Circular (Yet Sound) Proofs

  • Albert Atserias
  • Massimo LauriaEmail author
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 11628)


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.



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.


  1. 1.
    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
  2. 2.
    Atserias, A., Lauria, M.: Circular (yet sound) proofs. CoRR, abs/1802.05266 (2018)Google Scholar
  3. 3.
    Atserias, A., Lauria, M., Nordström, J.: Narrow proofs may be maximally long. ACM Trans. Comput. Log. 17(3), 19:1–19:30 (2016)MathSciNetCrossRefGoogle Scholar
  4. 4.
    Au, Y.H., Tunçel, L.: A comprehensive analysis of polyhedral lift-and-project methods. SIAM J. Discrete Math. 30(1), 411–451 (2016)MathSciNetCrossRefGoogle Scholar
  5. 5.
    Ben-Sasson, E., Wigderson, A.: Short proofs are narrow - resolution made simple. J. ACM 48(2), 149–169 (2001)MathSciNetCrossRefGoogle Scholar
  6. 6.
    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)Google Scholar
  7. 7.
    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)MathSciNetCrossRefGoogle Scholar
  8. 8.
    Brotherston, J.: Sequent calculus proof systems for inductive definitions. Ph.D. thesis, University of Edinburgh, November 2006Google Scholar
  9. 9.
    Brotherston, J., Simpson, A.: Sequent calculi for induction and infinite descent. J. Log. Comput. 21(6), 1177–1216 (2011)MathSciNetCrossRefGoogle Scholar
  10. 10.
    Buss, S.R.: Polynomial size proofs of the propositional pigeonhole principle. J. Symb. Log. 52(4), 916–927 (1987)MathSciNetCrossRefGoogle Scholar
  11. 11.
    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)Google Scholar
  12. 12.
    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)MathSciNetCrossRefGoogle Scholar
  13. 13.
    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). Scholar
  14. 14.
    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). Scholar
  15. 15.
    Grigoriev, D.: Linear lower bound on degrees of positivstellensatz calculus proofs for the parity. Theor. Comput. Sci. 259(1), 613–622 (2001)MathSciNetCrossRefGoogle Scholar
  16. 16.
    Haken, A.: The intractability of resolution. Theor. Comp. Sci. 39, 297–308 (1985)MathSciNetCrossRefGoogle Scholar
  17. 17.
    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). Scholar
  18. 18.
    Krajíček, J.: Bounded Arithmetic, Propositional Logic and Complexity Theory. Cambridge University Press, Cambridge (1994)zbMATHGoogle Scholar
  19. 19.
    Niwiński, D., Walukiewicz, I.: Games for the \(\mu \)-calculus. Theor. Comp. Sci. 163(1), 99–116 (1996)MathSciNetCrossRefGoogle Scholar
  20. 20.
    Pitassi, T., Segerlind, N.: Exponential lower bounds and integrality gaps for tree-like Lovász-Schrijver procedures. SIAM J. Comput. 41(1), 128–159 (2012)MathSciNetCrossRefGoogle Scholar
  21. 21.
    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)MathSciNetCrossRefGoogle Scholar
  22. 22.
    Shoesmith, J., Smiley, T.J.: Multiple-Conclusion Logic. Cambridge University Press, Cambridge (1978)CrossRefGoogle Scholar
  23. 23.
    Studer, T.: On the proof theory of the modal mu-calculus. Studia Logica 89(3), 343–363 (2008)MathSciNetCrossRefGoogle Scholar
  24. 24.
    Vinyals, M.: Personal communication (2018)Google Scholar

Copyright information

© Springer Nature Switzerland AG 2019

Authors and Affiliations

  1. 1.Universitat Politècnica de CatalunyaBarcelonaSpain
  2. 2.Sapienza - Università di RomaRomeItaly

Personalised recommendations