Recovering and Exploiting Structural Knowledge from CNF Formulas

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Recovering and Exploiting Structural
Knowledge from CNF Formulas

• Richard Ostrowski, Eric Gregoire, Bertrand
  Mazure, and Lakhdar Sais

Presenter Ashique
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What the paper talks about

• The motivation behind the research on efficient
  SAT solvers
• Useful structural knowledge can be lost due to
  CNF representation
• Extracting gates out of the CNF formula is a step
  to recover that underlying structure
• Graphical representation of clauses facilitates
  such extraction
• Other techniques that minimizes the number of
  Gates
• Techniques that eliminate clauses and variables

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Couple of definitions

• An equation or gate is of the form y f( x1 ,x2
  ,x3 ,xn) where f is a standard connective
  among ?, ?, ? and where y and xi are
  propositional variables.
• y is called the output variable (definable) and
  xi s are called input variables.
• A propositional variable z is an output variable
  for a set of gates iff z is an output variable of
  at least one gate in the set.
• An input variable for a set of gates is an input
  variable of a gate which is not an output
  variable of the set of gates.

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Graph of clauses
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Partial graph of clauses
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Extraction of Æ and Ç gates
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Computational complexity

• Building a graph is quadratic in the size of the
  set of the clauses ?
• Representation of the graph is space consuming
• A dynamic approach without representing the graph
  explicitly is taken

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Exploiting structure knowledge

• Once the gates are extracted various properties
  are used to minimize the number of gates,
  simplify them and reduce the number of clauses
  and variables
• All these reductions prune the search space

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Properties of , gates

• , is commutative and associative
• (a , a , B) is equivalent to B
• (a , b , c) is equivalent to ( a , b , c)
• ( a , b , c) is equivalent to ( a , b , c)
• (l , A1), (l , A2), (l , Am) is SAT iff (A1 ,
  A2) (Am-1 , Am) is SAT
• Let ? be a set of gates, B ½ ? a set of
  equivalence gates, b 2 B such that its output
  variable y occurs only in B and ? the set of
  gates obtained by the substitution of y with its
  definition and removing b from ? , then ? is
  satisfiable is ? is satisfiable
• Let S be a set of gates, any equivalence gate of
  S containing a literal which does not occur
  elsewhere in S, can be removed from S without
  loss of satisfiability.

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Properties Æ and Ç gates

• a f(b, c, b) with f ? ?, ? is equivalent to a
  f(b, c)
• a ?(b, c, b) (resp. a ?(b, c, b)) is
  equivalent to a (resp. a)
• a ?(b, c, d) (resp. a ?(b, c, d)) is
  equivalent to a ?(b, c, d)

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Simplication of remaining clause sets

• Blocked clause A clause c of a CNF formula S is
  blocked iff there is a literal l ? c such that
  for all c' ? S with l ? c' the resolvent of c
  and c' is tautological.
• Let c be a clause belonging to a CNF formula S
  such that c is blocked. S is satisfiable iff
  S\c is satisfiable.

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nf-blocked clause

• A clause c belonging to a CNF formula S is
  non-fundamental iff c is either tautological or
  is subsumed by another clause from S.
• A clause c belonging to a CNF formula S is
  nf-blocked iff there exists a literal l from c
  such that. there does not exist any resolvent in
  l, or such that all resolvents are not
  fundamental.
• Let c be a clause belonging to a CNF formula S
  s.t. c is nf-blocked. S is satifiable iff S\c
  is satisfiable.

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nf-blocked example continued

• Blocked clauses and clauses containing a pure
  literal are nf-blocked

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Variable elimination using nf-blocked clause

• Any clause c from a CNF formula S can be
  nf-blocked, introducing additional clauses in S.
  Transitive closure of two-literal clauses
• In order to eliminate a variable, we just need to
  nf-block all clauses where it occurs

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Some more reduction techniques

• A clause c belonging to a CNF formula ? is
  redundant iff ? \c ² c.
• A clause c from a CNF formula S is u-redundant
  iff the unsatisfiability of
• S ? c can be obtained using unit
  propagation.
• Let ? be a CNF formula, a subsuming resolvent is
  a resolvent from two clauses from ? that subsumes
  at least one clause of ? .

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How costly is it?

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Performance vis-à-vis other algorithms

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The reason behind such performance

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Future directions

• Branching heuristics that also take all the
  equations into account
• Exploiting the intrinsic property of each type of
  equation
• Simplification of Boolean formulas