Denis Kuperberg
Damien Pous
Skip Abstract Section
Abstract
We study a cyclic proof system C over regular expression types, inspired by linear logic and non-wellfounded proof theory. Proofs in C can be seen as strongly typed goto programs. We show that they denote computable total functions and we analyse the relative strength of C and Gödel’s system T. In the general case, we prove that the two systems capture the same functions on natural numbers. In the affine case, i.e., when contraction is removed, we prove that they capture precisely the primitive recursive functions—providing an alternative and more general proof of a result by Dal Lago, about an affine version of system T.
Without contraction, we manage to give a direct and uniform encoding of C into T, by analysing cycles and translating them into explicit recursions. Whether such a direct and uniform translation from C to T can be given in the presence of contraction remains open.
We obtain the two upper bounds on the expressivity of C using a different technique: we formalise weak normalisation of a small step reduction semantics in subsystems of second-order arithmetic: ACA0 and RCA0.
- Samson Abramsky, Esfandiar Haghverdi, and Philip Scott. 2002. Geometry of interaction and linear combinatory algebras. Mathematical Structures in Computer Science 12, 5 ( 2002 ), 625-665. https://doi.org/10.1017/S0960129502003730 Google Scholar
Digital Library
- Bahareh Afshari and Graham E. Leigh. 2017. Cut-free completeness for modal mu-calculus. In LiCS. IEEE, 1-12. https: //doi.org/10.1109/LICS. 2017.8005088 Google ScholarCross Ref
- Jeremy Avigad. 1996. Formalizing forcing arguments in subsystems of second-order arithmetic. Annals of Pure and Applied Logic 82, 2 ( 1996 ), 165-191. https://doi.org/10.1016/ 0168-0072 ( 96 ) 00003-6 Google ScholarCross Ref
- Jeremy Avigad and Solomon Feferman. 1998. Gödel's Functional Interpretation. In Handbook of Proof Theory, Samuel R. Buss (Ed.). Elsevier.Google Scholar
- Patrick Baillot and Marco Pedicini. 2001. Elementary complexity and geometry of interaction. Fundamenta Informaticae 45, 1-2 ( 2001 ), 1-31.Google ScholarDigital Library
- Stefano Berardi and Makoto Tatsuta. 2017a. Classical System of Martin-Löf's Inductive Definitions Is Not Equivalent to Cyclic Proof System. In FoSSaCS. Springer Verlag, 301-317. https://doi.org/10.1007/978-3-662-54458-7_18 Google ScholarDigital Library
- Stefano Berardi and Makoto Tatsuta. 2017b. Equivalence of inductive definitions and cyclic proofs under arithmetic. In LiCS. IEEE, 1-12. https://doi.org/10.1109/LICS. 2017.8005114 Google ScholarCross Ref
- James Brotherston. 2005. Cyclic Proofs for First-Order Logic with Inductive Definitions. In TABLEAUX (Lecture Notes in Artificial Intelligence, Vol. 3702 ). Springer Verlag, 78-92. https://doi.org/10.1007/11554554_8 Google ScholarDigital Library
- James Brotherston and Alex Simpson. 2011. Sequent calculi for induction and infinite descent. Journal of Logic and Computation 21, 6 ( 2011 ), 1177-1216. https://doi.org/10.1093/logcom/exq052 Google ScholarDigital Library
- Samuel R. Buss. 1995. The witness function method and provably recursive functions of Peano arithmetic. In Studies in Logic and the Foundations of Mathematics. Vol. 134. Elsevier, 29-68. https://doi.org/10.1016/ S0049-237X( 06 ) 80038-8 Google ScholarCross Ref
- Samuel R Buss. 1998. Handbook of proof theory. Vol. 137. Elsevier.Google Scholar
- Ugo Dal Lago. 2009. The geometry of linear higher-order recursion. ACM Trans. Comput. Log. 10, 2 ( 2009 ), 8 : 1-8 : 38. https://doi.org/10.1145/1462179.1462180 Google ScholarDigital Library
- Anupam Das. 2020. On the logical complexity of cyclic arithmetic. Logical Methods in Computer Science 16, 1 (Jan. 2020 ). https://doi.org/10.23638/LMCS-16 ( 1 :1) 2020 Google ScholarCross Ref
- Anupam Das, Amina Doumane, and Damien Pous. 2018. Left-handed completeness for Kleene algebra, via cyclic proofs. In LPAR (EPiC Series in Computing, Vol. 57 ). Easychair, 271-289. https://doi.org/10.29007/hzq3 Google ScholarCross Ref
- Anupam Das and Damien Pous. 2017. A cut-free cyclic proof system for Kleene algebra. In TABLEAUX (Lecture Notes in Computer Science, Vol. 10501 ). Springer Verlag, 261-277. https://doi.org/10.1007/978-3-319-66902-1_16 Google ScholarCross Ref
- Anupam Das and Damien Pous. 2018. Non-wellfounded proof theory for (Kleene+action)(algebras+lattices). In CSL (LIPIcs, Vol. 119 ). Schloss Dagstuhl-Leibniz-Zentrum für Informatik, 18 : 1-18 : 19. https://doi.org/10.4230/LIPIcs.CSL. 2018.19 Google ScholarCross Ref
- Amina Doumane. 2017. Constructive completeness for the linear-time-calculus. In LiCS. IEEE, 1-12. https://doi.org/10. 1109/LICS. 2017.8005075 Google ScholarCross Ref
