Cyclic Proofs with Ordering Constraints

  • Sorin StratulatEmail author
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 10501)


CLKID\(^{\omega }\) is a sequent-based cyclic inference system able to reason on first-order logic with inductive definitions. The current approach for verifying the soundness of CLKID\(^{\omega }\) proofs is based on expensive model-checking techniques leading to an explosion in the number of states.

We propose proof strategies that guarantee the soundness of a class of CLKID\(^{\omega }\) proofs if some ordering and derivability constraints are satisfied. They are inspired from previous works about cyclic well-founded induction reasoning, known to provide effective sets of ordering constraints. A derivability constraint can be checked in linear time. Under certain conditions, one can build proofs that implicitly satisfy the ordering constraints.


  1. 1.
    Aczel, P.: An introduction to inductive definitions. In: Barwise, J. (ed.) Handbook of Mathematical Logic, pp. 739–782. North Holland, Amsterdam (1977)CrossRefGoogle Scholar
  2. 2.
    Baader, F., Nipkow, T.: Term Rewriting and All That. Cambridge University Press, Cambridge (1998)CrossRefzbMATHGoogle Scholar
  3. 3.
    Barthe, G., Stratulat, S.: Validation of the JavaCard platform with implicit induction techniques. In: Nieuwenhuis, R. (ed.) RTA 2003. LNCS, vol. 2706, pp. 337–351. Springer, Heidelberg (2003). doi: 10.1007/3-540-44881-0_24 CrossRefGoogle Scholar
  4. 4.
    Bouhoula, A., Rusinowitch, M.: Implicit induction in conditional theories. J. Autom. Reason. 14(2), 189–235 (1995)CrossRefzbMATHMathSciNetGoogle Scholar
  5. 5.
    Bronsard, F., Reddy, U.S., Hasker, R.W.: Induction using term orderings. In: Bundy, A. (ed.) CADE 1994. LNCS, vol. 814, pp. 102–117. Springer, Heidelberg (1994). doi: 10.1007/3-540-58156-1_8 CrossRefGoogle Scholar
  6. 6.
    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
  7. 7.
    Brotherston, J.: Sequent calculus proof systems for inductive definitions. Ph.D. thesis, University of Edinburgh, November 2006Google Scholar
  8. 8.
    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
  9. 9.
    Brotherston, J., Simpson, A.: Sequent calculi for induction and infinite descent. J. Logic Comput. 21(6), 1177–1216 (2011)CrossRefzbMATHMathSciNetGoogle Scholar
  10. 10.
    Gentzen, G.: Untersuchungen über das logische Schließen. I. Mathematische Zeitschrift 39, 176–210 (1935)CrossRefzbMATHMathSciNetGoogle Scholar
  11. 11.
    Kupferman, O., Vardi, M.: Weak alternating automata are not that weak. ACM Trans. Comput. Logic (TOCL) 2(3), 408–429 (2001)CrossRefzbMATHMathSciNetGoogle Scholar
  12. 12.
    Michel, M.: Complementation is more difficult with automata on infinite words. Technical report, CNET (1988)Google Scholar
  13. 13.
    Negri, S., von Plato, J.: Structural Proof Theory. Cambridge University Press, Cambridge (2001)CrossRefzbMATHGoogle Scholar
  14. 14.
    Rusinowitch, M., Stratulat, S., Klay, F.: Mechanical verification of an ideal incremental ABR conformance algorithm. J. Autom. Reason. 30(2), 53–177 (2003)CrossRefzbMATHMathSciNetGoogle Scholar
  15. 15.
    Stratulat, S.: A unified view of induction reasoning for first-order logic. In: Voronkov, A. (ed.) Turing-100 (The Alan Turing Centenary Conference). EPiC Series, vol. 10, pp. 326–352. EasyChair (2012)Google Scholar
  16. 16.
    Stratulat, S.: Structural vs. cyclic induction: a report on some experiments with Coq. In: SYNASC International Symposium on Symbolic and Numeric Algorithms for Scientific Computing, pp. 27–34. IEEE Computer Society (2016)Google Scholar
  17. 17.
    Stratulat, S.: Mechanically certifying formula-based Noetherian induction reasoning. J. Symb. Comput. 80(Part 1), 209–249 (2017)CrossRefzbMATHMathSciNetGoogle Scholar
  18. 18.
    Tarjan, R.: Depth-first search and linear graph algorithms. SIAM J. Comput. 1(2), 146–160 (1972)CrossRefzbMATHMathSciNetGoogle Scholar
  19. 19.
    The Coq development team: The Coq Reference Manual. INRIA (2017)Google Scholar
  20. 20.
    Wirth, C.-P.: Descente infinie + deduction. Logic J. IGPL 12(1), 1–96 (2004)CrossRefzbMATHMathSciNetGoogle Scholar

Copyright information

© Springer International Publishing AG 2017

Authors and Affiliations

  1. 1.LORIA, Department of Computer ScienceUniversité de LorraineMetzFrance

Personalised recommendations