# Cyclic Proofs for First-Order Logic with Inductive Definitions

• James Brotherston
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 3702)

## Abstract

We consider a cyclic approach to inductive reasoning in the setting of first-order logic with inductive definitions. We present a proof system for this language in which proofs are represented as finite, locally sound derivation trees with a “repeat function” identifying cyclic proof sections. Soundness is guaranteed by a well-foundedness condition formulated globally in terms of traces over the proof tree, following an idea due to Sprenger and Dam. However, in contrast to their work, our proof system does not require an extension of logical syntax by ordinal variables.

A fundamental question in our setting is the strength of the cyclic proof system compared to the more familiar use of a non-cyclic proof system using explicit induction rules. We show that the cyclic proof system subsumes the use of explicit induction rules. In addition, we provide machinery for manipulating and analysing the structure of cyclic proofs, based primarily on viewing them as generating regular infinite trees, and also formulate a finitary trace condition sufficient (but not necessary) for soundness, that is computationally and combinatorially simpler than the general trace condition.

## Keywords

Proof System Predicate Symbol Trace Condition Sequent Calculus Induction Rule
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.

## References

1. 1.
Aczel, P.: An introduction to inductive definitions. In: Barwise, J. (ed.) Handbook of Mathematical Logic, pp. 739–782. North-Holland, Amsterdam (1977)
2. 2.
Bertot, Y., Castéran, P.: Interactive Theorem Proving and Program Development. In: EATCS: Texts in Theoretical Computer Science. Springer, Heidelberg (2004)Google Scholar
3. 3.
Bradfield, J., Stirling, C.: Local model checking for infinite state spaces. Theoretical Computer Science 96, 157–174 (1992)
4. 4.
Coquand, T.: Infinite objects in type theory. In: Barendregt, H., Nipkow, T. (eds.) Types for Proofs and Programs, pp. 62–78. Springer, Heidelberg (1993)Google Scholar
5. 5.
Dam, M., Gurov, D.: μ-calculus with explicit points and approximations. Journal of Logic and Computation 12(2), 255–269 (2002)
6. 6.
Gentzen, G.: Investigations into logical deduction. In: Szabo, M.E. (ed.) The Collected Papers of Gerhard Gentzen, pp. 68–131. North-Holland, Amsterdam (1969)Google Scholar
7. 7.
Giménez, E.: A Calculus of Infinite Constructions and its application to the verification of communicating systems. PhD thesis, Ecole Normale Supérieure de Lyon (1996)Google Scholar
8. 8.
Gordon, M.J.C., Melham, T.F.: Introduction to HOL: a theorem proving environment for higher order logic. Cambridge University Press, Cambridge (1993)
9. 9.
Hamilton, G.: Poítin: Distilling theorems from conjectures (to appear)Google Scholar
10. 10.
Kaufmann, M., Manolios, P., Moore, J.S.: Computer-Aided Reasoning: An Approach. Kluwer Academic Publishers, Dordrecht (2000)Google Scholar
11. 11.
Martin-Löf, P.: Haupstatz for the intuitionistic theory of iterated inductive definitions. In: Fenstad, J.E. (ed.) Proceedings of the Second Scandinavian Logic Symposium. North-Holland, Amsterdam (1971)Google Scholar
12. 12.
McDowell, R., Miller, D.: Cut-elimination for a logic with definitions and induction. Theoretical Computer Science 232, 91–119 (2000)
13. 13.
Nipkow, T., Paulson, L.C., Wenzel, M.T.: Isabelle/HOL. LNCS, vol. 2283. Springer, Heidelberg (2002)
14. 14.
Schöpp, U.: Formal verification of processes. Master’s thesis, University of Edinburgh (2001)Google Scholar
15. 15.
Schöpp, U., Simpson, A.: Verifying temporal properties using explicit approximants: Completeness for context-free processes. In: Nielsen, M., Engberg, U. (eds.) FOSSACS 2002, vol. 2303, pp. 372–386. Springer, Heidelberg (2002)
16. 16.
Schürmann, C.: Automating the Meta-Theory of Deductive Systems. PhD thesis, Carnegie-Mellon University (2000)Google Scholar
17. 17.
Sprenger, C., Dam, M.: A note on global induction mechanisms in a μ-calculus with explicit approximations. Theoretical Informatics and Applications (July 2003) Full version of FICS 2002 paperGoogle Scholar
18. 18.
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)
19. 19.
Turchin, V.: The concept of a supercompiler. ACM Transactions on Programming Languages and Systems 8, 90–121 (1986)
20. 20.
Vardi, M.Y., Wolper, P.: An automata-theoretic approach to automatic program verification. In: Logic in Computer Science, LICS 1986, pp. 322–331 (1986)Google Scholar