P(a2 True ?) ? 0.75, P(a2 False ?) ?
  0.25
• P(a3 True ?) ? 0.40, P(a3 False ?) ?
  0.60

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SAT ANP heuristic IV

• ANP heuristic choose the initial assignment with
  maximum approximate normalized probability
• ? ? a1 True, a2 True, and a3 False
• This assignment is in fact a model of ?

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SAT Experiment design

• Four benchmark problems from the 1996
  International Competition and Symposium on
  Satisfiability Testing
• - hardware verification
• Three order 15 balanced QWH problems
• 3, 4, 5 holes per row, column
• WalkSat solver

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SAT Results

• 100 trial runs per problem
• 3 Initial Assignment types
• - ANP
• - Random
• - Default (all False)

We didnt do so well
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SAT Conclusions

• Based on our initial experiments, stochastic
  solvers are harmed by heuristics.
• Quasigroup-based problems are likely to be the
  exception most variables are false in a model.
• More tests are needed possible domain dependence

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SAT Outline

• Definitions
• Solving SAT
• Testing SAT
• Attempted Contribution
• Conclusions

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Questions, Comments, Insults?
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