- Amina Doumane, David Baelde, and Alexis Saurin. 2016. Infinitary proof theory: the multiplicative additive case. In CSL (LIPIcs, Vol. 62 ). Schloss Dagstuhl-Leibniz-Zentrum für Informatik, 42 : 1-42 : 17. https://doi.org/10.4230/LIPIcs.CSL. 2016.42 Google ScholarCross Ref
- Jérôme Fortier and Luigi Santocanale. 2013. Cuts for circular proofs: semantics and cut-elimination. In CSL (LIPIcs, Vol. 23 ). Schloss Dagstuhl-Leibniz-Zentrum für Informatik, 248-262. https://doi.org/10.4230/LIPIcs.CSL. 2013.248 Google ScholarCross Ref
- Jean-Yves Girard. 1995. Geometry of interaction III: accommodating the additives. In workshop on Advances in linear logic. Cambridge University Press, 329-389.Google ScholarCross Ref
- Jean-Yves Girard, Paul Taylor, and Yves Lafont. 1989. Proofs and Types. Cambridge University Press, USA.Google Scholar
- Von Kurt Gödel. 1958. Über eine bisher noch nicht benütze erweiterung des finiten standpunktes. Dialectica 12, 3-4 ( 1958 ), 280-287. https://doi.org/10.1111/j.1746-8361. 1958.tb01464.x Google ScholarCross Ref
- Denis R. Hirschfeldt. 2014. Slicing the truth: On the computable and reverse mathematics of combinatorial principles. World Scientific.Google Scholar
- Naohiko Hoshino, Koko Muroya, and Ichiro Hasuo. 2014. Memoryful geometry of interaction: from coalgebraic components to algebraic efects. In CSL-LiCS. ACM, 52. https://doi.org/10.1145/2603088.2603124 Google ScholarDigital Library
- Leszek Kołodziejczyk, Henryk Michalewski, Pierre Pradic, and Michał Skrzypczak. 2019. The logical strength of Büchi's decidability theorem. Logical Methods in Computer Science 15, 2 (May 2019 ). https://doi.org/10.23638/LMCS-15 ( 2 :16) 2019 Google ScholarCross Ref
- Denis Kuperberg, Laureline Pinault, and Damien Pous. 2019. Cyclic Proofs and Jumping Automata. In FSTTCS (LIPIcs). Schloss Dagstuhl-Leibniz-Zentrum für Informatik, Bombay, India, 45 : 1-45 : 14. https://doi.org/10.4230/LIPIcs.FSTTCS. 2019.45 Google ScholarCross Ref
- Denis Kuperberg, Laureline Pinault, and Damien Pous. 2021. Cyclic proofs, system T, and the power of contraction-extended version, with appendices. https://hal.archives-ouvertes.fr/hal-02487175/documentGoogle Scholar
- Dorel Lucanu, Eugen-Ioan Goriac, Georgiana Caltais, and Grigore Rosu. 2009. CIRC: A Behavioral Verification Tool Based on Circular Coinduction. In CALCO (Lecture Notes in Computer Science, Vol. 5728 ). Springer Verlag, 433-442. https://doi.org/10.1007/978-3-642-03741-2_30 Google ScholarCross Ref
- Dorel Lucanu and Vlad Rusu. 2015. Program equivalence by circular reasoning. Formal Aspects of Computing 27, 4 ( 2015 ), 701-726. https://doi.org/10.1007/s00165-014-0319-6 Google ScholarDigital Library
- Charles Parsons. 1972. On n-quantifier induction. The Journal of Symbolic Logic 37, 3 ( 1972 ), 466-482.Google ScholarCross Ref
- Alex Simpson. 2017. Cyclic Arithmetic Is Equivalent to Peano Arithmetic. In FoSSaCS (Lecture Notes in Computer Science, Vol. 10203 ). Springer Verlag, 283-300. https://doi.org/10.1007/978-3-662-54458-7_17 Google ScholarDigital Library
- Stephen G. Simpson. 2009. Subsystems of second order arithmetic. Vol. 1. Cambridge University Press.Google Scholar
- William W. Tait. 1965. Infinitely Long Terms of Transfinite Type. In Formal Systems and Recursive Functions, J.N. Crossley and M.A.E. Dummett (Eds.). Studies in Logic and the Foundations of Mathematics, Vol. 40. Elsevier, 176-185. https: //doi.org/10.1016/ S0049-237X( 08 ) 71689-6 Google ScholarCross Ref
- William W. Tait. 1967. Intensional Interpretations of Functionals of Finite Type I. The Journal of Symbolic Logic 32, 2 ( 1967 ), 198-212. https://doi.org/10.2307/2271658 Google ScholarCross Ref
Index Terms
- Cyclic proofs, system t, and the power of contraction
Recommendations
Cyclic Proofs, Hypersequents, and Transitive Closure Logic
Automated ReasoningAbstractWe propose a cut-free cyclic system for Transitive Closure Logic (TCL) based on a form of hypersequents, suitable for automated reasoning via proof search. We show that previously proposed sequent systems are cut-free incomplete for basic ...
Cyclic Hypersequent Calculi for Some Modal Logics with the Master Modality
Automated Reasoning with Analytic Tableaux and Related MethodsAbstractAt LICS 2013, O. Lahav introduced a technique to uniformly construct cut-free hypersequent calculi for basic modal logics characterised by frames satisfying so-called ‘simple’ first-order conditions. We investigate the generalisation of this ...
Cyclic Hypersequent System for Transitive Closure Logic
AbstractWe propose a cut-free cyclic system for transitive closure logic (TCL) based on a form of hypersequents, suitable for automated reasoning via proof search. We show that previously proposed sequent systems are cut-free incomplete for basic ...
Comments