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Truth Assignments as Conditional Autarkies

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Automated Technology for Verification and Analysis (ATVA 2019)

Part of the book series: Lecture Notes in Computer Science ((LNPSE,volume 11781))

Abstract

An autarky for a formula in propositional logic is a truth assignment that satisfies every clause it touches, i.e., every clause for which the autarky assigns at least one variable. In this paper, we present how conditional autarkies, a generalization of autarkies, give rise to novel preprocessing techniques for SAT solving. We show that conditional autarkies correspond to a new type of redundant clauses, termed globally-blocked clauses, and that the elimination of these clauses can simulate existing circuit-simplification techniques on the CNF level.

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References

  1. Clarke, E.M., Biere, A., Raimi, R., Zhu, Y.: Bounded model checking using satisfiability solving. Form. Methods Syst. Des. 19(1), 7–34 (2001)

    Article  Google Scholar 

  2. Stevens, M., Bursztein, E., Karpman, P., Albertini, A., Markov, Y.: The first collision for full SHA-1. In: Katz, J., Shacham, H. (eds.) CRYPTO 2017. LNCS, vol. 10401, pp. 570–596. Springer, Cham (2017)

    Chapter  Google Scholar 

  3. Konev, B., Lisitsa, A.: A SAT attack on the Erdős discrepancy conjecture. In: Sinz, C., Egly, U. (eds.) SAT 2014. LNCS, vol. 8561, pp. 219–226. Springer, Cham (2014)

    Chapter  Google Scholar 

  4. Heule, M.J.H.: Schur number five. In: Proceedings of the 32nd AAAI Conference on Artificial Intelligence (AAAI 2018). AAAI Press (2018)

    Google Scholar 

  5. Järvisalo, M., Biere, A., Heule, M.J.H.: Simulating circuit-level simplifications on CNF. J. Autom. Reason. 49(4), 583–619 (2012)

    Article  MathSciNet  Google Scholar 

  6. Heule, M.J.H., Kiesl, B., Seidl, M., Biere, A.: PRuning through satisfaction. In: Strichman, O., Tzoref-Brill, R. (eds.) HVC 2017. LNCS, vol. 10629, pp. 179–194. Springer, Cham (2017)

    Chapter  Google Scholar 

  7. Monien, B., Speckenmeyer, E.: Solving satisfiability in less than \(2^n\) steps. Discrete Appl. Math. 10(3), 287–295 (1985)

    Article  MathSciNet  Google Scholar 

  8. Kleine Büning, H., Kullmann, O.: Minimal unsatisfiability and autarkies. In: Biere, A., Heule, M.J.H., van Maaren, H., Walsh, T. (eds.) Handbook of Satisfiability, pp. 399–401. IOS Press, Amsterdam (2009)

    Google Scholar 

  9. Kiesl, B., Seidl, M., Tompits, H., Biere, A.: Super-blocked clauses. In: Olivetti, N., Tiwari, A. (eds.) IJCAR 2016. LNCS (LNAI), vol. 9706, pp. 45–61. Springer, Cham (2016)

    Chapter  Google Scholar 

  10. Kullmann, O.: On a generalization of extended resolution. Discrete Appl. Math. 96–97, 149–176 (1999)

    Article  MathSciNet  Google Scholar 

  11. Järvisalo, M., Biere, A., Heule, M.J.H.: Blocked clause elimination. In: Esparza, J., Majumdar, R. (eds.) TACAS 2010. LNCS, vol. 6015, pp. 129–144. Springer, Heidelberg (2010)

    Chapter  Google Scholar 

  12. Kiesl, B., Suda, M., Seidl, M., Tompits, H., Biere, A.: Blocked clauses in first-order logic. In: Proceedings of the 21st International Conference on Logic for Programming, Artificial Intelligence and Reasoning (LPAR-21). EPiC Series in Computing, vol. 46, pp. 31–48. EasyChair (2017)

    Google Scholar 

  13. Biere, A., Lonsing, F., Seidl, M.: Blocked clause elimination for QBF. In: Bjørner, N., Sofronie-Stokkermans, V. (eds.) CADE 2011. LNCS (LNAI), vol. 6803, pp. 101–115. Springer, Heidelberg (2011)

    Chapter  Google Scholar 

  14. Lonsing, F., Bacchus, F., Biere, A., Egly, U., Seidl, M.: Enhancing search-based QBF solving by dynamic blocked clause elimination. In: Davis, M., Fehnker, A., McIver, A., Voronkov, A. (eds.) LPAR 2015. LNCS, vol. 9450, pp. 418–433. Springer, Heidelberg (2015)

