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DRAT Proofs, Propagation Redundancy, and Extended Resolution

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Theory and Applications of Satisfiability Testing – SAT 2019 (SAT 2019)

Part of the book series: Lecture Notes in Computer Science ((LNTCS,volume 11628))

Abstract

We study the proof complexity of RAT proofs and related systems including BC, SPR, and PR which use blocked clauses and (subset) propagation redundancy. These systems arise in satisfiability (SAT) solving, and allow inferences which preserve satisfiability but not logical implication. We introduce a new inference SR using substitution redundancy. We consider systems both with and without deletion. With new variables allowed, the systems are known to have the same proof theoretic strength as extended resolution. We focus on the systems that do not allow new variables to be introduced.

Our first main result is that the systems DRAT\({}^-\), DSPR\({}^-\) and DPR\({}^-\), which allow deletion but not new variables, are polynomially equivalent. By earlier work of Kiesl, Rebola-Pardo and Heule, they are also equivalent to DBC\({}^-\). Without deletion and without new variables, we show that SPR\({}^-\) can polynomially simulate PR\({}^-\) provided only short clauses are inferred by SPR inferences. Our next main results are that many of the well-known “hard” principles have polynomial size SPR\({}^-\) refutations (without deletions or new variables). These include the pigeonhole principle, bit pigeonhole principle, parity principle, Tseitin tautologies, and clique-coloring tautologies; SPR\({}^-\) can also handle or-fication and xor-ification. Our final result is an exponential size lower bound for RAT\({}^-\) refutations, giving exponential separations between RAT\({}^-\) and both DRAT\({}^-\) and SPR\({}^-\).

S. Buss—This work was initiated on a visit of the first author to the Czech Academy of Sciences in July 2018, supported by ERC advanced grant 339691 (FEALORA). Also supported by Simons Foundation grant 578919.

N. Thapen—Partially supported by GA ČR project 19-05497S and by ERC advanced grant 339691 (FEALORA) and RVO:67985840.

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References

  1. Ajtai, M.: Parity and the pigeonhole principle. In: Buss, S.R., Scott, P.J. (eds.) Feasible Mathematics, pp. 1–24. Birkhäuser, Boston (1990)

    Google Scholar 

  2. Beame, P., Impagliazzo, R., Krajíček, J., Pitassi, T., Pudlák, P.: Lower bounds on Hilbert’s Nullstellensatz and propositional proofs. Proc. Lond. Math. Soc. 73(3), 1–26 (1996)

    Article  MathSciNet  Google Scholar 

  3. Ben-Sasson, E.: Size space tradeoffs for resolution. SIAM J. Comput. 38(6), 2511–2525 (2009)

    Article  MathSciNet  Google Scholar 

  4. Ben-Sasson, E., Impagliazzo, R., Wigderson, A.: Near optimal separation of tree-like and general resolution. Combinatorica 24(4), 585–603 (2004)

    Article  MathSciNet  Google Scholar 

  5. Ben-Sasson, E., Wigderson, A.: Short proofs are narrow—resolution made simple. J. ACM 48, 149–169 (2001)

    Article  MathSciNet  Google Scholar 

  6. Cook, S.A., Reckhow, R.A.: On the lengths of proofs in the propositional calculus, preliminary version. In: Proceedings of the Sixth Annual ACM Symposium on the Theory of Computing, pp. 135–148 (1974)

    Google Scholar 

  7. Cook, S.A., Reckhow, R.A.: The relative efficiency of propositional proof systems. J. Symb. Log. 44, 36–50 (1979)

    Article  MathSciNet  Google Scholar 

  8. Goldberg, E.I., Novikov, Y.: Verification of proofs of unsatisfiability for CNF formulas. In: Design, Automation and Test in Europe Conference (DATE), pp. 10886–10891. IEEE Computer Society (2003)

    Google Scholar 

  9. 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). https://doi.org/10.1007/978-3-319-89963-3_5

    Chapter  Google Scholar 

  10. Heule, M.J.H., Hunt Jr., W.A., Wetzler, N.: Trimming while checking clausal proofs. In: Formal Methods in Computer-Aided Design (FMCAD), pp. 181–188. IEEE (2013)

    Google Scholar 

  11. Heule, M.J.H., Hunt Jr., W.A., Wetzler, N.: Verifying refutations with extended resolution. In: Bonacina, M.P. (ed.) CADE 2013. LNCS (LNAI), vol. 7898, pp. 345–359. Springer, Heidelberg (2013). https://doi.org/10.1007/978-3-642-38574-2_24

