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On the logical complexity of cyclic arithmetic. (English) Zbl 07155168
Summary: We study the logical complexity of proofs in cyclic arithmetic $$(\mathsf{CA})$$, as introduced by A. Simpson in [Lect. Notes Comput. Sci. 10203, 283–300 (2017; Zbl 06720996)], in terms of quantifier alternations of formulae occurring. Writing $$C\Sigma_n$$ for (the logical consequences of) cyclic proofs containing only $$\Sigma_n$$ formulae, our main result is that $$I\Sigma_{n+1}$$ and $$C\Sigma_n$$ prove the same $$\Pi_{n+1}$$ theorems, for all $$n\geq 0$$. Furthermore, due to the ‘uniformity’ of our method, we also show that $$\mathsf{CA}$$ and Peano Arithmetic $$( \mathsf{PA} )$$ proofs of the same theorem differ only exponentially in size.
The inclusion $$I\Sigma_{n+1} \subseteq C\Sigma_n$$ is obtained by proof theoretic techniques, relying on normal forms and structural manipulations of $$\mathsf{PA}$$ proofs. It improves upon the natural result that $$I\Sigma_n$$ is contained in $$C\Sigma_n$$. The converse inclusion, $$C\Sigma_n \subseteq I\Sigma_{n+1}$$, is obtained by calibrating the approach of [Simpson, loc. cit.] with recent results on the reverse mathematics of Büchi’s theorem in [L. Kołodziejczyk et al., Log. Methods Comput. Sci. 15, No. 2, Paper No. 16, 31 p. (2019; Zbl 07058775)], and specialising to the case of cyclic proofs. These results improve upon the bounds on proof complexity and logical complexity implicit in [Simpson, loc. cit.] and also an alternative approach due to S. Berardi and M. Tatsuta [“Equivalence of inductive definitions and cyclic proofs under arithmetic”, in: Proceedings of the 32nd annual ACM/IEEE symposium on logic in computer science, LICS 2017. IEEE Digital Library. 1–12 (2017; doi:10.1109/LICS.2017.8005114)].
The uniformity of our method also allows us to recover a metamathematical account of fragments of $$\mathsf{CA}$$; in particular we show that, for $$n\geq 0$$, the consistency of $$C\Sigma_n$$ is provable in $$I\Sigma_{n+2}$$ but not $$I\Sigma_{n+1}$$. As a result, we show that certain versions of McNaughton’s theorem (the determinisation of $$\omega$$-word automata) are not provable in $$\mathsf{RCA}_0$$, partially resolving an open problem from [Kołodziejczyk et al., loc. cit.].

##### MSC:
 03B70 Logic in computer science 68 Computer science
Cyclist
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##### References:
 [1] [AL17]Bahareh Afshari and Graham E. Leigh. Cut-free completeness for modalµ-calculus. In32nd Annual ACM/IEEE Symposium on Logic in Computer Science, LICS 2017, Reykjavik, Iceland, June 20-23, 2017, pages 1-12, 2017. [2] [AL18]Albert Atserias and Massimo Lauria. Circular (yet sound) proofs.CoRR, abs/1802.05266, 2018. [3] [BBC08]James Brotherston, Richard Bornat, and Cristiano Calcagno. Cyclic proofs of program termination in separation logic. InProceedings of the 35th ACM SIGPLAN-SIGACT Symposium on Principles · Zbl 1295.68156 [4] [BDP11]James Brotherston, Dino Distefano, and Rasmus Lerchedahl Petersen. Automated cyclic entailment proofs in separation logic. InCADE-23 - 23rd International Conference on Automated Deduction, · Zbl 1341.68184 [5] [BDS16]David Baelde, Amina Doumane, and Alexis Saurin. Infinitary proof theory: the multiplicative additive case. In25th EACSL Annual Conference on Computer Science Logic, CSL 2016, August · Zbl 1370.03077 [6] [BGP12]James Brotherston, Nikos Gorogiannis, and Rasmus