1. Inference and search for the propositional satisfiability problem
Lyndon Drake
University of York
July 2003
Swansea University

2. Inference and search for SAT
Introduction
Propositional satisfiability (SAT)
Theoretical interest
Archetypal NP-complete problem (Cook 1971)
Practical applications
Hardware verification
Often solved using a search procedure, which can be augmented with inference
July 2003
Inference and search for SAT

3. Inference and search for SAT
Outline
Overview of SAT and my thesis
Neighbour resolution during search
Preprocessing techniques
Interaction between inference and branching heuristics
Conclusions
July 2003
Inference and search for SAT

4. Satisfiable SAT instance
Example:
Alternative form:
Satisfying assignment:
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Inference and search for SAT

5. Inference and search for SAT
CNF SAT instances
Boolean variable
Literal
Positive or negative occurrence of a variable
Clause
Disjunctions of literals
Formula
Conjunction of clauses
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Inference and search for SAT

6. Inference and search for SAT
Assignments
Mapping from variables to values
Can be partial
Applying an assignment to a formula modifies the formula
Deletes false literals
Deletes true clauses
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Inference and search for SAT

7. Example application: BMC
Bounded model checking (Biere 1999)
Used for verification, especially of hardware circuit designs
Model encodes the circuit design
Specification of necessary properties
Map both model and specification into a SAT formula
If a satisfying assignment exists, there is probably a bug in the model
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Inference and search for SAT

8. Solving SAT: resolution
The Davis-Putnam method (1960)
Uses resolution to eliminate variables
Resolution is intractable
E.g. best case exponential space complexity on pigeon-hole problems
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Inference and search for SAT

9. Solving SAT: backtracking searchF c c a a a d d d
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Inference and search for SAT

10. Combining inference and search : Unit propagation
Given:
We know that:
We must assign c true in order for the clause to be satisfied
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Inference and search for SAT

11. Effect of unit propagationb T F a c c d d
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Inference and search for SAT

12. Inference and search for SAT
Unit propagation
During search, if a clause contains exactly one unassigned literal, that literal must be assigned true if a satisfying assignment exists
Unit propagations typically outnumber branches by 3:1
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Inference and search for SAT

13. Evaluating unit propagation
Unit propagation is an effective inference technique to add to search
Benefit: Substantial pruning
Cost: Performing simple reasoning
Implementation details have a massive impact on performance
July 2003
Inference and search for SAT

14. Inference and search for SAT
Thesis
The performance of backtracking search-based SAT solvers can be improved by adding more inference to the search procedure
The effect of inference depends on:
Branching heuristic
Other inference in the search procedure
Problem class
July 2003
Inference and search for SAT

15. Evaluating inference techniques
We need to consider:
Benefit: How much the search space will be pruned
Cost: How much time and space the inference technique requires
Implementation details cannot be avoided
July 2003
Inference and search for SAT

16. Inference and search for SAT
Outline
Overview of SAT and my thesis
Neighbour resolution during search
Preprocessing techniques
Interaction between inference and branching heuristics
Conclusions
July 2003
Inference and search for SAT

17. Inference and search for SAT
Neighbour resolution
July 2003
Inference and search for SAT

18. Neighbour resolution during search
Two clauses are neighbours if:
All but one of the literals are identical
The differing literal is a variable which appears positively in one clause and negatively in the other
At each decision node in the search space, do all neighbour resolutions
July 2003
Inference and search for SAT

19. Inference and search for SAT
Results
On many problems, the search space is significantly pruned
Identifying the neighbouring clauses to resolve on takes a long time
The time cost outweighs the benefit
This implementation of neighbour resolution during search is not an effective technique for improving performance
July 2003
Inference and search for SAT

20. Preprocessing neighbour resolution
Binary resolution before search produces the same resolvents as neighbour resolution during search
Much cheaper to do a single stage of resolution before search
Less pruning, as no subsumption is carried out
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Inference and search for SAT

21. Inference and search for SAT
Results: pruning
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Inference and search for SAT

22. Inference and search for SAT
Results: runtime
July 2003
Inference and search for SAT

23. Neighbour resolution: Conclusions
Neighbour resolution during search
Can result in substantial pruning
In its current implementation is too expensive to be effective
Preprocessing resolution
Can be effective
July 2003
Inference and search for SAT

24. Inference and search for SAT
Outline
Overview of SAT and my thesis
Neighbour resolution during search
Preprocessing techniques
Interaction between inference and branching heuristics
Conclusions
July 2003
Inference and search for SAT

25. Inference and search for SAT
Preprocessing
Inference before search has two main advantages
Implementation is simpler
Conflict recording can be used during search
July 2003
Inference and search for SAT

26. Two preprocessing techniques
Length-bounded resolution
Binary equivalence finding
Effective when possible, but rarely possible
July 2003
Inference and search for SAT

27. Length-bounded resolution
Short clauses are more likely to unit propagate
Can perform resolution to closure on the binary clauses in the formula
During search (van Gelder 2001)
Can do a single stage of resolution with the resolvent limited to at most three literals
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Inference and search for SAT

28. Binary equivalence finding
Given a  b and b  a
We can replace all occurrences of b with a
Delete duplicate literals (a  a)
Delete tautologies (a  a)
Simplifies the formula
Low cost inference
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Inference and search for SAT

