Solving%20Non-clausal%20Formulas%20with%20DPLL%20search

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Solving Non-clausal Formulas with DPLL search

• Christian Thiffault Fahiem Bacchus
• University of Toronto

Toby Walsh UNSW
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CNF

• The classical DPLL algorithm (most modern SAT
  solvers) work with Conjunctive Normal Form (CNF)
• However, CNF is not the most natural
  representation.
• Hence, to use modern SAT solver technology
  problems must be converted to CNF
• In this work we demonstrate that such a
  conversion is unnecessary.
• For a number of reasons such a conversion is also
  undesirable.
• We demonstrate that original structure lost in
  conversion can be utilized to significantly
  improve the performance of SAT solvers.

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Tseitin Encodings

• The most commonly used CNF encoding is the
  Tseitin encoding Tseitin 1970.
• This encoding converts a propositional formula
  recursively by adding a new variable for every
  subformula.

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Tseitin Encodings

• A ? (C D)
• V1 (C D)
• (V1, C), (V1, D), (C,D,V1)

(V1, C) (V1, D) (C,D,V1)
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Tseitin Encodings

• A ? (C D)
• V1 (C D)
• (V1, C), (V1, D), (C,D,V1)
• V2 (A ? V1)
• (V2, A,V1), (A,V2), (V1,V2)

(V1, C) (V1, D) (C,D,V1) 4. (V2, A,V1) 5. (A,V2) 6. (V1,V2)
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Tseitin Encodings

• A ? (C D)
• V1 (C D)
• (V1, C), (V1, D), (C,D,V1)
• V2 (A ? V1)
• (V2, A,V1), (A,V2), (V1,V2)
• The formula must be true (V2)

(V1, C) (V1, D) (C,D,V1) 4. (V2, A,V1) 5. (A,V2) 6. (V1,V2) 7. (V2)
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Converting to CNF is Undesirable

• There are two obvious problems arising from this
  conversion.
• Structural information is lost
• The fact that a particular group of clauses were
  generated from the same subformula is lost.
• The global interconnections between the various
  subformulas are lost.
• Additional variables are added. This potentially
  increases the size of the DPLL search.

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Structural Information

• Some of this information could be recovered from
  the CNF encoding, but not all of it Lang
  Marquis, 1989.
• There is good empirical evidence that recovering
  structural information can yield considerable
  benefits in solving performance. EqSatZ, LSAT.
• But why lose this information in the first place?
  
• We will show conversion to CNF is not necessary.
• We also develop techniques which confirm the
  benefits of retaining the original structure.

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Extra Variables

• Various works have noted the potential difficulty
  of introducing new variables.
• Some have suggested restricting the DPLL search
  so that it cannot branch on any on the newly
  introduced subformula variables.
• However, it is not difficult to show that
  restricting branching in this manner causes an
  exponential slowdown on some problems Jarvisalo
  et al. 2004
• Solvers that attempt to restrict their branching
  in this manner can perform very poorly on many
  problems.

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Extra Variables

• The alternative is unrestricted branching.
• However, with unrestricted branching a CNF solver
  can waste a lot of time branching on variables
  that have dynamically become irrelevant.

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Irrelevant Variables
AFalse. Formula satisfied

• A ? (C D)

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Irrelevant Variables CNF

• A ? (C D)
• V1 (C D)
• V2 (A ? V1)

Solver must still determine that the remaining
clauses are SAT
(V1, C) (V1, D) (C,D,V1) 4. (V2, A,V1) 5. (A,V2) 6. (V1,V2) 7. (V2) 8. (A)
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Converting to CNF is Unnecessary

• DPLL search can be performed on the original
  formula.
• This has been noted in previous work on circuit
  based solvers Ganai et al. 2002

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DPLL on formulas

• Convert formula to a dag.
• A ? (C D) \/ B ? (C D)

\/
?
?
B
A

C
D
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DPLL on formulas

• Associate a truth value (True/ False/ Unassigned)
  with every node of the DAG.
• Now perform DPLL search by splitting on the truth
  value of an unassigned node.
• Use the logic of the boolean operators to
  propagate truth values to neighboring nodes.
• A solution is found when all nodes have been
  labeled with consistent truth values (i.e., the
  parent and child values are consistent with the
  logic of the parents operator.)
• A contradiction occurs when True and False are
  propagated to the same node.

