Efficient SAT Solving: Beyond Supercubes

1
Efficient SAT SolvingBeyond Supercubes

• Domagoj Babic, Jesse Bingham, Alan J. Hu
• Department of Computer Science
• University of British Columbia

2
Outline

• Motivation and Background
• Moving Beyond Supercubing
• Naïve Algorithm
• Standard Approach
• Supercubing
• B-Cubing
• Approximating B-Cubing
• Experimental Results

3
Motivation

• SAT has become dominant Boolean reasoning engine
  for EDA applications. Therefore, we need
  efficient SAT solvers.
• SAT is NP-complete, so unlikely that one approach
  will always work well. Therefore, we should
  explore a variety of different, but efficient
  approaches.
• Supercubing Goldberg, Prasad, Brayton 2002 is a
  fresh theoretical insight. We have been working
  on making supercubing-style SAT solvers
  performance-competitive with state-of-the-art
  standard solvers.

4
Background

• SAT Boolean Satisfiability
• Given a formula in CNF (product of sums), find a
  satisfying assignment, or determine that none
  exists.

Negative Literal
Positive Literal
Clause
5
Naïve Algorithm

• Try assigning values to variables one-by-one.
• (Propagate unit clauses.)
• Backtrack when a conflict is found.
• Visualize as a search tree.

6
Naïve Algorithm
a
7
Naïve Algorithm
a
b
8
Naïve Algorithm
a
b
c
9
Naïve Algorithm
d
10
Naïve Algorithm
d
e
f
11
Naïve Algorithm
d
e
f
x implied. Conflict!
12
Naïve Algorithm
d
e
f
x implied. Conflict!
x implied. Conflict!
13
Standard Algorithm

• Breakthrough for SAT solving, pioneered by
  leading SAT solvers (e.g., GRASP Marques-Silva
  and Sakallah, 99, CHAFF Moskewicz et al.,
  00, etc.)
• Analyze conflict which decisions really
  mattered?
• Backtrack non-chronologically.
• Learn a new clause to prevent same conflict from
  reoccurring.

14
Standard Algorithm
d
e
f
15
Standard Algorithm
d
e
f
16
Supercubing

• Fresh theoretical insight to SAT-solvingGoldberg
  , Prasad, Brayton, 02
• Demonstrated reduction in number of
  decisions,but did not achieve speedup over
  standard solvers.
• When exploring one subtree under literal x, if
  all conflicts involving x also contain literal y,
  then we can immediately assign literal y after
  flipping x.

17
Supercubing
a
b
c
18
Supercubing
a
b
c
d
Pruned
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Why Supercubing Works

• Complement of conflict clause is a cube.
• Cubes for part of search tree collectively
  certify that that part is unsatisfiable.
• Supercube is a coarse overapproximation of
  remaining search space not covered by conflict
  cubes.

20
B-Cubing

• Generalization of Supercubing
• Ideally, remember all information derived from
  conflicts, not just an over-approximated cube.
• In practice, we must prune the informationkeep
  more than just a cube, but not too much.
• Must be implemented efficiently. The savings
  from exploiting the information must outweigh the
  overhead of maintaining it.

21
B-Cubing (Beyond Cubes)
a
b
c
d
e
f
f
Pruned
Pruned
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B-Cubing Theory
x
Conflict cubes that involve x. Conflict cubes
that dont involve x.
Any over-approximation is safe!
(Doing this recursively is tricky!)
23
Approximating B-Cubing

• Any over-approximation of ideal B-cubing
  information is correct.
• Supercube is smallest over-approximate cube.
• We introduce Boolean Constraint Trees to
  summarize a set of conflict cubes
• Basically a decision tree
• Heuristics to maximize search-space pruning
• Different variable order allowed on different
  paths

f
d
e
!f
24
Experimental Results

• B-Cubing obviously will reduce number of
  decisions, but is the overhead excessive?
• We have implemented B-cubing in an experimental
  solver.
• Integration with learning is tricky. See Babic,
  Hu, ASPDAC05 for combining supercubing with
  learning.
• Run on an extensive array of benchmarks.
• Compared against ZChaff II, one of the leading
  solvers using the standard approach.

25
PicoJava BMC
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IBM BMC
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CMU BMC
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FPGA Routing SAT
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FPGA Routing UnSAT
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Integer Factorization
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Constraint Satisfaction
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IBM FV Benchmark Library
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Conclusion and Future Work

• Generalized supercubing-style solvers beyond
  cubes.
• Preliminary implementation competitive with
  top-notch standard solver.
• Sometimes much better/worse.
• Important to have alternative solvers.
• Enormous range of possibilities
• Supercubing coarse approx., easier, little
  memory
• JeruSAT Nadel, 02 all cubes within window
• B-Cubing general framework, BCTs provide good
  balance between accuracy and speed?
• Good engineering and implementation