Cited By
View all- Bonacina IBonet MLevy J(2024)Polynomial calculus for optimizationArtificial Intelligence10.1016/j.artint.2024.104208337:COnline publication date: 1-Dec-2024
Circular (Yet Sound) Proofs in Propositional Logic
Article No.: 20, Pages 1 - 26
Abstract
Proofs in propositional logic are typically presented as trees of derived formulas or, alternatively, as directed acyclic graphs of derived formulas. This distinction between tree-like vs. dag-like structure is particularly relevant when making quantitative considerations regarding, for example, proof size. Here we analyze a more general type of structural restriction for proofs in rule-based proof systems. In this 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 with single or multiple conclusions, circular proofs are sound. We start the study of the proof complexity of circular proofs at Circular Resolution, the circular version of Resolution. We immediately see that Circular Resolution is stronger than dag-like Resolution since, as we show, the propositional encoding of the pigeonhole principle has circular Resolution proofs of polynomial size. Furthermore, for derivations of clauses from clauses, we show that Circular Resolution is, surprisingly, 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 proofs with at most polynomial overhead.
References
[1]
Miklos Ajtai. 1988. The complexity of the pigeonhole principle. In 29th Annual Symposium on Foundations of Computer Science. 346–355. DOI:
[2]
A. Atserias and T. Hakoniemi. 2019. Size-degree trade-offs for sums-of-squares and positivstellensatz proofs. In Proceedings of 34th Annual Conference on Computational Complexity (CCC’19). Vol. 137, Schloss Dagstuhl - Leibniz Center for Informatics (LZI), 24:1–24:20.
[3]
Albert Atserias and Massimo Lauria. 2019. Circular (yet sound) proofs. In 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), Mikolás Janota and Inês Lynce (Eds.), Vol. 11628. Springer, 1–18. DOI:
[4]
A. Atserias, M. Lauria, and J. Nordström. 2016. Narrow proofs may be maximally long. ACM Trans. Comput. Log. 17, 3 (2016), 19:1–19:30. DOI:
[5]
Yu Hin Au and L. Tunçel. 2016. A comprehensive analysis of polyhedral lift-and-project methods. SIAM J. Discrete Math. 30, 1 (2016), 411–451.
[6]
E. Ben-Sasson and A. Wigderson. 2001. Short proofs are narrow - resolution made simple. J. ACM 48, 2 (2001), 149–169. DOI:
[7]
M. L. Bonet, S. Buss, A. Ignatiev, J. Marques-Silva, and A. Morgado. 2018. MaxSAT resolution with the dual rail encoding. In Proc. 32nd AAAI Conference on Artificial Intelligence. https://www.aaai.org/ocs/index.php/AAAI/AAAI18/paper/view/16782.
[8]
M. L. Bonet, J. L. Esteban, N. Galesi, and J. Johannsen. 2000. On the relative complexity of resolution refinements and cutting planes proof systems. SIAM J. Comp. 30, 5 (2000). DOI:
[9]
Maria Luisa Bonet and Jordi Levy. 2020. Equivalence between systems stronger than resolution. In Theory and Applications of Satisfiability Testing - SAT 2020-23rd International Conference, Alghero, Italy, July 3–10, 2020, Proceedings (Lecture Notes in Computer Science), Luca Pulina and Martina Seidl (Eds.), Vol. 12178. Springer, 166–181. DOI:
[10]
J. Brotherston. 2006. Sequent Calculus Proof Systems for Inductive Definitions. Ph.D. Dissertation. University of Edinburgh.
[11]
J. Brotherston and A. Simpson. 2011. Sequent calculi for induction and infinite descent. Journal of Logic and Computation 21, 6 (2011), 1177–1216. DOI: arXiv:.
