Abstract
We introduce a new way of composing proofs in rule-based proof systems that generalizes tree-like and dag-like proofs. In the new definition, proofs are directed graphs of derived formulas, in which cycles are allowed as long as every formula is derived at least as many times as it is required as a premise. We call such proofs circular. We show that, for all sets of standard inference rules, circular proofs are sound. We first focus on the circular version of Resolution, and see that it is stronger than Resolution since, as we show, the pigeonhole principle has circular Resolution proofs of polynomial size. Surprisingly, as proof systems for deriving clauses from clauses, Circular Resolution turns out to be equivalent to Sherali-Adams, a proof system for reasoning through polynomial inequalities that has linear programming at its base. As corollaries we get: (1) polynomial-time (LP-based) algorithms that find circular Resolution proofs of constant width, (2) examples that separate circular from dag-like Resolution, such as the pigeonhole principle and its variants, and (3) exponentially hard cases for circular Resolution. Contrary to the case of circular resolution, for Frege we show that circular proofs can be converted into tree-like ones with at most polynomial overhead.
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Atserias, A., Hakoniemi, T.: Size-degree trade-offs for sums-of-squares and Positivstellensatz proofs. To appear in Proceedings of 34th Annual Conference on Computational Complexity (CCC 2019) (2019). Long version in arXiv:1811.01351 [cs.CC] 2018
Atserias, A., Lauria, M.: Circular (yet sound) proofs. CoRR, abs/1802.05266 (2018)
Atserias, A., Lauria, M., Nordström, J.: Narrow proofs may be maximally long. ACM Trans. Comput. Log. 17(3), 19:1–19:30 (2016)
Au, Y.H., Tunçel, L.: A comprehensive analysis of polyhedral lift-and-project methods. SIAM J. Discrete Math. 30(1), 411–451 (2016)
Ben-Sasson, E., Wigderson, A.: Short proofs are narrow - resolution made simple. J. ACM 48(2), 149–169 (2001)
Bonet, M.L., Buss, S., Ignatiev, A., Marques-Silva, J., Morgado, A.: MaxSAT resolution with the dual rail encoding. In: Proceedings of the 32nd AAAI Conference on Artificial Intelligence (2018)
Bonet, M.L., Esteban, J.L., Galesi, N., Johannsen, J.: On the relative complexity of resolution refinements and cutting planes proof systems. SIAM J. Comput. 30(5), 1462–1484 (2000)
Brotherston, J.: Sequent calculus proof systems for inductive definitions. Ph.D. thesis, University of Edinburgh, November 2006
Brotherston, J., Simpson, A.: Sequent calculi for induction and infinite descent. J. Log. Comput. 21(6), 1177–1216 (2011)
Buss, S.R.: Polynomial size proofs of the propositional pigeonhole principle. J. Symb. Log. 52(4), 916–927 (1987)
Dantchev, S.S.: Rank complexity gap for Lovász-Schrijver and Sherali-Adams proof systems. In: Proceedings of the 39th Annual ACM Symposium on Theory of Computing, pp. 311–317 (2007)
Dantchev, S.S., Martin, B., Rhodes, M.N.C.: Tight rank lower bounds for the Sherali-Adams proof system. Theor. Comput. Sci. 410(21–23), 2054–2063 (2009)
Dax, C., Hofmann, M., Lange, M.: A proof system for the linear time \({\mu }\)-calculus. In: Arun-Kumar, S., Garg, N. (eds.) FSTTCS 2006. LNCS, vol. 4337, pp. 273–284. Springer, Heidelberg (2006). https://doi.org/10.1007/11944836_26
Goerdt, A.: Cutting plane versus frege proof systems. In: Börger, E., Kleine Büning, H., Richter, M.M., Schönfeld, W. (eds.) CSL 1990. LNCS, vol. 533, pp. 174–194. Springer, Heidelberg (1991). https://doi.org/10.1007/3-540-54487-9_59
Grigoriev, D.: Linear lower bound on degrees of positivstellensatz calculus proofs for the parity. Theor. Comput. Sci. 259(1), 613–622 (2001)
Haken, A.: The intractability of resolution. Theor. Comp. Sci. 39, 297–308 (1985)
Ignatiev, A., Morgado, A., Marques-Silva, J.: On tackling the limits of resolution in SAT solving. In: Gaspers, S., Walsh, T. (eds.) SAT 2017. LNCS, vol. 10491, pp. 164–183. Springer, Cham (2017). https://doi.org/10.1007/978-3-319-66263-3_11
Krajíček, J.: Bounded Arithmetic, Propositional Logic and Complexity Theory. Cambridge University Press, Cambridge (1994)
Niwiński, D., Walukiewicz, I.: Games for the \(\mu \)-calculus. Theor. Comp. Sci. 163(1), 99–116 (1996)
Pitassi, T., Segerlind, N.: Exponential lower bounds and integrality gaps for tree-like Lovász-Schrijver procedures. SIAM J. Comput. 41(1), 128–159 (2012)
Sherali, H.D., Adams, W.P.: A hierarchy of relaxations between the continuous and convex hull representations for zero-one programming problems. SIAM J. Disc. Math. 3(3), 411–430 (1990)
Shoesmith, J., Smiley, T.J.: Multiple-Conclusion Logic. Cambridge University Press, Cambridge (1978)
Studer, T.: On the proof theory of the modal mu-calculus. Studia Logica 89(3), 343–363 (2008)
Vinyals, M.: Personal communication (2018)
Acknowledgments
Both authors were partially funded by European Research Council (ERC) under the European Union’s Horizon 2020 research and innovation programme, grant agreement ERC-2014-CoG 648276 (AUTAR). First author partially funded by MICINN through TIN2016-76573-C2-1P (TASSAT3). We acknowledge the work of Jordi Coll who conducted experimental results for finding and visualizing actual circular resolution proofs of small instances of the sparse pigeonhole principle.
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Atserias, A., Lauria, M. (2019). Circular (Yet Sound) Proofs. 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_1
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