Satisfiability Checking of Nonclausal Formulas using General Matings

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Satisfiability Checking of Non-clausal Formulas
using General Matings

• Himanshu Jain
• Constantinos Bartzis
• Edmund Clarke

Carnegie Mellon University
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Boolean Satisfiability

• The first problem to be proven NP-complete
• Applications in verification
• equivalence checking
• bounded model checking
• predicate abstraction
• theorem proving
• test generation
• Applications in AI
• planning

3
Current state of SAT solving

• State-of-the-art complete SAT solvers
• Davis-Putnam-Logemann-Loveland (DPLL) algorithm
• Require the input formula to be in clausal form
  (CNF)
• MiniSat, BerkMin, Siege, zChaff, Limmat, GRASP,
  SATO
• Conversion to CNF by adding new variables
• Linear size but exponential state-space
• In practice does not seem to hurt

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Non-Clausal (non-cnf) SAT solving

• DPLL on circuit representation of formula
• Ganai et al. (DAC 2002)
• Lu et al. (CSAT, DAC 2003)
• Thiffault et al. (NoClause, SAT 2004)

This work Non-clausal SAT-solver based on
DPLL
General Matings
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Outline

• Introduction
• General Matings
• Search space pruning
• Learning
• Non-chronological backtracking
• Experimental results

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General Matings example
Formula F (((p ? q) ? ?r ? ?q) ? (?p ? (r ? ?s)
? q))
Vertical path
?
?
?
Vertical path form (vpgraph) of F
Each vertical path corresponds to a term in the
DNF form of F
F is satisfiable iff there exists a vertical path
without opposite literals
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Basic search
vpgraph
c
a
Partial solution
-c
b
-a
b
a
-a
-c
partial assignment
Satisfiable!
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Basic Idea

• Given negation normal formula (NNF) F
• Obtain vpgraph of F (O(k2), k F)
• Find vertical path without opposite literals
• Due to P. Andrews , W. Bibel 1981
• Focus on higher order theorem proving
• Quantifier instantiation is the main problem

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Main problem exponentially many vertical paths
in the vpgraph of F
Sample vpgraph
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Our contributions

• Preventing enumeration of vertical paths
• Search space pruning
• Learning
• Non-chronological backtracking

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Outline

• Introduction
• General Matings
• Search space pruning
• Learning
• Non-chronological backtracking
• Experimental results

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Search space pruning on vpgraph
Avoids enumeration of exponentially many paths
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Conflict when pruning vpgraph
Local conflict
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Local learning
Locally learned clause
(?a ? ?b)
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Non-chronological backtracking
a
b
x
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Non-chronological backtracking
a
b
x
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Limitations of local learning

• Learned clause is useful at a particular node
• Can learn same clause multiple times

Need an equivalent of learning in CNF SAT solvers
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Dual of vpgraph hpgraph
Formula F (((p ? q) ? ?r ? ?q) ? (?p ? (r ? ?s)
? q))
horizontal path
p q
p q
?p
?p
r ?s
r ?s
?r
?r
?q
q
?q
q
hpgraph of F
vpgraph of F
Each horizontal path corresponds to a clause in
the CNF representation of F
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Conflicts and implications in hpgraph
Conflict clause ?r ? ?p
Global conflict
Unit clause p ? q ? r ? ?s Implied literal
r
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Conflicts and implications in hpgraph

• Detecting conflict and implications
• Can be done in linear time
• Why use hpgraph
• Globally learned clauses
• Obtain implications efficiently

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Putting vpgraph and hpgraph together
Formula F (((p ? q) ? ?r ? ?q) ? (?p ? (r ? ?s)
? q))
p, ?r
Global Conflict
Unit clauses (implications)
Local conflict ?
Aim to find a vertical path in vpgraph without
opposite literals
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Outline

• Introduction
• General Matings
• Search space pruning
• Learning
• Non-chronological backtracking
• Experimental results

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Experimental results
Bench mark
Prob- lems
Classification theorems for quasigroups Sorge et
al. SAT 2005
Timeout of 10 minutes per problem per solver
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Performance on individual benchmarks
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Conclusion

• SAT solver based on General Matings
• Graphical representations vpgraph, hpgraph
• Preventing enumeration of vertical paths
• Experiments show promise of this technique

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Questions?
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Conversion to CNF

• Results in more complex formula. Either
• Same variables but exponential size
• Linear size but exponential state-space
• Doesnt seem to hurt in practice
• Original formula
• (a ? b) ? (?c ? d) ? (e ? ?f)
• CNF using expansion
• (a ? ?c ? e) ? (a ? ?c ? ?f) ? (a ? d ? e) ? (a ?
  d ? ?f) ?
• (b ? ?c ? e) ? (b ? ?c ? ?f) ? (b ? d ? e) ? (b ?
  d ? ?f)
• CNF using new variables
• (x1 ? x2 ? x3) ?
• (x1 ? ?a ? ?b) ? (?x1 ? a) ? (?x1 ? b) ?
• (x2 ? a ? ?d) ? (?x2 ? ?c) ? (?x2 ? d) ?
• (x3 ? f ? ?e) ? (?x3 ? ?f ) ? (?x3 ? e)

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Using hpgraph for conflict detection
Partial truth assignment rtrue, ptrue
Conflict clause ?r ? ?p
Global conflict
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Using hpgraph to produce implications
Unit clause p ? q ? r ? ?s
Partial truth assignment pfalse, qfalse,
strue
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Obtaining vpgraph

• Constructed directly from a NNF formula
• Time/Space required O(k2)
• k is the size of given formula
• Recently improved to O(k)
• Directed acyclic graph