DRAT Proofs of Unsatisfiability for SAT Modulo Monotonic Theories
Authors: 
Nick Feng, Alan J. Hu, Sam Bayless, Syed M. Iqbal, Patrick Trentin, Mike Whalen, Lee Pike, John BackesAuthors Info & Claims
Nick Feng, Alan J. Hu, Sam Bayless, Syed M. Iqbal, Patrick Trentin, Mike Whalen, Lee Pike, John BackesAuthors Info & ClaimsPages 3 - 23
Abstract
Generating proofs of unsatisfiability is a valuable capability of most SAT solvers, and is an active area of research for SMT solvers. This paper introduces the first method to efficiently generate proofs of unsatisfiability specifically for an important subset of SMT: SAT Modulo Monotonic Theories (SMMT), which includes many useful finite-domain theories (e.g., bit vectors and many graph-theoretic properties) and is used in production at Amazon Web Services. Our method uses propositional definitions of the theory predicates, from which it generates compact Horn approximations of the definitions, which lead to efficient DRAT proofs, leveraging the large investment the SAT community has made in DRAT. In experiments on practical SMMT problems, our proof generation overhead is minimal (7.41% geometric mean slowdown, 28.8% worst-case), and we can generate and check proofs for many problems that were previously intractable.
References
[1]
Armand, M., Faure, G., Grégoire, B., Keller, C., Théry, L., Werner, B.: A Modular Integration of SAT/SMT Solvers to Coq through Proof Witnesses. In: Jouannaud, J., Shao, Z. (eds.) Certified Programs and Proofs - First International Conference, CPP 2011, Kenting, Taiwan, December 7-9, 2011. Proceedings. Lecture Notes in Computer Science, vol. 7086, pp. 135–150. Springer (2011).,
[2]
Backes, J., Bayless, S., Cook, B., Dodge, C., Gacek, A., Hu, A.J., Kahsai, T., Kocik, B., Kotelnikov, E., Kukovec, J., McLaughlin, S., Reed, J., Rungta, N., Sizemore, J., Stalzer, M., Srinivasan, P., Subotić, P., Varming, C., Whaley, B.: Reachability analysis for AWS-based networks. In: Dillig, I., Tasiran, S. (eds.) International Conference on Computer Aided Verification (CAV). pp. 231–241. Springer (2019)
[3]
Baek, S., Carneiro, M., Heule, M.J.H.: A Flexible Proof Format for SAT Solver-Elaborator Communication. In: Groote, J.F., Larsen, K.G. (eds.) Tools and Algorithms for the Construction and Analysis of Systems — 27th International Conference, TACAS 2021. Lecture Notes in Computer Science, vol. 12651, pp. 59–75. Springer (2021).,
[4]
Barbosa, H., Blanchette, J., Fleury, M., Fontaine, P., Schurr, H.J.: Better SMT proofs for easier reconstruction. In: AITP 2019-4th Conference on Artificial Intelligence and Theorem Proving (2019)
[5]
Barbosa, H., Blanchette, J.C., Fontaine, P.: Scalable Fine-Grained Proofs for Formula Processing. In: de Moura, L. (ed.) Automated Deduction - CADE 26 - 26th International Conference on Automated Deduction, Gothenburg, Sweden, August 6-11, 2017, Proceedings. Lecture Notes in Computer Science, vol. 10395, pp. 398–412. Springer (2017).,
[6]
Barrett, C., De Moura, L., Fontaine, P.: Proofs in satisfiability modulo theories. All about proofs, Proofs for all 55(1), 23–44 (2015)
[7]
Barrett, C.W., Conway, C.L., Deters, M., Hadarean, L., Jovanovic, D., King, T., Reynolds, A., Tinelli, C.: CVC4. In: Gopalakrishnan, G., Qadeer, S. (eds.) Computer Aided Verification - 23rd International Conference, CAV 2011, Snowbird, UT, USA, July 14-20, 2011. Proceedings. Lecture Notes in Computer Science, vol. 6806, pp. 171–177. Springer (2011).,
[8]