    Chapter  Google Scholar 

  12. 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). https://doi.org/10.1007/978-3-319-63046-5_9

    Chapter  Google Scholar 

  13. Heule, M.J.H., Kiesl, B., Biere, A.: Strong extension-free proof systems. J. Autom. Reason. 1–22 (2019). https://doi.org/10.1007/s10817-019-09516-0. Extended version of [12]

  14. Heule, M.J.H., Kiesl, B., Seidl, M., Biere, A.: PRuning through satisfaction. Hardware and Software: Verification and Testing. LNCS, vol. 10629, pp. 179–194. Springer, Cham (2017). https://doi.org/10.1007/978-3-319-70389-3_12

    Chapter  Google Scholar 

  15. 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). https://doi.org/10.1007/978-3-642-31365-3_28

    Chapter  Google Scholar 

  16. Kiesl, B., Rebola-Pardo, A., Heule, M.J.H.: Extended resolution simulates DRAT. In: Galmiche, D., Schulz, S., Sebastiani, R. (eds.) IJCAR 2018. LNCS (LNAI), vol. 10900, pp. 516–531. Springer, Cham (2018). https://doi.org/10.1007/978-3-319-94205-6_34

    Chapter  Google Scholar 

  17. Krajíček, J.: Interpolation theorems, lower bounds for proof systems, and independence results for bounded arithmetic. J. Symb. Log. 62, 457–486 (1997)

    Article  MathSciNet  Google Scholar 

  18. Krajíček, J., Pudlák, P., Woods, A.: Exponential lower bound to the size of bounded depth Frege proofs of the pigeonhole principle. Random Struct. Algorithms 7, 15–39 (1995)

    Article  MathSciNet  Google Scholar 

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

    Article  Google Scholar 

  20. Pitassi, T., Beame, P., Impagliazzo, R.: Exponential lower bounds for the pigeonhole principle. Comput. Complex. 3, 97–140 (1993)

    Article  MathSciNet  Google Scholar 

  21. Pudlák, P.: Lower bounds for resolution and cutting planes proofs and monotone computations. J. Symb. Log. 62, 981–998 (1997)

    Article  MathSciNet  Google Scholar 

  22. Rebola-Pardo, A., Suda, M.: A theory of satisfiability-preserving proofs in SAT solving. In: Proceedings 22nd International Conference on Logic for Programming, Artificial Intelligence and Reasoning (LPAR-22). EPiC Series in Computing, vol. 57, pp. 583–603. EasyChair (2018)

    Google Scholar 

  23. Siekmann, J., Wrightson, G.: Automation of Reasoning, vol. 1&2. Springer, Berlin (1983)

    Book  Google Scholar 

  24. Tsejtin, G.S.: On the complexity of derivation in propositional logic. In: Studies in Constructive Mathematics and Mathematical Logic, part 2, pp. 115–125 (1968). Reprinted in: [23, vol. 2], pp. 466–483

    Google Scholar 

  25. Urquhart, A.: Hard examples for resolution. J. ACM 34, 209–219 (1987)

    Article  MathSciNet  Google Scholar 

  26. Urquhart, A.: A near-optimal separation of regular and general resolution. SIAM J. Comput. 40(1), 107–121 (2011)

    Article  MathSciNet  Google Scholar 

  27. Van Gelder, A.: Verifying RUP proofs of propositional unsatisfiability. In: 10th International Symposium on Artificial Intelligence and Mathematics (ISAIM) (2008). http://isaim2008.unl.edu/index.php?page=proceedings

  28. Wetzler, N., Heule, M.J.H., Hunt, 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). https://doi.org/10.1007/978-3-319-09284-3_31

    Chapter  MATH  Google Scholar 

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Authors and Affiliations

  1. Department of Mathematics, U.C. San Diego, La Jolla, CA, USA

    Sam Buss

  2. Institute of Mathematics, Czech Academy of Sciences, Prague, Czech Republic

    Neil Thapen

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  1. Sam Buss
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Correspondence to Sam Buss .

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Editors and Affiliations

  1. University of Lisbon, Lisbon, Portugal

    Mikoláš Janota

  2. University of Lisbon, Lisbon, Portugal

    Inês Lynce

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Buss, S., Thapen, N. (2019). DRAT Proofs, Propagation Redundancy, and Extended Resolution. In: Janota, M., Lynce, I. (eds) Theory and Applications of Satisfiability Testing – SAT 2019. SAT 2019. Lecture Notes in Computer Science(), vol 11628. Springer, Cham. https://doi.org/10.1007/978-3-030-24258-9_5

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  • DOI: https://doi.org/10.1007/978-3-030-24258-9_5

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