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May 27, 2023 184 likes •535 views

DRAT-trim: Efficient Checking and Trimming Using Expressive Clausal Proofs Nathan Wetzler Marijn J.H. Heule Warren A. Hunt, Jr. The University of Texas at Austin Theory and Applications of Satisfiability Solving (SAT)

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DRAT-trim: Efficient Checking and Trimming Using Expressive Clausal Proofs

Nathan Wetzler Marijn J.H. Heule Warren A. Hunt, Jr.

  • The University of Texas at Austin
  • Theory and Applications of Satisfiability Solving (SAT)

Vienna, Austria July 14, 2014

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Nathan Wetzler DRAT-trim: Efficient Checking and Trimming Using Expressive Clausal Proofs / 13 2

Outline

DRAT-trim is an unsatisfiability proof checking utility that can validate all presently-known SAT solving and preprocessing techniques and optionally produce trimmed formulas, optimized proofs, and resolution and dependency graphs.

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Nathan Wetzler DRAT-trim: Efficient Checking and Trimming Using Expressive Clausal Proofs / 13 2

  • Motivation
  • Approach
  • Proof Formats
  • Input, Output, and Applications
  • Evaluation

Outline

DRAT-trim is an unsatisfiability proof checking utility that can validate all presently-known SAT solving and preprocessing techniques and optionally produce trimmed formulas, optimized proofs, and resolution and dependency graphs.

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Nathan Wetzler DRAT-trim: Efficient Checking and Trimming Using Expressive Clausal Proofs / 13 3

Motivation

Satisfiability solvers are used in amazing ways...

  • Hardware verification: Centaur x86 verification
  • Combinatorial problems:
  • van der Waerden numbers


[Dransfield, Marek, and Truszczynski, 2004; Kouril and Paul, 2008]

  • Gardens of Eden in Conway’s Game of Life


[Hartman, Heule, Kwekkeboom, and Noels, 2013]

  • Erdős Discrepancy Problem


[Konev and Lisitsa, 2014]

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Nathan Wetzler DRAT-trim: Efficient Checking and Trimming Using Expressive Clausal Proofs / 13 3

Motivation

Satisfiability solvers are used in amazing ways...

  • Hardware verification: Centaur x86 verification
  • Combinatorial problems:
  • van der Waerden numbers


[Dransfield, Marek, and Truszczynski, 2004; Kouril and Paul, 2008]

  • Gardens of Eden in Conway’s Game of Life


[Hartman, Heule, Kwekkeboom, and Noels, 2013]

  • Erdős Discrepancy Problem


[Konev and Lisitsa, 2014]

..., but satisfiability solvers have errors.

  • Documented bugs in SAT, SMT, and QBF solvers


[Brummayer and Biere, 2009; Brummayer et al., 2010]

  • Competition winners have contradictory results


(HWMCC winners from 2011 and 2012)

  • Implementation errors often imply conceptual errors
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Nathan Wetzler DRAT-trim: Efficient Checking and Trimming Using Expressive Clausal Proofs / 13 4

Approach

Verify SAT solvers

  • Requires verification of all crucial search techniques
  • Delicate balance between efficiency and ease of verification
  • Verification is specific to one solver
  • New developments in solving require additional proof effort
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Nathan Wetzler DRAT-trim: Efficient Checking and Trimming Using Expressive Clausal Proofs / 13 4

Approach

Verify SAT solvers

  • Requires verification of all crucial search techniques
  • Delicate balance between efficiency and ease of verification
  • Verification is specific to one solver
  • New developments in solving require additional proof effort
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Nathan Wetzler DRAT-trim: Efficient Checking and Trimming Using Expressive Clausal Proofs / 13 4

Approach

Verify SAT solvers

  • Requires verification of all crucial search techniques
  • Delicate balance between efficiency and ease of verification
  • Verification is specific to one solver
  • New developments in solving require additional proof effort

