Advertisement

A Generic Cyclic Theorem Prover

  • James Brotherston
  • Nikos Gorogiannis
  • Rasmus L. Petersen
Part of the Lecture Notes in Computer Science book series (LNCS, volume 7705)

Abstract

We describe the design and implementation of an automated theorem prover realising a fully general notion of cyclic proof. Our tool, called \(\textsc{Cyclist}\), is able to construct proofs obeying a very general cycle scheme in which leaves may be linked to any other matching node in the proof, and to verify the general, global infinitary condition on such proof objects ensuring their soundness. \(\textsc{Cyclist}\) is based on a new, generic theory of cyclic proofs that can be instantiated to a wide variety of logics. We have developed three such concrete instantiations, based on: (a) first-order logic with inductive definitions; (b) entailments of pure separation logic; and (c) Hoare-style termination proofs for pointer programs. Experiments run on these instantiations indicate that \(\textsc{Cyclist}\) offers significant potential as a future platform for inductive theorem proving.

Keywords

Inference Rule Theorem Prover Inductive Rule Derivation Tree Separation Logic 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Cyclist framework download, http://www.cs.ucl.ac.uk/staff/ngorogia/
  2. 2.
    Avenhaus, J., Kühler, U., Schmidt-Samoa, T., Wirth, C.-P.: How to Prove Inductive Theorems? QuodLibet! In: Baader, F. (ed.) CADE-19. LNCS (LNAI), vol. 2741, pp. 328–333. Springer, Heidelberg (2003)CrossRefGoogle Scholar
  3. 3.
    Berdine, J., Cook, B., Distefano, D., O’Hearn, P.W.: Automatic Termination Proofs for Programs with Shape-Shifting Heaps. In: Ball, T., Jones, R.B. (eds.) CAV 2006. LNCS, vol. 4144, pp. 386–400. Springer, Heidelberg (2006)CrossRefGoogle 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)CrossRefGoogle Scholar
  5. 5.
    Brotherston, J.: Sequent Calculus Proof Systems for Inductive Definitions. Ph.D. thesis, University of Edinburgh (November 2006)Google Scholar
  6. 6.
    Brotherston, J.: Formalised Inductive Reasoning in the Logic of Bunched Implications. In: Riis Nielson, H., Filé, G. (eds.) SAS 2007. LNCS, vol. 4634, pp. 87–103. Springer, Heidelberg (2007)CrossRefGoogle Scholar
  7. 7.
    Brotherston, J., Bornat, R., Calcagno, C.: Cyclic proofs of program termination in separation logic. In: POPL-35, pp. 101–112. ACM (2008)Google Scholar
  8. 8.
    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)CrossRefGoogle Scholar
  9. 9.
    Brotherston, J., Simpson, A.: Sequent calculi for induction and infinite descent. Journal of Logic and Computation 21(6), 1177–1216 (2011)MathSciNetzbMATHCrossRefGoogle Scholar
  10. 10.
    Bundy, A.: The automation of proof by mathematical induction. In: Handbook of Automated Reasoning, vol. I, ch. 13, pp. 845–911. Elsevier Science (2001)Google Scholar
  11. 11.
    Couvreur, J.-M.: On-the-fly Verification of Linear Temporal Logic. In: Wing, J., Woodcock, J., Davies, J. (eds.) FM 1999. LNCS, vol. 1708, pp. 253–271. Springer, Heidelberg (1999)CrossRefGoogle Scholar
  12. 12.
    Dixon, L., Fleuriot, J.: IsaPlanner: A Prototype Proof Planner in Isabelle. In: Baader, F. (ed.) CADE-19. LNCS (LNAI), vol. 2741, pp. 279–283. Springer, Heidelberg (2003)CrossRefGoogle Scholar
  13. 13.
    Duret-Lutz, A., Poitrenaud, D.: Spot: An extensible model checking library using transition-based generalized Büchi automata. In: MASCOTS, pp. 76–83 (2004)Google Scholar
  14. 14.
    Fogarty, S., Vardi, M.Y.: Büchi Complementation and Size-Change Termination. In: Kowalewski, S., Philippou, A. (eds.) TACAS 2009. LNCS, vol. 5505, pp. 16–30. Springer, Heidelberg (2009)CrossRefGoogle Scholar
  15. 15.
    Friedgut, E., Kupferman, O., Vardi, M.Y.: Büchi Complementation Made Tighter. In: Wang, F. (ed.) ATVA 2004. LNCS, vol. 3299, pp. 64–78. Springer, Heidelberg (2004)CrossRefGoogle Scholar
  16. 16.
    Johansson, M., Dixon, L., Bundy, A.: Conjecture synthesis for inductive theories. Journal of Automated Reasoning 47(3) (October 2011)Google Scholar
  17. 17.
    Kaufmann, M., Manolios, P., Moore, J.S.: Computer-Aided Reasoning: An Approach. Kluwer (2000)Google Scholar
  18. 18.
    Lee, C.S., Jones, N.D., Ben-Amram, A.M.: The size-change principle for program termination. In: POPL-28, pp. 81–92. ACM (2001)Google Scholar
  19. 19.
    Nguyen, H.H., Chin, W.N.: Enhancing Program Verification with Lemmas. In: Gupta, A., Malik, S. (eds.) CAV 2008. LNCS, vol. 5123, pp. 355–369. Springer, Heidelberg (2008)CrossRefGoogle Scholar
  20. 20.
    Schöpp, U., Simpson, A.: Verifying Temporal Properties Using Explicit Approximants: Completeness for Context-free Processes. In: Nielsen, M., Engberg, U. (eds.) FOSSACS 2002. LNCS, vol. 2303, pp. 372–386. Springer, Heidelberg (2002)CrossRefGoogle Scholar
  21. 21.
    Sprenger, C., Dam, M.: A note on global induction mechanisms in a μ-calculus with explicit approximations. Theor. Informatics and Applications 37, 365–399 (2003)MathSciNetzbMATHCrossRefGoogle Scholar
  22. 22.
    Sprenger, C., Dam, M.: On the Structure of Inductive Reasoning: Circular and Tree-Shaped Proofs in the μ-Calculus. In: Gordon, A.D. (ed.) FOSSACS 2003. LNCS, vol. 2620, pp. 425–440. Springer, Heidelberg (2003)CrossRefGoogle Scholar
  23. 23.
    Stirling, C., Walker, D.: Local model checking in the modal μ-calculus. Theoretical Computer Science 89, 161–177 (1991)MathSciNetzbMATHCrossRefGoogle Scholar
  24. 24.
    Wirth, C.P.: Descente infinie + Deduction. Logic J. of the IGPL 12(1), 1–96 (2004)MathSciNetzbMATHCrossRefGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2012

Authors and Affiliations

  • James Brotherston
    • 1
  • Nikos Gorogiannis
    • 1
  • Rasmus L. Petersen
    • 2
  1. 1.Dept. of Computer ScienceUniversity College LondonUK
  2. 2.Microsoft Research CambridgeUK

Personalised recommendations