Abstract
In 2006, Biere, Jussila, and Sinz made the key observation that the underlying logic behind algorithms for constructing Reduced, Ordered Binary Decision Diagrams (BDDs) can be encoded as steps in a proof in the extended resolution logical framework. Through this, a BDD-based Boolean satisfiability (SAT) solver can generate a checkable proof of unsatisfiability. Such proofs indicate that the formula is truly unsatisfiable without requiring the user to trust the BDD package or the SAT solver built on top of it.
We extend their work to enable arbitrary existential quantification of the formula variables, a critical capability for BDD-based SAT solvers. We demonstrate the utility of this approach by applying a prototype solver to obtain polynomially sized proofs on benchmarks for the mutilated chessboard and pigeonhole problems—ones that are very challenging for search-based SAT solvers.
Keywords
- extended resolution
- binary decision diagrams
- mutilated chessboard
- pigeonhole problem
M. J. H. Heule—Supported by the National Science Foundation under grant CCF-2010951
Download conference paper PDF
References
Alekhnovich, M.: Mutilated chessboard problem is exponentially hard for resolution. Theoretical Computer Science 310(1-3), 513–525 (Jan 2004)
Andersen, H.R.: An introduction to binary decision diagrams. Tech. rep., Technical University of Denmark (October 1997)
Biere, A., Fazekas, K., Fleury, M., Heisinger, M.: CaDiCaL, Kissat, Paracooba, Plingeling, and Treengeling entering the SAT competition 2020 (2020), unpublished
Bryant, R.E.: Graph-based algorithms for Boolean function manipulation. IEEE Trans. Computers 35(8), 677–691 (1986)
Bryant, R.E.: Symbolic Boolean manipulation with ordered binary decision diagrams. ACM Computing Surveys 24(3), 293–318 (September 1992)
Bryant, R.E.: Binary decision diagrams. In: Clarke, E.M., Henzinger, T.A., Veith, H., Bloem, R. (eds.) Handbook of Model Checking, pp. 191–217. Springer (2018)
Burch, J.R., Clarke, E.M., McMillan, K.L., Dill, D.L., Hwang, L.J.: Symbolic model checking: \(10^{20}\) states and beyond. Information and Computation 98(2), 142–170 (1992)
Cook, S.A.: A short proof of the pigeon hole principle using extended resolution. SIGACT News 8(4), 28–32 (1976)
Cruz-Filipe, L., Heule, M.J.H., Hunt, W.A., Kaufmann, M., Schneider-Kamp, P.: Efficient certified RAT verification. In: Automated Deduction (CADE). LNCS, vol. 10395, pp. 220–236 (2017)
Cruz-Filipe, L., Marques-Silva, J., Schneider-Kamp, P.: Efficient certified resolution proof checking. In: Tools and Algorithms for the Construction and Analysis of Systems (TACAS). LNCS, vol. 10205, pp. 118–135 (2017)
Davis, M., Putnam, H.: A computing procedure for quantification theory. J. ACM 7(3), 201–215 (1960)
Dechter, R.: Bucket elimination: A unifying framework for reasoning. Artificial Intelligence 113(1–2), 41–85 (1999)
Franco, J., Kouril, M., Schlipf, J., Ward, J., Weaver, S., Dransfield, M., Vanfleet, W.M.: SBSAT: a state-based, BDD-based satisfiability solver. In: Theory and Applications of Satisfiability Testing (SAT). LNCS, vol. 2919, pp. 398–410 (2004)
Groote, J.F., Zantema, H.: Resolution and binary decision diagrams cannot simulate each other polynomially. Discrete Applied Mathematics 130(2), 157–171 (2003)
Haken, A.: The intractability of resolution. Theoretical Computer Science 39, 297–308 (1985)
Heule, M.J.H., Biere, A.: Proofs for satisfiability problems. In: All about Proofs, Proofs for All (APPA), Math. Logic and Foundations, vol. 55. College Pub. (2015)
Heule, M.J.H., Biere, A.: What a difference a variable makes. In: Tools and Algorithms for the Construction and Analysis of Systems (TACAS). LNCS, vol. 10806, pp. 75–92 (2018)
Heule, M.J.H., Hunt, W.A., Kaufmann, M., Wetzler, N.D.: Efficient, verified checking of propositional proofs. In: Interactive Theorem Proving. LNCS, vol. 10499, pp. 269–284 (2017)