Validate SAT solver output

  • Emit a “proof” of unsatisfiability from SAT solver
  • A proof checker can validate results from many 


state-of-the-art solvers

  • Proof checker uses limited number of techniques and can be

mechanically verified

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Nathan Wetzler DRAT-trim: Efficient Checking and Trimming Using Expressive Clausal Proofs / 13 5

Proof Formats

Easy to Emit Compact Checked Efficiently Expressive Resolution Proofs

[Zhang and Malik, 2003] [Van Gelder, 2008] [Biere, 2008]

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Nathan Wetzler DRAT-trim: Efficient Checking and Trimming Using Expressive Clausal Proofs / 13 5

Proof Formats

Easy to Emit Compact Checked Efficiently Expressive Resolution Proofs

[Zhang and Malik, 2003] [Van Gelder, 2008] [Biere, 2008]

Clausal (RUP) Proofs

[Goldberg and Novikov, 2003] [Van Gelder, 2008]

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Nathan Wetzler DRAT-trim: Efficient Checking and Trimming Using Expressive Clausal Proofs / 13 5

Proof Formats

Easy to Emit Compact Checked Efficiently Expressive Resolution Proofs

[Zhang and Malik, 2003] [Van Gelder, 2008] [Biere, 2008]

Clausal (RUP) Proofs

[Goldberg and Novikov, 2003] [Van Gelder, 2008]

RAT Proofs

[Heule, Hunt, Jr., and Wetzler, CADE 2013]

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Nathan Wetzler DRAT-trim: Efficient Checking and Trimming Using Expressive Clausal Proofs / 13 5

Proof Formats

Easy to Emit Compact Checked Efficiently Expressive Resolution Proofs

[Zhang and Malik, 2003] [Van Gelder, 2008] [Biere, 2008]

Clausal (RUP) Proofs

[Goldberg and Novikov, 2003] [Van Gelder, 2008]

DRUP (DRUP-Trim)

[Heule, Hunt, Jr., and Wetzler, FMCAD 2013]

RAT Proofs

[Heule, Hunt, Jr., and Wetzler, CADE 2013]

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Nathan Wetzler DRAT-trim: Efficient Checking and Trimming Using Expressive Clausal Proofs / 13 5

Proof Formats

Easy to Emit Compact Checked Efficiently Expressive Resolution Proofs

[Zhang and Malik, 2003] [Van Gelder, 2008] [Biere, 2008]

Clausal (RUP) Proofs

[Goldberg and Novikov, 2003] [Van Gelder, 2008]

DRUP (DRUP-Trim)

[Heule, Hunt, Jr., and Wetzler, FMCAD 2013]

DRAT (DRAT-Trim)

[Wetzler, Heule, and Hunt, Jr., SAT 2014]

RAT Proofs

[Heule, Hunt, Jr., and Wetzler, CADE 2013]

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Nathan Wetzler DRAT-trim: Efficient Checking and Trimming Using Expressive Clausal Proofs / 13 6

Overview

p cnf 4 10 1 2 -3 0

  • 1 -2 3 0
  • 1 -2 -3 0

2 3 -4 0

  • 2 -3 4 0
  • 1 -3 -4 0

1 3 4 0

  • 1 2 4 0

1 -2 -4 0

  • 1 2 -4 0

CNF Formula

  • 1 0

d -1 2 4 0 2 0

DRAT Proof

p cnf 4 8 1 2 -3 0

  • 1 -2 3 0

2 3 -4 0

  • 2 -3 4 0
  • 1 -3 -4 0

1 3 4 0

  • 1 2 4 0

1 -2 -4 0

Trimmed Formula

  • 1 0

d -1 -2 3 0 d -1 -3 -4 0 d -1 2 4 0 2 0 d 1 2 -3 0 d 2 3 -4 0

Optimized Proof

1 1 2 -3 0 2 -1 -2 3 0 4 2 3 -4 0 5 -2 -3 4 0 6 -1 -3 -4 0 7 1 3 4 0 8 -1 2 4 0 9 1 -2 -4 0 11 -1 0 2 6 8 0 12 2 0 1 4 7 11 0 13 0 5 7 9 11 12 0