Bayless, S., Backes, J., DaCosta, D., Jones, B., Launchbury, N., Trentin, P., Jewell, K., Joshi, S., Zeng, M., Mathews, N.: Debugging Network Reachability with Blocked Paths. In: International Conference on Computer Aided Verification (CAV). pp. 851–862. Springer (2021)
[9]
Bayless, S., Bayless, N., Hoos, H., Hu, A.: SAT modulo monotonic theories. In: Proceedings of the AAAI Conference on Artificial Intelligence. vol. 29 (2015)
[10]
Bayless, S., Bayless, N., Hoos, H.H., Hu, A.J.: SAT Modulo Monotonic Theories. In: Bonet, B., Koenig, S. (eds.) Proceedings of the Twenty-Ninth AAAI Conference on Artificial Intelligence, January 25-30, 2015, Austin, Texas, USA. pp. 3702–3709. AAAI Press (2015), http://www.aaai.org/ocs/index.php/AAAI/AAAI15/paper/view/9951
[11]
Bayless, S., Hoos, H.H., Hu, A.J.: Scalable, high-quality, SAT-based multi-layer escape routing. In: Liu, F. (ed.) Proceedings of the 35th International Conference on Computer-Aided Design, ICCAD 2016, Austin, TX, USA, November 7-10, 2016. p. 22. ACM (2016).,
[12]
Bayless, S., Kodirov, N., Iqbal, S.M., Beschastnikh, I., Hoos, H.H., Hu, A.J.: Scalable Constraint-Based Virtual Data Center Allocation. Artif. Intell. 278(C) (jan 2020).,
[13]
Biere, A., Fazekas, K., Fleury, M., Heisinger, M.: CaDiCaL, Kissat, Paracooba, Plingeling, and Treengeling Entering the SAT Competition 2020. In: Balyo, T., Froleyks, N., Heule, M., Iser, M., Järvisalo, M., Suda, M. (eds.) SAT Competition 2020 — Solver and Benchmark Descriptions. Department of Computer Science Report Series B, vol. B-2020-1, pp. 51–53. University of Helsinki (2020)
[14]
Böhme, S.: Proof reconstruction for Z3 in Isabelle/HOL. In: 7th International Workshop on Satisfiability Modulo Theories (SMT’09) (2009)
[15]
Böhme, S., Weber, T.: Fast LCF-Style Proof Reconstruction for Z3. In: Kaufmann, M., Paulson, L.C. (eds.) Interactive Theorem Proving, First International Conference, ITP 2010, Edinburgh, UK, July 11-14, 2010. Proceedings. Lecture Notes in Computer Science, vol. 6172, pp. 179–194. Springer (2010).,
[16]
Bouton, T., Oliveira, D.C.B.D., Déharbe, D., Fontaine, P.: veriT: An Open, Trustable and Efficient SMT-Solver. In: Schmidt, R.A. (ed.) Automated Deduction - CADE-22, 22nd International Conference on Automated Deduction, Montreal, Canada, August 2-7, 2009. Proceedings. Lecture Notes in Computer Science, vol. 5663, pp. 151–156. Springer (2009).,
[17]
Christ, J., Hoenicke, J., Nutz, A.: SMTInterpol: An Interpolating SMT Solver. In: Donaldson, A.F., Parker, D. (eds.) Model Checking Software - 19th International Workshop, SPIN 2012, Oxford, UK, July 23-24, 2012. Proceedings. Lecture Notes in Computer Science, vol. 7385, pp. 248–254. Springer (2012).,
[18]
Cruz-Filipe, L., Heule, M.J.H., Jr., W.A.H., Kaufmann, M., Schneider-Kamp, P.: Efficient Certified RAT Verification. In: de Moura, L. (ed.) Automated Deduction - CADE 26 - 26th International Conference on Automated Deduction, Gothenburg, Sweden, August 6-11, 2017, Proceedings. Lecture Notes in Computer Science, vol. 10395, pp. 220–236. Springer (2017).,
[19]
Cruz-Filipe, L., Marques-Silva, J., Schneider-Kamp, P.: Efficient Certified Resolution Proof Checking. In: Legay, A., Margaria, T. (eds.) Tools and Algorithms for the Construction and Analysis of Systems. pp. 118–135. Springer Berlin Heidelberg, Berlin, Heidelberg (2017)
[20]
Davis, M., Putnam, H.: A Computing Procedure for Quantification Theory. J. ACM 7(3), 201–215 (1960)., http://doi.acm.org/10.1145/321033.321034
[21]
Fleury, M., Schurr, H.: Reconstructing veriT Proofs in Isabelle/HOL. In: Reis, G., Barbosa, H. (eds.) Proceedings Sixth Workshop on Proof eXchange for Theorem Proving, PxTP 2019, Natal, Brazil, August 26, 2019. EPTCS, vol. 301, pp. 36–50 (2019).,
