Clausal Proofs for Pseudo-Boolean Reasoning

Bryant, Randal E.; Biere, Armin; Heule, Marijn J. H.

doi:10.1007/978-3-030-99524-9_25

Cited by 5 publications

(3 citation statements)

References 42 publications

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“…In this paper and in our linked generator, we manually construct the shortest known DRAT proofs of the unsatisfiability of the Pigeonhole Problem formula PHP(n). Our proofs combine the O(n 3 ) scaling of recent solvers [7,15] with the immediate generation and small leading constant of Cook's proof.…”

Section: Background On the Pigeonhole Principlementioning

confidence: 99%

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Towards the shortest DRAT proof of the Pigeonhole Principle

Grosof¹,

Zhang²,

Heule³

2022

Preprint

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The Pigeonhole Principle (PHP) has been heavily studied in automated reasoning, both theoretically and in practice. Most solvers have exponential runtime and proof length, while some specialized techniques achieve polynomial runtime and proof length. Several decades ago, Cook manually constructed O(n 4 ) extended resolution proofs, where n denotes the number of pigeons. Existing automated techniques only surpass Cook's proofs in similar proof systems for large n. We construct the shortest known proofs of PHP in the standard proof format of modern SAT solving, DRAT. Using auxiliary variables and by recursively decomposing the original program into smaller sizes, we manually obtain proofs having length O(n 3 ) and leading coefficient 5/2.

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Section: Background On the Pigeonhole Principlementioning

confidence: 99%

“…Heule and Biere [14] then give a method to convert these PR proofs into O(n 3 ) length DRAT proofs. In a recently published paper, Bryant, Biere, and Heule [7] describe a solver based on Pseudo-Boolean binary decision diagrams (PGBDD), which outputs DRAT proofs of the Pigeonhole Problem whose clause length empirically scales as O(n 3 ).…”

Section: Introductionmentioning

confidence: 99%

Towards the shortest DRAT proof of the Pigeonhole Principle

Grosof¹,

Zhang²,

Heule³

2022

Preprint

Self Cite

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“…The DRAT proof logging method has recently been extended to FRAT [BCH21], which allows to integrate different forms of reasoning. Proof logging using binary decision diagrams (BDDs) [Bry22], generating proofs in all of the DRAT , LRAT , and FRAT formats, has also been developed for pseudo-Boolean reasoning [BBH22] and parity reasoning [SB22]. Further evaluation will be needed to decide whether such clausal proof logging methods can be truly competitive with pseudo-Boolean proof logging.…”

Section: Subsequent Developmentsmentioning

confidence: 99%

Certifying Parity Reasoning Efficiently Using Pseudo-Boolean Proofs

Gocht¹,

Nordström

2021

AAAI

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The dramatic improvements in combinatorial optimization algorithms over the last decades have had a major impact in artificial intelligence, operations research, and beyond, but the output of current state-of-the-art solvers is often hard to verify and is sometimes wrong. For Boolean satisfiability (SAT) solvers proof logging has been introduced as a way to certify correctness, but the methods used seem hard to generalize to stronger paradigms. What is more, even for enhanced SAT techniques such as parity (XOR) reasoning, cardinality detection, and symmetry handling, it has remained beyond reach to design practically efficient proofs in the standard DRAT format. In this work, we show how to instead use pseudo-Boolean inequalities with extension variables to concisely justify XOR reasoning. Our experimental evaluation of a SAT solver integration shows a dramatic decrease in proof logging and verification time compared to existing DRAT methods. Since our method is a strict generalization of DRAT, and readily lends itself to expressing also 0-1 programming and even constraint programming problems, we hope this work points the way towards a unified approach for efficient machine-verifiable proofs for a rich class of combinatorial optimization paradigms.

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Making $$\textsf {IP}=\textsf {PSPACE}$$ Practical: Efficient Interactive Protocols for BDD Algorithms

Couillard

Czerner

Esparza

et al. 2023

Lecture Notes in Computer Science

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We show that interactive protocols between a prover and a verifier, a well-known tool of complexity theory, can be used in practice to certify the correctness of automated reasoning tools.Theoretically, interactive protocols exist for all $$\textsf {PSPACE}$$ PSPACE problems. The verifier of a protocol checks the prover’s answer to a problem instance in probabilistic polynomial time, with polynomially many bits of communication, and with exponentially small probability of error. (The prover may need exponential time.) Existing interactive protocols are not used in practice because their provers use naive algorithms, inefficient even for small instances, that are incompatible with practical implementations of automated reasoning.We bridge the gap between theory and practice by means of an interactive protocol whose prover uses BDDs. We consider the problem of counting the number of assignments to a QBF instance ($$\#\text {CP}$$ # CP ), which has a natural BDD-based algorithm. We give an interactive protocol for $$\#\text {CP}$$ # CP whose prover is implemented on top of an extended BDD library. The prover has only a linear overhead in computation time over the natural algorithm.We have implemented our protocol in $$\textsf {blic}$$ blic , a certifying tool for $$\#\text {CP}$$ # CP . Experiments on standard QBF benchmarks show that is competitive with state-of-the-art QBF-solvers. The run time of the verifier is negligible. While loss of absolute certainty can be concerning, the error probability in our experiments is at most $$10^{-10}$$ 10 - 10 and reduces to $$10^{-10k}$$ 10 - 10 k by repeating the verification k times.

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Clausal Proofs for Pseudo-Boolean Reasoning

Cited by 5 publications

References 42 publications

Towards the shortest DRAT proof of the Pigeonhole Principle

Towards the shortest DRAT proof of the Pigeonhole Principle

Certifying Parity Reasoning Efficiently Using Pseudo-Boolean Proofs

Making $$\textsf {IP}=\textsf {PSPACE}$$ Practical: Efficient Interactive Protocols for BDD Algorithms

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