Heule, M.J.H., Kiesl, B., Biere, A.: Clausal proofs of mutilated chessboards. In: NASA Formal Methods. LNCS, vol. 11460, pp. 204–210 (2019)
Heule, M.J.H., Kiesl, B., Seidl, M., Biere, A.: PRuning through satisfaction. In: Haifa Verification Conference (HVC). LNCS, vol. 10629, pp. 179–194 (2017)
Jussila, T., Sinz, C., Biere, A.: Extended resolution proofs for symbolic SAT solving with quantification. In: Theory and Applications of Satisfiability Testing (SAT). LNCS, vol. 4121, pp. 54–60 (2006)
Kullmann, O.: On a generalization of extended resolution. Discrete Applied Mathematics 96-97, 149–176 (1999)
Lammich, P.: Efficient verified (UN)SAT certificate checking. Journal of Automated Reasoning 64, 513–532 (2020)
Minato, S.I., Ishiura, N., Yajima, S.: Shared binary decision diagrams with attributed edges for efficient Boolean function manipulation. In: 27th ACM/IEEE Design Automation Conference. pp. 52–57 (June 1990)
Pan, G., Vardi, M.Y.: Search vs. symbolic techniques in satisfiability solving. In: Theory and Applications of Satisfiability Testing (SAT). LNCS, vol. 3542, pp. 235–250 (2005)
Robinson, J.A.: A machine-oriented logic based on the resolution principle. J. ACM 12(1), 23–41 (January 1965)
Sinz, C.: Towards an optimal CNF encoding of Boolean cardinality constraints. In: Principles and Practice of Constraint Programming (CP). LNCS, vol. 3709, pp. 827–831 (2005)
Sinz, C., Biere, A.: Extended resolution proofs for conjoining BDDs. In: Computer Science Symposium in Russia (CSR). LNCS, vol. 3967, pp. 600–611 (2006)
Tan, Y.K., Heule, M.J.H., Myreen, M.O.: cake\_lpr: Verified propagation redundancy checking in CakeML. In: Tools and Algorithms for the Construction and Analysis of Systems (TACAS) (2021)
Tseitin, G.S.: On the complexity of derivation in propositional calculus. In: Automation of Reasoning: 2: Classical Papers on Computational Logic 1967–1970. pp. 466–483. Springer (1983)
Uribe, T.E., Stickel, M.E.: Ordered binary decision diagrams and the Davis-Putnam procedure. In: Constraints in Computational Logics. LNCS, vol. 845, pp. 34–49 (1994)
Wetzler, N.D., Heule, M.J.H., Hunt Jr., W.A.: DRAT-trim: Efficient checking and trimming using expressive clausal proofs. In: Theory and Applications of Satisfiability Testing (SAT). LNCS, vol. 8561, pp. 422–429 (2014)
Zhang, L., Malik, S.: Validating SAT solvers using an independent resolution-based checker: Practical implementations and other applications. In: Design, Automation and Test in Europe (DATE) Volume 1. p. 10880. IEEE Computer Society (2003)
Author information
Authors and Affiliations
Corresponding author
Editor information
Editors and Affiliations
Rights and permissions
Open Access This chapter is licensed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license and indicate if changes were made.
The images or other third party material in this chapter are included in the chapter's Creative Commons license, unless indicated otherwise in a credit line to the material. If material is not included in the chapter's Creative Commons license and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder.
Copyright information
© 2021 The Author(s)
About this paper
Cite this paper
Bryant, R.E., Heule, M.J.H. (2021). Generating Extended Resolution Proofs with a BDD-Based SAT Solver. In: Groote, J.F., Larsen, K.G. (eds) Tools and Algorithms for the Construction and Analysis of Systems. TACAS 2021. Lecture Notes in Computer Science(), vol 12651. Springer, Cham. https://doi.org/10.1007/978-3-030-72016-2_5
Download citation
DOI: https://doi.org/10.1007/978-3-030-72016-2_5
Published:
Publisher Name: Springer, Cham
Print ISBN: 978-3-030-72015-5
Online ISBN: 978-3-030-72016-2
eBook Packages: Computer ScienceComputer Science (R0)
-
Published in cooperation with
http://www.etaps.org/