[22]
Ganzinger, H., Hagen, G., Nieuwenhuis, R., Oliveras, A., Tinelli, C.: DPLL (T): Fast decision procedures. In: International Conference on Computer Aided Verification. pp. 175–188. Springer (2004)
[23]
Giesl, J., Aschermann, C., Brockschmidt, M., Emmes, F., Frohn, F., Fuhs, C., Hensel, J., Otto, C., Plücker, M., Schneider-Kamp, P., Ströder, T., Swiderski, S., Thiemann, R.: Analyzing Program Termination and Complexity Automatically with AProVE. J. Autom. Reason. 58(1), 3–31 (2017).,
[24]
Goldberg, E., Novikov, Y.: Verification of Proofs of Unsatisfiability for CNF Formulas. In: Proceedings of the Conference on Design, Automation and Test in Europe - Volume 1. p. 10886. DATE ’03, IEEE Computer Society, USA (2003)
[25]
Gurfinkel, A., Vizel, Y.: DRUPing for interpolates. In: Formal Methods in Computer-Aided Design, FMCAD 2014, Lausanne, Switzerland, October 21-24, 2014. pp. 99–106. IEEE (2014).,
[26]
Heule, M.J.H., Kiesl, B., Biere, A.: Strong Extension-Free Proof Systems. J. Automated Reasoning 64, 533–554 (2020).
[27]
Heule, M.J., Hunt, W.A., Wetzler, N.: Trimming while checking clausal proofs. In: 2013 Formal Methods in Computer-Aided Design. pp. 181–188. IEEE (2013)
[28]
Klenze, T., Bayless, S., Hu, A.J.: Fast, Flexible, and Minimal CTL Synthesis via SMT. In: Chaudhuri, S., Farzan, A. (eds.) Computer Aided Verification. pp. 136–156. Springer International Publishing, Cham (2016)
[29]
Luckow, K.S., Dimjasevic, M., Giannakopoulou, D., Howar, F., Isberner, M., Kahsai, T., Rakamaric, Z., Raman, V.: JDart: A Dynamic Symbolic Analysis Framework. In: Chechik, M., Raskin, J. (eds.) Tools and Algorithms for the Construction and Analysis of Systems - 22nd International Conference, TACAS 2016, Held as Part of the European Joint Conferences on Theory and Practice of Software, ETAPS 2016, Eindhoven, The Netherlands, April 2-8, 2016, Proceedings. Lecture Notes in Computer Science, vol. 9636, pp. 442–459. Springer (2016).,
[30]
McMillan, K.L.: Interpolation and SAT-Based Model Checking. In: Jr., W.A.H., Somenzi, F. (eds.) Computer Aided Verification, 15th International Conference, CAV 2003, Boulder, CO, USA, July 8-12, 2003, Proceedings. Lecture Notes in Computer Science, vol. 2725, pp. 1–13. Springer (2003).,
[31]
de Moura, L.M., Bjørner, N.: Proofs and Refutations, and Z3. In: Rudnicki, P., Sutcliffe, G., Konev, B., Schmidt, R.A., Schulz, S. (eds.) Proceedings of the LPAR 2008 Workshops, Knowledge Exchange: Automated Provers and Proof Assistants, and the 7th International Workshop on the Implementation of Logics, Doha, Qatar, November 22, 2008. CEUR Workshop Proceedings, vol. 418. CEUR-WS.org (2008), http://ceur-ws.org/Vol-418/paper10.pdf
[32]
de Moura, L.M., Bjørner, N.: Z3: An Efficient SMT Solver. In: Ramakrishnan, C.R., Rehof, J. (eds.) Tools and Algorithms for the Construction and Analysis of Systems, 14th International Conference, TACAS 2008, Held as Part of the Joint European Conferences on Theory and Practice of Software, ETAPS 2008, Budapest, Hungary, March 29-April 6, 2008. Proceedings. Lecture Notes in Computer Science, vol. 4963, pp. 337–340. Springer (2008).,
[33]
Otoni, R., Blicha, M., Eugster, P., Hyvärinen, A.E.J., Sharygina, N.: Theory-Specific Proof Steps Witnessing Correctness of SMT Executions. In: 58th ACM/IEEE Design Automation Conference, DAC 2021, San Francisco, CA, USA, December 5-9, 2021. pp. 541–546. IEEE (2021).,
[34]