L. Petersen. A generic cyclic theorem prover. InProgramming Languages and Systems - 10th Asian Symposium, APLAS 2012, Kyoto, Japan, [7] [Bro05]James Brotherston. Cyclic proofs for first-order logic with inductive definitions. InAutomated Reasoning with Analytic Tableaux and Related Methods, International Conference, TABLEAUX 2005, Koblenz, Germany, September 14-17, 2005, Proceedings, pages 78-92, 2005. · Zbl 1142.03366 [8] [Bro06]James Brotherston.Sequent calculus proof systems for inductive definitions. PhD thesis, University of Edinburgh, 2006. [9] [BS07]James Brotherston and Alex Simpson. Complete sequent calculi for induction and infinite descent. In22nd IEEE Symposium on Logic in Computer Science (LICS 2007), 10-12 July 2007, Wroclaw, [10] [BS11]James Brotherston and Alex Simpson. Sequent calculi for induction and infinite descent.J. Log. Comput., 21(6):1177-1216, 2011. · Zbl 1242.03084 [11] [BT17a]Stefano Berardi and Makoto Tatsuta. Classical system of Martin-L¨of’s inductive definitions is not equivalent to cyclic proof system. InFoundations of Software Science and Computation · Zbl 06720997 [12] [BT17b]Stefano Berardi and Makoto Tatsuta. Equivalence of inductive definitions and cyclic proofs under arithmetic. In32nd Annual ACM/IEEE Symposium on Logic in Computer Science, LICS 2017, · Zbl 06720997 [13] [Bus95]Samuel R. Buss. The witness function method and provably recursive functions of Peano arithmetic. InStudies in Logic and the Foundations of Mathematics, volume 134, pages 29-68. Elsevier, 1995. · Zbl 0829.03036 [14] [Bus98]Samuel R. Buss, editor.Handbook of Proof Theory. Studies in Logic and the Foundations of Mathematics 137. Elsevier, 1998. · Zbl 0898.03001 [15] [CK02]Peter Clote and Evangelos Kranakis.Boolean Functions and Computation Models. Texts in Theoretical Computer Science. An EATCS Series. Springer, 2002. · Zbl 1016.94046 [16] [CN10]Stephen Cook and Phuong Nguyen.Logical Foundations of Proof Complexity. Cambridge University Press, New York, NY, USA, 1st edition, 2010. · Zbl 1206.03051 [17] [DBHS16] Amina Doumane, David Baelde, Lucca Hirschi, and Alexis Saurin. Towards completeness via proof search in the linear timeµ-calculus: The case of B¨uchi inclusions. InProceedings of the 31st · Zbl 1401.68193 [18] [DHL06]Christian Dax, Martin Hofmann, and Martin Lange. A proof system for the linear timeµ-calculus. InFSTTCS 2006: Foundations of Software Technology and Theoretical Computer Science, 26th · Zbl 1163.03308 [19] [Dou17]Amina Doumane. Constructive completeness for the linear-timeµ-calculus. In32nd Annual ACM/IEEE Symposium on Logic in Computer Science, LICS 2017, Reykjavik, Iceland, June 20-23, 2017, pages 1-12, 2017. [20] [DP17]Anupam Das and Damien Pous. A cut-free cyclic proof system for Kleene algebra. InAutomated Reasoning with Analytic Tableaux and Related Methods - 26th International Conference, [21] [For14]J´erˆome Fortier.Puissance expressive des preuves circulaires. (Expressive Power of Circular Proofs). PhD thesis, Aix-Marseille University, Aix-en-Provence, France, 2014. [22] [FS13]J´erˆome Fortier and Luigi Santocanale. Cuts for circular proofs: semantics and cut-elimination. In Computer Science Logic 2013 (CSL 2013), September 2-5, 2013, Torino, Italy, pages 248-262, · Zbl 1356.03098 [23] [Hir14]Denis R. Hirschfeldt.Slicing the truth: On the computable and reverse mathematics of combinatorial principles. World Scientific, 2014. · Zbl 1304.03001 [24] [HP93]Petr H´ajek and Pavel