    Chapter  Google Scholar 

  15. Marques Silva, J.P., Sakallah, K.A.: GRASP: a search algorithm for propositional satisfiability. IEEE Trans. Comput. 48(5), 506–521 (1999)

    Article  MathSciNet  Google Scholar 

  16. Kiesl, B.: Structural reasoning methods for satisfiability solving and beyond. Ph.D. thesis, TU Wien (2019)

    Google Scholar 

  17. Järvisalo, M., Heule, M.J.H., Biere, A.: Inprocessing rules. In: Gramlich, B., Miller, D., Sattler, U. (eds.) IJCAR 2012. LNCS (LNAI), vol. 7364, pp. 355–370. Springer, Heidelberg (2012)

    Chapter  Google Scholar 

  18. Eén, N., Biere, A.: Effective preprocessing in SAT through variable and clause elimination. In: Bacchus, F., Walsh, T. (eds.) SAT 2005. LNCS, vol. 3569, pp. 61–75. Springer, Heidelberg (2005)

    Chapter  Google Scholar 

  19. Heule, M.J.H., Kullmann, O., Marek, V.W.: Solving and verifying the Boolean Pythagorean Triples problem via Cube-and-Conquer. In: Creignou, N., Le Berre, D. (eds.) SAT 2016. LNCS, vol. 9710, pp. 228–245. Springer, Cham (2016)

    Chapter  Google Scholar 

  20. Heule, M., Järvisalo, M., Biere, A.: Clause elimination procedures for CNF formulas. In: Fermüller, C.G., Voronkov, A. (eds.) LPAR 2010. LNCS, vol. 6397, pp. 357–371. Springer, Heidelberg (2010)

    Chapter  Google Scholar 

  21. Heule, M., Järvisalo, M., Biere, A.: Covered clause elimination. In: Short papers for the 17th International Conference on Logic for Programming, Artificial intelligence, and Reasoning (LPAR-17-short). EPiC Series, vol. 13, pp. 41–46. EasyChair (2010)

    Google Scholar 

  22. Heule, M.J.H., Järvisalo, M., Lonsing, F., Seidl, M., Biere, A.: Clause elimination for SAT and QSAT. J. Artif. Intell. Res. 53, 127–168 (2015)

    Article  MathSciNet  Google Scholar 

  23. Wetzler, N.D., Heule, M.J.H., Hunt Jr., W.A.: DRAT-trim: efficient checking and trimming using expressive clausal proofs. In: Sinz, C., Egly, U. (eds.) SAT 2014. LNCS, vol. 8561, pp. 422–429. Springer, Cham (2014)

    Chapter  Google Scholar 

  24. Cruz-Filipe, L., Heule, M.J.H., Hunt Jr., W.A., Kaufmann, M., Schneider-Kamp, P.: Efficient certified RAT verification. In: de Moura, L. (ed.) CADE 2017. LNCS (LNAI), vol. 10395, pp. 220–236. Springer, Cham (2017)

    Chapter  Google Scholar 

  25. Lammich, P.: Efficient verified (UN)SAT certificate checking. In: de Moura, L. (ed.) CADE 2017. LNCS (LNAI), vol. 10395, pp. 237–254. Springer, Cham (2017)

    Chapter  Google Scholar 

  26. Heule, M.J.H., Järvisalo, M., Suda, M.: SAT competition 2018 (2019)

    Google Scholar 

  27. Heule, M.J.H., Kiesl, B., Biere, A.: Short proofs without new variables. In: de Moura, L. (ed.) CADE 2017. LNCS (LNAI), vol. 10395, pp. 130–147. Springer, Cham (2017)

    Chapter  Google Scholar 

  28. Heule, M.J.H., Biere, A.: What a difference a variable makes. In: Beyer, D., Huisman, M. (eds.) TACAS 2018. LNCS, vol. 10806, pp. 75–92. Springer, Cham (2018)

    Chapter  Google Scholar 

  29. Heule, M.J.H., Kiesl, B., Biere, A.: Clausal proofs of mutilated chessboards. In: Badger, J.M., Rozier, K.Y. (eds.) NFM 2019. LNCS, vol. 11460, pp. 204–210. Springer, Cham (2019)

    Chapter  Google Scholar 

  30. Heule, M.J.H., Kiesl, B., Biere, A.: Encoding redundancy for satisfaction-driven clause learning. In: Vojnar, T., Zhang, L. (eds.) TACAS 2019. LNCS, vol. 11427, pp. 41–58. Springer, Cham (2019)