Ozdemir, A., Niemetz, A., Preiner, M., Zohar, Y., Barrett, C.W.: DRAT-based Bit-Vector Proofs in CVC4. In: Janota, M., Lynce, I. (eds.) Theory and Applications of Satisfiability Testing - SAT 2019 - 22nd International Conference, SAT 2019, Lisbon, Portugal, July 9-12, 2019, Proceedings. Lecture Notes in Computer Science, vol. 11628, pp. 298–305. Springer (2019).,
[35]
Selman, B., Kautz, H.A.: Knowledge Compilation using Horn Approximations. In: Dean, T.L., McKeown, K.R. (eds.) Proceedings of the 9th National Conference on Artificial Intelligence, Anaheim, CA, USA, July 14-19, 1991, Volume 2. pp. 904–909. AAAI Press / The MIT Press (1991), http://www.aaai.org/Library/AAAI/1991/aaai91-140.php
[36]
Stump, A., Oe, D., Reynolds, A., Hadarean, L., Tinelli, C.: SMT proof checking using a logical framework. Formal Methods Syst. Des. 42(1), 91–118 (2013).,
[37]
Wetzler, N., Heule, M., Jr., W.A.H.: DRAT-trim: Efficient Checking and Trimming Using Expressive Clausal Proofs. In: Sinz, C., Egly, U. (eds.) Theory and Applications of Satisfiability Testing - SAT 2014 - 17th International Conference, Held as Part of the Vienna Summer of Logic, VSL 2014, Vienna, Austria, July 14-17, 2014. Proceedings. Lecture Notes in Computer Science, vol. 8561, pp. 422–429. Springer (2014).,
[38]
Wetzler, N., Heule, M.J.H., Hunt, W.A.: Mechanical Verification of SAT Refutations with Extended Resolution. In: Blazy, S., Paulin-Mohring, C., Pichardie, D. (eds.) Interactive Theorem Proving. pp. 229–244. Springer Berlin Heidelberg, Berlin, Heidelberg (2013)
[39]
Zhang, L., Malik, S.: Validating SAT Solvers Using an Independent Resolution-Based Checker: Practical Implementations and Other Applications. In: 2003 Design, Automation and Test in Europe Conference and Exposition (DATE 2003), 3-7 March 2003, Munich, Germany. pp. 10880–10885. IEEE Computer Society (2003)., http://doi.ieeecomputersociety.org/10.1109/DATE.2003.10014
Recommendations
SAT modulo monotonic theories
AAAI'15: Proceedings of the Twenty-Ninth AAAI Conference on Artificial IntelligenceBoolean satisfiability (SAT) solvers have been successfully applied to a wide variety of difficult combinatorial problems. Many further problems can be solved by SAT Modulo Theory (SMT) solvers, which extend SAT solvers to handle additional types of ...
Solving SAT and SAT Modulo Theories: From an abstract Davis--Putnam--Logemann--Loveland procedure to DPLL(T)
We first introduce Abstract DPLL, a rule-based formulation of the Davis--Putnam--Logemann--Loveland (DPLL) procedure for propositional satisfiability. This abstract framework allows one to cleanly express practical DPLL algorithms and to formally reason ...
Splitting on demand in SAT modulo theories
LPAR'06: Proceedings of the 13th international conference on Logic for Programming, Artificial Intelligence, and ReasoningLazy algorithms for Satisfiability Modulo Theories (SMT) combine a generic DPLL-based SAT engine with a theory solver for the given theory T that can decide the T-consistency of conjunctions of ground literals. For many theories of interest, theory ...
Comments
Information & Contributors
Information
Published In
Apr 2024
412 pages
© The Author(s) 2024.
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.
Publisher
Springer-Verlag
Berlin, Heidelberg
Publication History
Published: 06 April 2024
Qualifiers
- Article
Contributors
Other Metrics
Bibliometrics & Citations
Bibliometrics
Article Metrics
- 0Total Citations
- 0Total Downloads
- Downloads (Last 12 months)0
- Downloads (Last 6 weeks)0
Reflects downloads up to 28 Dec 2024