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Example
\/
?
?
B
A

C
D
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Example
\/
?
?
B
A

C
D
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Example
\/
?
?
B
A

C
D
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Comparison

• Assigning a truth value to a node in the DAG is
  equivalent to assigning a truth value to the
  corresponding variable in the CNF encoding.
• Propagation in the DAG is equivalent to unit
  propagation in the CNF encoding.

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Comparison

• The rule that propagated a truth value can be
  recorded. Once a contradiction is detected a
  conflict clause can be learned (a set of
  impossible node assignments). This clause can be
  made equivalent to the 1-UIP clauses learned by
  CNF solvers.
• The learned clauses can be stored and used to
  unit propagate node truth values in the rest of
  the search.

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Comparison

• Complex gates, e.g., n-ary XOR can be more
  efficiently propagated in the DAG representation
  (these gates require a number of clauses
  exponential in the number of their inputs).

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Efficient Propagation

• Efficient Unit Propagation via lazy data
  structures is a key element of modern SAT
  solvers.
• One new feature of our circuit based solver over
  previous circuit solvers is that it utilizes
  similar lazy data structures to make its
  propagation as efficient as CNF solvers.

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Efficient Propagation

• E.g., an AND gate becomes true only when all of
  its children become true.
• Assign one child as a true watch, and dont check
  for true propagating to the AND gate node unless
  its true watch becomes true.
• Some benchmarks have AND gates with thousands of
  children.
• Previous table-lookup circuit solvers required
  some computation each time a child became true.
• Using these techniques there is no intrinsic loss
  of efficiency in using the DAG over CNF.

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Structure Based Optimizations

• Since no penalty is being paid for using the
  original formula, we can turn to exploiting the
  extra structural information it provides.
• We implemented two structure based optimizations.
• Dont care propagation to deal with irrelevant
  variables.
• Conflict Clause reduction.

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Dont Care Propagation

• We propagate a third truth value through the
  DAG dont cares.
• A node C is dont care wrt a particular parent P
• if its truth value can no longer affect the truth
  value of P nor any of its P siblings.
• or P is dont care.
• A node C is dont care if it is dont care wrt to
  all of its parents.

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Dont Care Propagation

• Assign a dont care watch parent for each node.
• C becoming dont care wrt to its watch parent P
  can be detected when truth values are propagated
  to P.
• If C becomes dont care wrt to its dont care
  watch we look for another watch.
• If we cant find one we know that C has become
  dont care.
• Finally, we stop the search from branching on
  dont care nodes. This eliminates branching of
  irrelevant values.

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Conflict Clause Reductions

• If one learns a conflict clause (L1,L2,...) and
  one has L1 ? L2, then we can reduce the conflict
  clause by a resolution step
• (-L1,L2) (L1,L2,...) ? (L2,...)
• The resolvant subsumes the original conflict
  clause.
• In a CNF encoding one would have to search the
  clauses to detect this situationprobably not
  going to be effective.
• In the DAG one can examine the neighbors of each
  node assignment in the conflict clause to see if
  any of the other assignments in the clause are
  implied by it.
• If so we can remove the implying node assignment.
• If this is done with discrimination it can yield
  a significant decrease in the size of the learned
  clauses with little overhead.

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Empirical Results.

• We compared with Zchaff.
• Tried to make the two solvers as close as
  possible, same magic numbers (e.g., clause
  database cleanup criteria, restart intervals
  etc.), same branching heuristics.
• Hence, we tried to isolate the impact of the
  non-clausal representation and the structure
  based optimizations.
• We believe that the same improvements could be
  obtained with others CNF solvers via this
  technique.

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Empirical Results caveats

• Our experiments were hampered by a lack of
  non-clausal benchmarks.
• The performance of our solver was also limited by
  the fact that the benchmarks we did obtain has
  already been transformed into simpler formulas,
  e.g., no complex XOR of IFF gates were present in
  the benchmarks.

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FVP-UNSAT-2.0 (Velev) Time
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FVP-UNSAT-2.0 Decisions
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FVP-UNSAT-2.0 Dont Cares
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FVP-UNSAT-2.0 Clause Reduction
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Other Series
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Conclusions

• No intrinsic reason to convert to CNF.
• Many other structure based optimizations remain
  to be investigated, e.g.
• better branching
• non-clausal representation of conflicts
• more complex gates.