TraceCheck+

c 8 of 10 clauses in core c 3 of 3 lemmas in core using 15 resolution steps c 1 RAT lemmas in core; 0 redundant literals in core lemmas s VERIFIED

Console Output

DRAT-trim

Input Output

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Nathan Wetzler DRAT-trim: Efficient Checking and Trimming Using Expressive Clausal Proofs / 13 7

Input

p cnf 4 10 1 2 -3 0

  • 1 -2 3 0
  • 1 -2 -3 0

2 3 -4 0

  • 2 -3 4 0
  • 1 -3 -4 0

1 3 4 0

  • 1 2 4 0

1 -2 -4 0

  • 1 2 -4 0

CNF Formula

  • 1 0

d -1 2 4 0 2 0

DRAT Proof

DRAT-trim

Input Output

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Nathan Wetzler DRAT-trim: Efficient Checking and Trimming Using Expressive Clausal Proofs / 13 7

Input

p cnf 4 10 1 2 -3 0

  • 1 -2 3 0
  • 1 -2 -3 0

2 3 -4 0

  • 2 -3 4 0
  • 1 -3 -4 0

1 3 4 0

  • 1 2 4 0

1 -2 -4 0

  • 1 2 -4 0

CNF Formula

  • 1 0

d -1 2 4 0 2 0

DRAT Proof

DRAT-trim

Input Output

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Nathan Wetzler DRAT-trim: Efficient Checking and Trimming Using Expressive Clausal Proofs / 13 7

Input

p cnf 4 10 1 2 -3 0

  • 1 -2 3 0
  • 1 -2 -3 0

2 3 -4 0

  • 2 -3 4 0
  • 1 -3 -4 0

1 3 4 0

  • 1 2 4 0

1 -2 -4 0

  • 1 2 -4 0

CNF Formula

  • 1 0

d -1 2 4 0 2 0

DRAT Proof

  • DIMACS format

DRAT-trim

Input Output

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Nathan Wetzler DRAT-trim: Efficient Checking and Trimming Using Expressive Clausal Proofs / 13 7

Input

p cnf 4 10 1 2 -3 0

  • 1 -2 3 0
  • 1 -2 -3 0

2 3 -4 0

  • 2 -3 4 0
  • 1 -3 -4 0

1 3 4 0

  • 1 2 4 0

1 -2 -4 0

  • 1 2 -4 0

CNF Formula

  • 1 0

d -1 2 4 0 2 0

DRAT Proof

  • DIMACS format
  • DRAT format
  • Each line is either:
  • Lemma (clause) with

RUP or RAT property

  • Deletion instruction with

“d ” prefix

DRAT-trim

Input Output

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Nathan Wetzler DRAT-trim: Efficient Checking and Trimming Using Expressive Clausal Proofs / 13 8

Output and Applications

DRAT-trim

Input Output

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Nathan Wetzler DRAT-trim: Efficient Checking and Trimming Using Expressive Clausal Proofs / 13 8

Output and Applications

DRAT-trim

Input Output

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Nathan Wetzler DRAT-trim: Efficient Checking and Trimming Using Expressive Clausal Proofs / 13 8

Output and Applications

c 8 of 10 clauses in core c 3 of 3 lemmas in core using 15 resolution steps c 1 RAT lemmas in core; 0 redundant literals in core lemmas s VERIFIED

Console Output

DRAT-trim

Input Output

  • Basic output about

success/failure

  • Provide assurance to user
  • Diagnose bugs in solving
  • Verify competition results
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Nathan Wetzler DRAT-trim: Efficient Checking and Trimming Using Expressive Clausal Proofs / 13 8

