Cyclic Arithmetic Is Equivalent to Peano Arithmetic

Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 10203)


Cyclic proof provides a style of proof for logics with inductive (and coinductive) definitions, in which proofs are cyclic graphs representing a form of argument by infinite descent. It is easily shown that cyclic proof subsumes proof by (co)induction. So cyclic proof systems are at least as powerful as the corresponding proof systems with explicit (co)induction rules. Whether or not the converse inclusion holds is a non-trivial question. In this paper, we resolve this question in one interesting case. We show that a cyclic formulation of first-order arithmetic is equivalent in power to Peano Arithmetic. The proof involves formalising the meta-theory of cyclic proof in a subsystem of second-order arithmetic.


  1. 1.
    Baelde, D.: On the proof theory of regular fixed points. In: Giese, M., Waaler, A. (eds.) TABLEAUX 2009. LNCS (LNAI), vol. 5607, pp. 93–107. Springer, Heidelberg (2009). doi: 10.1007/978-3-642-02716-1_8 CrossRefGoogle Scholar
  2. 2.
    Baelde, D., Doumane, A., Saurin, A.: Infinitary proof theory: the multiplicative additive case. In: Talbot, J.-M., Regnier, L. (eds.) 25th EACSL Annual Conference on Computer Science Logic (CSL 2016). Leibniz International Proceedings in Informatics (LIPIcs), Dagstuhl, Germany, vol. 62, pp. 42:1–42:17. Schloss Dagstuhl-Leibniz-Zentrum fuer Informatik (2016)Google Scholar
  3. 3.
    Berardi, S., Tatsuta, M.: Classical system of Martin-Löf’s inductive definitions is not equivalent to cyclic proof system. In: Esparza, J., Murawski, A.S. (Eds.) FOSSACS 2017. LNCS, vol. 10203, pp. 301–317. Springer, Heidelberg (2017)Google Scholar
  4. 4.
    Brotherston, J.: Cyclic proofs for first-order logic with inductive definitions. In: Beckert, B. (ed.) TABLEAUX 2005. LNCS (LNAI), vol. 3702, pp. 78–92. Springer, Heidelberg (2005). doi: 10.1007/11554554_8 CrossRefGoogle Scholar
  5. 5.
    Brotherston, J.: Sequent calculus proof systems for inductive definitions. Ph.D. thesis, University of Edinburgh, November 2006Google Scholar
  6. 6.
    Brotherston, J., Bornat, R., Calcagno, C.: Cyclic proofs of program termination in separation logic. In: Proceedings of POPL-35, pp. 101–112. ACM (2008)Google Scholar
  7. 7.
    Brotherston, J., Distefano, D., Petersen, R.L.: Automated cyclic entailment proofs in separation logic. In: Bjørner, N., Sofronie-Stokkermans, V. (eds.) CADE 2011. LNCS (LNAI), vol. 6803, pp. 131–146. Springer, Heidelberg (2011). doi: 10.1007/978-3-642-22438-6_12 CrossRefGoogle Scholar
  8. 8.
    Brotherston, J., Gorogiannis, N.: Cyclic abduction of inductively defined safety and termination preconditions. In: Müller-Olm, M., Seidl, H. (eds.) SAS 2014. LNCS, vol. 8723, pp. 68–84. Springer, Cham (2014). doi: 10.1007/978-3-319-10936-7_5 Google Scholar
  9. 9.
    Brotherston, J., Gorogiannis, N., Petersen, R.L.: A generic cyclic theorem prover. In: Jhala, R., Igarashi, A. (eds.) APLAS 2012. LNCS, vol. 7705, pp. 350–367. Springer, Heidelberg (2012). doi: 10.1007/978-3-642-35182-2_25 CrossRefGoogle Scholar
  10. 10.
    Brotherston, J., Simpson, A.: Complete sequent calculi for induction and infinite descent. In: Proceedings of LICS-22, pp. 51–60. IEEE Computer Society, July 2007Google Scholar
  11. 11.
    Brotherston, J., Simpson, A.: Sequent calculi for induction and infinite descent. J. Logic Comput. 21(6), 1177–1216 (2011)MathSciNetCrossRefMATHGoogle Scholar