[12]
Samuel R. Buss. 1987. Polynomial size proofs of the propositional pigeonhole principle. Journal of Symbolic Logic 52, 4 (1987), 916–927. DOI:
[13]
Stephen A. Cook and Robert A. Reckhow. 1979. The relative efficiency of propositional proof systems. J. Symb. Log. 44, 1 (1979), 36–50. DOI:
[14]
S. S. Dantchev. 2007. Rank complexity gap for Lovász-Schrijver and Sherali-Adams proof systems. In Proc. 39th Annual ACM Symposium on Theory of Computing. 311–317. DOI:
[15]
S. S. Dantchev, B. Martin, and M. N. C. Rhodes. 2009. Tight rank lower bounds for the Sherali-Adams proof system. Theor. Comput. Sci. 410, 21-23 (2009), 2054–2063. DOI:
[16]
Anupam Das. 2020. On the logical complexity of cyclic arithmetic. Log. Methods Comput. Sci. 16, 1 (2020). DOI:
[17]
C. Dax, M. Hoffman, and M. Lange. 2006. A Proof System for the Linear Time \(\mu\)-Calculus. Springer Berlin, Berlin, 273–284. DOI:
[18]
Jérôme Fortier. 2014. Puissance Expressive des Preuves Circulaires. (Expressive Power of Circular Proofs). Ph.D. Dissertation. Aix-Marseille University, Aix-en-Provence, France. https://tel.archives-ouvertes.fr/tel-01154972.
[19]
Andreas Goerdt. 1991. Cutting plane versus Frege proof systems. In Computer Science Logic: 4th Workshop, CSL’90 Heidelberg, Germany, October 1–5, 1990 Proceedings, Egon Börger, Hans Kleine Büning, Michael M. Richter, and Wolfgang Schönfeld (Eds.). Springer Berlin, Berlin, 174–194.
[20]
D. Grigoriev. 2001. Linear lower bound on degrees of Positivstellensatz calculus proofs for the parity. Theoretical Computer Science 259, 1 (2001), 613–622. DOI:
[21]
A. Haken. 1985. The intractability of resolution. Theor. Comp. Sci. 39 (1985), 297–308. DOI:
[22]
Alexey Ignatiev, António Morgado, and João Marques-Silva. 2017. On tackling the limits of resolution in SAT solving. In Proc. 20th International Conference on Theory and Applications of Satisfiability Testing - SAT 2017. 164–183. DOI:
[23]
J. Krajíček. 1994. Bounded Arithmetic, Propositional Logic, Complexity Theory. Cambridge.
[24]
Jan Krajíček, Pavel Pudlák, and Alan Woods. 1995. An exponential lower bound to the size of bounded depth Frege proofs of the pigeonhole principle. Random Structures and Algorithms 7, 1 (1995), 15–39. DOI:
[25]
Javier Larrosa and Emma Rollon. 2020. Towards a better understanding of (partial weighted) MaxSAT proof systems. In Theory and Applications of Satisfiability Testing - SAT 2020-23rd International Conference, Alghero, Italy, July 3–10, 2020, Proceedings (Lecture Notes in Computer Science), Luca Pulina and Martina Seidl (Eds.), Vol. 12178. Springer, 218–232. DOI:
[26]
D. Niwiński and I. Walukiewicz. 1996. Games for the \(\mu\)-calculus. Theor. Comp. Sci. 163, 1 (1996). DOI:
[27]
Toniann Pitassi, Paul Beame, and Russell Impagliazzo. 1993. Exponential lower bounds for the pigeonhole principle. Computational Complexity 3 (1993), 97–140. DOI:
[28]
T. Pitassi and N. Segerlind. 2012. Exponential lower bounds and integrality gaps for tree-like Lovász-Schrijver procedures. SIAM J. Comput. 41, 1 (2012), 128–159. DOI:
[29]
Alexander Schrijver. 2003. Combinatorial Optimization: Polyhedra and Efficiency. Vol. 24. Springer Science & Business Media.
[30]
Daniyar S. Shamkanov. 2020. Non-well-founded derivations in the Gödel-löB provability logic. Rev. Symb. Log. 13, 4 (2020), 776–796. DOI:Early draft as http://arxiv.org/abs/1401.4002 in 2014.