Pudl´ak.Metamathematics of First-Order Arithmetic. Perspectives in mathematical logic. Springer, 1993. [25] [Kay91]Richard Kaye.Models of Peano Arithmetic. Oxford Logic Guides 15. Oxford University Press, 1991. [26] [KMPS19] Leszek Ko lodziejczyk, Henryk Michalewski, Pierre Pradic, and Micha l Skrzypczak. The logical strength of B¨uchi’s decidability theorem. volume Volume 15, Issue 2, May 2019. · Zbl 07058775 [27] [KPW95]Jan Kraj´ıˇcek, Pavel Pudl´ak, and Alan Woods. An exponential lower bound to the size of bounded depth Frege proofs of the pigeonhole principle.Random Structures & Algorithms, 7(1):15-39, [28] [Kra95]Jan Kraj´ıˇcek.Bounded arithmetic, propositional logic, and complexity theory. Cambridge University [29] [McN66]Robert McNaughton. Testing and generating infinite sequences by a finite automaton.Information and Control, 9(5):521-530, 1966. · Zbl 0212.33902 [30] [ML71]Per Martin-L¨of. Hauptsatz for the intuitionistic theory of iterated inductive definitions. In J.E. Fenstad, editor,Proceedings of the Second Scandinavian Logic Symposium, volume 63 ofStudies · Zbl 0231.02040 [31] [NST18]R´emi Nollet, Alexis Saurin, and Christine Tasson. Local validity for circular proofs in linear logic with fixed points. In27th EACSL Annual Conference on Computer Science Logic, CSL 2018, September 4-7, 2018, Birmingham, UK, pages 35:1-35:23, 2018. [32] [NW96]Damian Niwinski and Igor Walukiewicz. Games for theµ-calculus.Theor. Comput. Sci., 163(1&2):99-116, 1996. · Zbl 0872.03017 [33] [Par71]Rohit Parikh. Existence and feasibility in arithmetic.J. Symb. Log., 36(3):494-508, 1971. · Zbl 0243.02037 [34] [Par72]Charles Parsons. On n-quantifier induction.The Journal of Symbolic Logic, 37(3):466-482, 1972. [35] [PBI93]Toniann Pitassi, Paul Beame, and Russell Impagliazzo. Exponential lower bounds for the pigeonhole principle.Computational Complexity, 3:97-140, 1993. 10.1007/BF01200117. · Zbl 0784.03034 [36] [PW81]Jeff B. Paris and Alex J. Wilkie. ∆0sets and induction.Open Days in Model Theory and Set Theory, W. Guzicki, W. Marek, A. Pelc, and C. Rauszer, eds, pages 237-248, 1981. [37] [RB17]Reuben N. S. Rowe and James Brotherston. Automatic cyclic termination proofs for recursive procedures in separation logic. InProceedings of the 6th ACM SIGPLAN Conference on Certified [38] [San02]Luigi Santocanale. A calculus of circular proofs and its categorical semantics. InFoundations of Software Science and Computation Structures, 5th International Conference, FOSSACS 2002, · Zbl 1077.03515 [39] [Sch77]Kurt Sch¨utte.Proof Theory. Grundlehren der mathematischen Wissenschaften 225. Springer Berlin Heidelberg, 1977. Translation of Beweistheorie, 1968. [40] [SD03]Christoph Sprenger and Mads Dam. On the structure of inductive reasoning: Circular and tree-shaped proofs in theµ-calculus. InFoundations of Software Science and Computational · Zbl 1029.03016 [41] [Sim09]Stephen G. Simpson.Subsystems of second order arithmetic, volume 1. Cambridge University Press, 2009. · Zbl 1181.03001 [42] [Sim17]Alex Simpson. Cyclic arithmetic is equivalent to Peano arithmetic. InFoundations of Software Science and Computation Structures - 20th International Conference, FOSSACS 2017, Proceedings, · Zbl 06720996 [43] [Str17]Sorin Stratulat. Cyclic proofs with ordering constraints. InAutomated Reasoning with Analytic Tableaux and Related Methods - 26th International Conference, TABLEAUX 2017, Bras´ılia, Brazil,
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