State-of-the-art in SAT solvers

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State-of-the-art in SAT solvers

• Vibhav Gogate

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SAT formulas

• A set of propositional variables and clauses
  involving variables
• (x1x2x3) and (x2x1x4)
• x1,x2, x3 and x4 are variables (True or false)
• Literals Variable and its negation
• x1 and x1
• A clause is satisfied if one of the literals is
  true
• x1true satisfies clause 1
• x1false satisfies clause 2
• Solution An assignment that satisfies all clauses

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SAT solvers

• Given 10 minutes of time
• Started with DPLL (1962)
• Able to solve 10-15 variable problems
• Satz (Chu Min Li, 1995)
• Able to solve some 1000 variable problems
• Chaff (Malik et al., 2001)
• Intelligently hacked DPLL , Won the 2004
  competition
• Able to solve some 10000 variable problems
• Current state-of-the-art
• Minisat and SATELITEGTI (Chalmers university,
  2004-2006)
• Jerusat and Haifasat (Intel Haifa, 2002)
• Ace (UCLA, 2004-2006)

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DPLL Example
p,r,?p,?q,r,p,?r
pT
pF
T,r,?T,?q,r,T,?r
F,r,?F,?q,r,F,?r
SIMPLIFY
SIMPLIFY
?q,r
r,?r
SIMPLIFY

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DPLL Algorithm as seen by SAT solver

• While (1)
• if (decide_next_branch()) //1. Branching
• while (deduce()conflict) //2. Deducing
• blevelanalyze_conflicts() // 3. Learning
• if (blevel lt 0)
• return UNSAT
• else backtrack(blevel) // 4. Backtracking
• else RETURN UNSAT

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Chaff implementation

• While (1)
• if (decide_next_branch()) //1. Branching
• while (deduce()conflict) //2. Deducing
• blevelanalyze_conflicts() // 3. Learning
• if (blevel lt 0)
• return UNSAT
• else backtrack(blevel) // 4. Backtracking
• else RETURN UNSAT

Use conflict-directed backjumping Learning
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Learning

• Adding information about the instance into the
  solution process without changing the
  satisfiability of the problem.
• In CNF representation it is accomplished by
  adding clauses into the clause database
• Knowledge of failure may help search in other
  spaces
• Learning is very effective in pruning the search
  space for structured problems
• It is of limited use for random instances
• Why? Still an open question

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Chaff implementation

• While (1)
• if (decide_next_branch()) //1. Branching
• while (deduce()conflict) //2. Deducing
• blevelanalyze_conflicts() // 3. Learning
• if (blevel lt 0)
• return UNSAT
• else backtrack(blevel) // 4. Backtracking
• else RETURN UNSAT

Boolean constraint propagation Main factor
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Naive Implementation of Deduce or Unit propagation

• Check every clause after an assignment is made
  and reduce it if possible
• Repeat if a unit clause is generated
  (implication)
• After backtrack, revert all clauses to their
  original form as they were before.
• Very slow.
• A solver would spend 85-90 of the time doing
  unit propagation
• Why not speed it up?

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Chaff implementation

• While (1)
• if (decide_next_branch()) //1. Branching
• while (deduce()conflict) //2. Deducing
• blevelanalyze_conflicts() // 3. Learning
• if (blevel lt 0)
• return UNSAT
• else backtrack(blevel) // 4. Backtracking
• else RETURN UNSAT

Variable ordering heuristics
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Other issues

• Clause Deletion
• Learned clauses slows down the bcp and eat up
  memory
• Delete clauses periodically
• Various heuristics for this purpose
• Winner of SAT competition 2004.

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But Chaff no longer State of the Art

• More hacking
• Minisat (2006, winner of SAT RACE 2006)
• Based on chaff but a better faster implementation
• Some new things like conflict analysis and
  minimization but basically same as chaff

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Benchmarks

• Random
• Crafted
• Industrial

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SAT race 2006
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SAT race 2006