[31]
H. D. Sherali and W. P. Adams. 1990. A hierarchy of relaxations between the continuous and convex hull representations for zero-one programming problems. SIAM J. Disc. Math. 3, 3 (1990), 411–430.
[32]
J. Shoesmith and T. J. Smiley. 1978. Multiple-Conclusion Logic. Cambridge.
[33]
Alex Simpson. 2017. Cyclic arithmetic is equivalent to peano arithmetic. In Foundations of Software Science and Computation Structures - 20th International Conference, FOSSACS 2017, Held as Part of the European Joint Conferences on Theory and Practice of Software, ETAPS 2017, Uppsala, Sweden, April 22–29, 2017, Proceedings (Lecture Notes in Computer Science), Javier Esparza and Andrzej S. Murawski (Eds.), Vol. 10203. 283–300. DOI:
[34]
T. Studer. 2008. On the proof theory of the modal mu-calculus. Studia Logica 89, 3 (2008). DOI:
[35]
W. W. Tait. 1968. Normal derivability in classical logic. In The Syntax and Semantics of Infinitary Languages, J. Barwise (Ed.). Lecture Notes in Mathematics, Vol. 72. Springer-Verlag, 204–236.
[36]
M. Vinyals. 2018. Personal communication. (2018).
Index Terms
- Circular (Yet Sound) Proofs in Propositional Logic
Recommendations
Quasipolynomial size proofs of the propositional pigeonhole principle
Cook and Reckhow proved in 1979 that the propositional pigeonhole principle has polynomial size extended Frege proofs. Buss proved in 1987 that it also has polynomial size Frege proofs; these Frege proofs used a completely different proof method based ...
Parameterized Bounded-Depth Frege Is not Optimal
A general framework for parameterized proof complexity was introduced by Dantchev et al. [2007]. There, the authors show important results on tree-like Parameterized Resolution---a parameterized version of classical Resolution---and their gap complexity ...
Polynomial-size Frege and resolution proofs of st-connectivity and Hex tautologies
Clifford lectures and the mathematical foundations of programming semanticsA grid graph has rectangularly arranged vertices with edges permitted only between orthogonally adjacent vertices. The st- connectivity principle states that it is not possible to have a red path of edges and a green path of edges which connect ...
Comments
Information & Contributors
Information
Published In
July 2023
268 pages
Permission to make digital or hard copies of all or part of this work for personal or classroom use is granted without fee provided that copies are not made or distributed for profit or commercial advantage and that copies bear this notice and the full citation on the first page. Copyrights for components of this work owned by others than the author(s) must be honored. Abstracting with credit is permitted. To copy otherwise, or republish, to post on servers or to redistribute to lists, requires prior specific permission and/or a fee. Request permissions from [email protected].
Publisher
Association for Computing Machinery
New York, NY, United States
Publication History
Published: 07 April 2023
Online AM: 30 January 2023
Accepted: 01 November 2022
Revised: 30 November 2021
Received: 03 September 2019
Published in TOCL Volume 24, Issue 3
Check for updates
Author Tags
Qualifiers
- Research-article
Funding Sources
- European Research Council (ERC)
- European Union’s Horizon 2020 research and innovation programme
- MINECO
Contributors
Other Metrics
Bibliometrics & Citations
Bibliometrics
Article Metrics
- View Citations1Total Citations
- 215Total Downloads
- Downloads (Last 12 months)55
- Downloads (Last 6 weeks)2
Reflects downloads up to 18 Jan 2025
Other Metrics
Citations
Cited By
View all- Bonacina IBonet MLevy J(2024)Polynomial calculus for optimizationArtificial Intelligence10.1016/j.artint.2024.104208337:COnline publication date: 1-Dec-2024
View Options
Login options
Check if you have access through your login credentials or your institution to get full access on this article.
Sign inFull Access
View options
View or Download as a PDF file.
PDFeReader
View online with eReader.
eReaderFull Text
View this article in Full Text.
Full TextHTML Format
View this article in HTML Format.
HTML Format