    Chapter  Google Scholar 

  31. Heule, M.J.H., Seidl, M., Biere, A.: Solution validation and extraction for QBF preprocessing. J. Autom. Reason. 58, 1–29 (2016)

    MathSciNet  MATH  Google Scholar 

  32. Heule, M., Seidl, M., Biere, A.: Blocked literals are universal. In: Havelund, K., Holzmann, G., Joshi, R. (eds.) NFM 2015. LNCS, vol. 9058, pp. 436–442. Springer, Cham (2015)

    Chapter  Google Scholar 

  33. Lonsing, F., Egly, U.: QRAT+: generalizing QRAT by a more powerful QBF redundancy property. In: Galmiche, D., Schulz, S., Sebastiani, R. (eds.) IJCAR 2018. LNCS (LNAI), vol. 10900, pp. 161–177. Springer, Cham (2018)

    Chapter  Google Scholar 

  34. Lonsing, F., Egly, U.: QRATPre+: effective QBF preprocessing via strong redundancy properties. In: Janota, M., Lynce, I. (eds.) SAT 2019. LNCS, vol. 11628, pp. 203–210. Springer, Cham (2019)

    Chapter  Google Scholar 

  35. Kiesl, B., Suda, M.: A unifying principle for clause elimination in first-order logic. In: de Moura, L. (ed.) CADE 2017. LNCS (LNAI), vol. 10395, pp. 274–290. Springer, Cham (2017)

    Chapter  Google Scholar 

  36. Fujiwara, H.: FAN: a fanout-oriented test pattern generation algorithm. In: Proceedings of the IEEE International Symposium on Circuits and Systems (ISCAS 1985), pp. 671–674 (1985)

    Google Scholar 

  37. Biere, A.: CaDiCaL, Lingeling, Plingeling, Treengeling and YalSAT entering the SAT competition 2018. In: Heule, M., Järvisalo, M., Suda, M. (eds.) Proceedings of SAT Competition 2018 – Solver and Benchmark Descriptions, pp. 13–14. Volume B-2018-1 of Department of Computer Science Series of Publications B, University of Helsinki (2018)

    Google Scholar 

  38. Fazekas, K., Biere, A., Scholl, C.: Incremental inprocessing in SAT solving. In: Janota, M., Lynce, I. (eds.) SAT 2019. LNCS, vol. 11628, pp. 136–154. Springer, Cham (2019)

    Chapter  Google Scholar 

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Acknowledgment

This work has been supported by the National Science Foundation under grant CCF-1618574 and by the Austrian Science Fund (FWF) under project W1255 (LogiCS) and S11409-N23 (RiSE).

We want to thank Oliver Kullmann, who explained to the second author an (as far as we know unpublished) algorithm to compute the maximal autarky of a total assignment, which is a special case of \(\mathsf {LeastConditionalPart}\) in Fig. 1.

This work was triggered by Gianpiero Cabodi who asked the last author whether there is a CNF level version of unique sensitization as explained in Sect. 3 with potential applications in SAT-based model checking.

We would finally also like to thank the organizers of ATVA’19 for inviting the last author to present these ideas as invited talk and include this invited paper in the proceedings.

Author information

Authors and Affiliations

  1. CISPA Helmholtz Center for Information Security, Saarbrücken, Germany

    Benjamin Kiesl

  2. Computer Science Department, CMU, Pittsburgh, USA

    Marijn J. H. Heule

  3. Institute for Formal Models and Verification, JKU, Linz, Austria

    Armin Biere

Authors
  1. Benjamin Kiesl
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  2. Marijn J. H. Heule
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  3. Armin Biere
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Correspondence to Armin Biere .

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Editors and Affiliations

  1. Institute of Information Science, Academia Sinica, Taipei, Taiwan

    Yu-Fang Chen

  2. DENSO AUTOMOTIVE Deutschland GmbH, Eching, Germany

    Chih-Hong Cheng

  3. Institute of Computer Science, TU München, Munich, Germany

    Javier Esparza

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Kiesl, B., Heule, M.J.H., Biere, A. (2019). Truth Assignments as Conditional Autarkies. In: Chen, YF., Cheng, CH., Esparza, J. (eds) Automated Technology for Verification and Analysis. ATVA 2019. Lecture Notes in Computer Science(), vol 11781. Springer, Cham. https://doi.org/10.1007/978-3-030-31784-3_3

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  • DOI: https://doi.org/10.1007/978-3-030-31784-3_3

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