Output and Applications

p cnf 4 8 1 2 -3 0

  • 1 -2 3 0

2 3 -4 0

  • 2 -3 4 0
  • 1 -3 -4 0

1 3 4 0

  • 1 2 4 0

1 -2 -4 0

Trimmed Formula

DRAT-trim

Input Output

  • Subset of original formula
  • DIMACS format
  • Minimal unsatisfiable

subset (MUS) preprocessing

  • Craig interpolants
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Nathan Wetzler DRAT-trim: Efficient Checking and Trimming Using Expressive Clausal Proofs / 13 8

Output and Applications

  • 1 0

d -1 -2 3 0 d -1 -3 -4 0 d -1 2 4 0 2 0 d 1 2 -3 0 d 2 3 -4 0

Optimized Proof

DRAT-trim

Input Output

  • Subset of original

proof with extra deletion information

  • DRAT format
  • Proof replay
  • Mechanically-verified

proof checking

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Nathan Wetzler DRAT-trim: Efficient Checking and Trimming Using Expressive Clausal Proofs / 13 8

Output and Applications

1 1 2 -3 0 2 -1 -2 3 0 4 2 3 -4 0 5 -2 -3 4 0 6 -1 -3 -4 0 7 1 3 4 0 8 -1 2 4 0 9 1 -2 -4 0 11 -1 0 2 6 8 0 12 2 0 1 4 7 11 0 13 0 5 7 9 11 12 0

TraceCheck+

DRAT-trim

Input Output

  • Resolution and

dependency graph

  • TraceCheck+ format
  • Better input to MUS than a

formula

  • Craig interpolants
  • Detailed account of solving

for debugging

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Nathan Wetzler DRAT-trim: Efficient Checking and Trimming Using Expressive Clausal Proofs / 13 8

Output and Applications

DRAT-trim

Input Output

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Nathan Wetzler DRAT-trim: Efficient Checking and Trimming Using Expressive Clausal Proofs / 13 9

Overview

p cnf 4 10 1 2 -3 0

  • 1 -2 3 0
  • 1 -2 -3 0

2 3 -4 0

  • 2 -3 4 0
  • 1 -3 -4 0

1 3 4 0

  • 1 2 4 0

1 -2 -4 0

  • 1 2 -4 0

CNF Formula

  • 1 0

d -1 2 4 0 2 0

DRAT Proof

p cnf 4 8 1 2 -3 0

  • 1 -2 3 0

2 3 -4 0

  • 2 -3 4 0
  • 1 -3 -4 0

1 3 4 0

  • 1 2 4 0

1 -2 -4 0

Trimmed Formula

  • 1 0

d -1 -2 3 0 d -1 -3 -4 0 d -1 2 4 0 2 0 d 1 2 -3 0 d 2 3 -4 0

Optimized Proof

1 1 2 -3 0 2 -1 -2 3 0 4 2 3 -4 0 5 -2 -3 4 0 6 -1 -3 -4 0 7 1 3 4 0 8 -1 2 4 0 9 1 -2 -4 0 11 -1 0 2 6 8 0 12 2 0 1 4 7 11 0 13 0 5 7 9 11 12 0

TraceCheck+

c 8 of 10 clauses in core c 3 of 3 lemmas in core using 15 resolution steps c 1 RAT lemmas in core; 0 redundant literals in core lemmas s VERIFIED

Console Output

DRAT-trim

Input Output

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Nathan Wetzler DRAT-trim: Efficient Checking and Trimming Using Expressive Clausal Proofs / 13 10

Example Benchmark

Bioinformatics Benchmark from SAT Competition 2009 rbcl_xits_07_UNSAT.cnf

c Solving time: 110 seconds c Validation time: 194 seconds c 31676 of 57446 clauses in core c 1072932 of 1225647 lemmas in core using 37392110 resolution steps c 0 RAT lemmas in core; 688434 redundant literals in core lemmas s VERIFIED