29. Binary equivalence chains
Can find loops of binary implications
Tarjan’s SCC algorithm finds these equivalence chains in linear time a b c e d e f
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Inference and search for SAT

30. Two preprocessing techniques: Conclusions
Both methods are cheap to apply
Both can result in significant pruning
Length-bounded resolution
Can generate too many clauses
Binary equivalence finding
Rare to find many before search
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Inference and search for SAT

31. Hyper-binary resolution
HypBinRes for short
Resolution rule first defined by Bacchus (2002)
Generates only binary clauses
Binary clauses are likely to propagate
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Inference and search for SAT

32. Inference and search for SAT
HypBinRes example
From:
Infer:
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Inference and search for SAT

33. HypBinRes preprocessing
Apply HypBinRes before search
HypBinRes during search is effective
HypBinRes can add support clauses to a directly encoded CSP
Expectation is that HypBinRes preprocessing will result in improved performance
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Inference and search for SAT

34. Inference and search for SAT
Results
Runtime in seconds, before and after preprocessing
Points are all on the line
No real improvement in performance
July 2003
Inference and search for SAT

35. Inference and search for SAT
Results
Tested on several benchmark sets
Using ZChaff as the solver
Efficient preprocessing algorithm
No significant runtime required
No significant performance improvement
Often much worse after preprocessing
Search space correspondingly larger
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Inference and search for SAT

36. Preprocessing: Conclusions
Many preprocessing techniques are generally fast enough to apply
Some have nice worst-case bounds
Their effectiveness in improving performance depends on the problem class
July 2003
Inference and search for SAT

37. Inference and search for SAT
Outline
Overview of SAT and my thesis
Neighbour resolution during search
Preprocessing techniques
Interaction between inference and branching heuristics
Conclusions
July 2003
Inference and search for SAT

38. Interaction between inference and heuristics
HypBinRes preprocessing
Adding implied clauses can cause poor decisions by the branching heuristic
When implied clauses added by HypBinRes improve performance, they do so by guiding the heuristic
July 2003
Inference and search for SAT

39. Inference and search for SAT
Solver 1: ZChaff
Runtime in seconds, before and after preprocessing
Points are all on the line
No real improvement in performance
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Inference and search for SAT

40. Inference and search for SAT
Analysis
ZChaff uses:
VSIDS heuristic
Conflict recording
Additional clauses could result in a larger search space because:
Heuristic is confused
Different conflict clauses generated
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Inference and search for SAT

41. Inference and search for SAT
Solver 2: Swan
Lots of points below the line
Nice performance improvement
July 2003
Inference and search for SAT

42. Inference and search for SAT
Analysis
Swan uses:
Static version of the VSIDS heuristic
No conflict recording
Intuition for poor ZChaff results:
Heuristic at fault
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Inference and search for SAT

43. Adding implied clauses
Before preproc
After preproc
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Inference and search for SAT

44. Inference and search for SAT
Conflict recording
Before search
During search
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Inference and search for SAT

45. Inference and search for SAT
VSIDS heuristic
Distinguishes between clauses in the formula before search and those added by conflict recording
Gives higher priority to literals in conflict clauses
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Inference and search for SAT

46. Isolating implied clauses
Tag the implied clauses
Optionally prevent the heuristic counting literals in the implied clauses
Conflict recording is complicated, so done in Swan
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Inference and search for SAT

47. Results: implied clauses visible
Lots of points below the line
Nice performance improvement
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Inference and search for SAT

48. Results: implied clauses hidden
No performance improvement
These results look very similar to those obtained in ZChaff
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Inference and search for SAT

49. Implied clauses visible Implied clauses hidden
Results
Implied clauses visible
Implied clauses hidden
50% improvement
No improvement
July 2003
Inference and search for SAT

50. Inference and search for SAT
Analysis
Intuition at least partially correct
Heuristic strongly affected by implied clause visibility
Heuristic operates best when implied clauses are visible
VSIDS will discount literals in the implied clauses
July 2003
Inference and search for SAT

51. Inference and search for SAT
Pruning or guiding?
Do implied clauses prune branches?
Modified solver that gives statistics on branch pruning
Implied clauses rarely prune
Extra conflict clauses will not be generated
Pruned branches are rarely large
Need to do more experiments
July 2003
Inference and search for SAT

52. Heuristic interaction: Conclusions
Adding implied clauses can result in the branching heuristic making poor decisions
Tentative: implied clauses improve performance primarily by guiding the heuristic
July 2003
Inference and search for SAT

53. Inference and search for SAT
Outline
Overview of SAT and my thesis
Neighbour resolution during search
Preprocessing techniques
Interaction between inference and branching heuristics
Conclusions
July 2003
Inference and search for SAT

54. Inference and search for SAT
Future work
Extending the work on interaction between inference and heuristics to other inference techniques
The interaction between inference and conflict recording
July 2003
Inference and search for SAT

55. Inference and search for SAT
Conclusions
Neighbour resolution is effective as a preprocessing method, but not during search
Other preprocessing methods can also be effective
Effectiveness depends strongly on interactions with the branching heuristic
July 2003
Inference and search for SAT