  12. 12.
    Dax, C., Hofmann, M., Lange, M.: A proof system for the linear time \(\upmu \)-calculus. In: Arun-Kumar, S., Garg, N. (eds.) FSTTCS 2006. LNCS, vol. 4337, pp. 273–284. Springer, Heidelberg (2006). doi: 10.1007/11944836_26 CrossRefGoogle Scholar
  13. 13.
    Fortier, J., Santocanale, L.: Cuts for circular proofs: semantics and cut-elimination. In: Computer Science Logic 2013 (CSL 2013), CSL 2013, 2–5 September 2013, Torino, Italy, pp. 248–262 (2013)Google Scholar
  14. 14.
    Kaye, R.: Models of Peano Arithmetic. Number 15 in Oxford Logic Guides. Oxford University Press, Oxford (1991)Google Scholar
  15. 15.
    Kolodziejczyk, L.A., Michalewski, H., Pradic, P., Skrzypczak, M.: The logical strength of Büchi’s decidability theorem. In: Talbot, J.-M., Regnier, L. (eds.) 25th EACSL Annual Conference on Computer Science Logic (CSL 2016). Leibniz International Proceedings in Informatics (LIPIcs), Dagstuhl, Germany, vol. 62, pp. 36:1–36:16. Schloss Dagstuhl-Leibniz-Zentrum fuer Informatik (2016)Google Scholar
  16. 16.
    Martin-Löf, P.: Haupstatz for the intuitionistic theory of iterated inductive def initions. In: Fenstad, J.E. (ed.) Proceedings of the Second Scandinavian Logic Symposium, pp. 179–216. North Holland (1971)Google Scholar
  17. 17.
    Mio, M., Simpson, A.: A proof system for compositional verification of probabilistic concurrent processes. In: Pfenning, F. (ed.) FoSSaCS 2013. LNCS, vol. 7794, pp. 161–176. Springer, Heidelberg (2013). doi: 10.1007/978-3-642-37075-5_11 CrossRefGoogle Scholar
  18. 18.
    Niwiński, D., Walukiewicz, I.: Games for the \(\mu \)-calculus. Theoret. Comput. Sci. 163, 99–116 (1997)MathSciNetCrossRefMATHGoogle Scholar
  19. 19.
    Santocanale, L.: A calculus of circular proofs and its categorical semantics. In: Nielsen, M., Engberg, U. (eds.) FoSSaCS 2002. LNCS, vol. 2303, pp. 357–371. Springer, Heidelberg (2002). doi: 10.1007/3-540-45931-6_25 CrossRefGoogle Scholar
  20. 20.
    Simpson, S.G.: Subsystems of Second Order Arithmetic. Perspectives in Logic. Association for Symbolic Logic, New York (2009)CrossRefMATHGoogle Scholar
  21. 21.
    Sprenger, C., Dam, M.: A note on global induction mechanisms in a \(\mu \)-calculus with explicit approximations. Theor. Inform. Appl. 37, 365–399 (2003)MathSciNetCrossRefMATHGoogle Scholar
  22. 22.
    Sprenger, C., Dam, M.: On the structure of inductive reasoning: circular and tree-shaped proofs in the \(\upmu \)calculus. In: Gordon, A.D. (ed.) FoSSaCS 2003. LNCS, vol. 2620, pp. 425–440. Springer, Heidelberg (2003). doi: 10.1007/3-540-36576-1_27 CrossRefGoogle Scholar
  23. 23.
    Stratulat, S.: Structural vs. cyclic induction: a report on some experiments with Coq. In: SYNASC 2016: Proceedings of the 18th International Symposium on Symbolic and Numeric Algorithms for Scientific Computing, pp. 27–34. IEEE Computer Society (2016)Google Scholar
  24. 24.
    Studer, T.: On the proof theory of the modal mu-calculus. Stud. Logica. 89(3), 343–363 (2008)MathSciNetCrossRefMATHGoogle Scholar
  25. 25.
    Thomas, W.: Automata on infinite objects. In: van Leeuwen, J. (ed.) Handbook of Theoretical Computer Science. Formal Models and Semantics, vol. B, pp. 133–192. Elsevier Science Publishers, Amsterdam (1990)Google Scholar

Copyright information

© Springer-Verlag GmbH Germany 2017

Authors and Affiliations

  1. 1.Faculty of Mathematics and PhysicsUniversity of LjubljanaLjubljanaSlovenia

Personalised recommendations