Solving with Glucose 3.0

c Solving time: 4 seconds c Validation time: 25 seconds c 39673 of 57446 clauses in core c 41359 of 59616 lemmas in core using 1316017 resolution steps c 13039 RAT lemmas in core; 30449 redundant literals in core lemmas s VERIFIED

Solving with Coprocessor and Glucose 3.0

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Nathan Wetzler DRAT-trim: Efficient Checking and Trimming Using Expressive Clausal Proofs / 13 11

Solving versus Validation

Validation Time (seconds) Solving Time (seconds)

Results of SAT Competition 2014

Application Hard-Combinatorial

  • DRAT-trim used to check

results of this year’s competition

  • All techniques used in

competition are supported by DRAT-trim

  • Validation time is within

limits of competition

  • Several solvers reported

UNSAT on SAT benchmarks

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Nathan Wetzler DRAT-trim: Efficient Checking and Trimming Using Expressive Clausal Proofs / 13 12

  • DRAT-trim is an unsatisfiability proof checker designed

to validate all contemporary SAT preprocessing and solving techniques

  • The tool can optionally emit trimmed formulas,
  • ptimized proofs, and dependency graphs
  • Solvers can take more risks when results are validated
  • Interested to see what uses the SAT community can find

for DRAT-trim

Conclusions

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Nathan Wetzler DRAT-trim: Efficient Checking and Trimming Using Expressive Clausal Proofs / 13 12

  • DRAT-trim is an unsatisfiability proof checker designed

to validate all contemporary SAT preprocessing and solving techniques

  • The tool can optionally emit trimmed formulas,
  • ptimized proofs, and dependency graphs
  • Solvers can take more risks when results are validated
  • Interested to see what uses the SAT community can find

for DRAT-trim

Conclusions

http://cs.utexas.edu/~marijn/drat-trim/

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Nathan Wetzler DRAT-trim: Efficient Checking and Trimming Using Expressive Clausal Proofs / 13 13

Recent Work

Verifying Refutations with Extended Resolution

Published: Conference on Automated Deduction (CADE 2013)

Mechanical Verification of SAT Refutations with Extended Resolution

Published: Interactive Theorem Proving (ITP 2013)

Trimming while Checking Clausal Proofs

Published: Formal Methods in Computer-Aided Design (FMCAD 2013)

DRAT-trim: Efficient Checking and Trimming Using Expressive Clausal Proofs

Published: Theory and Applications of Satisfiability Testing (SAT 2014)

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Nathan Wetzler DRAT-trim: Efficient Checking and Trimming Using Expressive Clausal Proofs / 13 13

Recent Work

Verifying Refutations with Extended Resolution

Published: Conference on Automated Deduction (CADE 2013)

Mechanical Verification of SAT Refutations with Extended Resolution

Published: Interactive Theorem Proving (ITP 2013)

Trimming while Checking Clausal Proofs

Published: Formal Methods in Computer-Aided Design (FMCAD 2013)

DRAT-trim: Efficient Checking and Trimming Using Expressive Clausal Proofs

Published: Theory and Applications of Satisfiability Testing (SAT 2014)

http://cs.utexas.edu/~marijn/drat-trim/

QRAT-trim (QBF proof checking)
 IJCAR, July 19, 15:00

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Nathan Wetzler DRAT-trim: Efficient Checking and Trimming Using Expressive Clausal Proofs / 13 13

Recent Work

Verifying Refutations with Extended Resolution

Published: Conference on Automated Deduction (CADE 2013)

Mechanical Verification of SAT Refutations with Extended Resolution

Published: Interactive Theorem Proving (ITP 2013)

Trimming while Checking Clausal Proofs

Published: Formal Methods in Computer-Aided Design (FMCAD 2013)

DRAT-trim: Efficient Checking and Trimming Using Expressive Clausal Proofs

Published: Theory and Applications of Satisfiability Testing (SAT 2014)

http://cs.utexas.edu/~marijn/drat-trim/

Thank you for your attention!

QRAT-trim (QBF proof checking)
 IJCAR, July 19, 15:00