On global induction mechanisms in a μ-calculus with explicit approximations
RAIRO - Theoretical Informatics and Applications - Informatique Théorique et Applications, Tome 37 (2003) no. 4, pp. 365-391.

We investigate a Gentzen-style proof system for the first-order μ-calculus based on cyclic proofs, produced by unfolding fixed point formulas and detecting repeated proof goals. Our system uses explicit ordinal variables and approximations to support a simple semantic induction discharge condition which ensures the well-foundedness of inductive reasoning. As the main result of this paper we propose a new syntactic discharge condition based on traces and establish its equivalence with the semantic condition. We give an automata-theoretic reformulation of this condition which is more suitable for practical proofs. For a detailed comparison with previous work we consider two simpler syntactic conditions and show that they are more restrictive than our new condition.

DOI : 10.1051/ita:2003024
Classification : 68Q60, 03F07, 03B35
Mots clés : inductive reasoning, circular proofs, well-foundedness, global consistency condition, $\mu $-calculus, approximants
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Sprenger, Christoph; Dam, Mads. On global induction mechanisms in a $\mu $-calculus with explicit approximations. RAIRO - Theoretical Informatics and Applications - Informatique Théorique et Applications, Tome 37 (2003) no. 4, pp. 365-391. doi : 10.1051/ita:2003024. http://archive.numdam.org/articles/10.1051/ita:2003024/

[1] T. Arts, M. Dam, L. Fredlund and D. Gurov, System description: Verification of distributed Erlang programs. Lecture Notes in Artificial Intelligence 1421 (1998) 38-41.

[2] J. Bradfield and C. Stirling, Local model checking for infinite state spaces. Theor. Comput. Sci. 96 (1992) 157-174. | MR | Zbl

[3] M. Dam, Proving properties of dynamic process networks. Inf. Comput. 140 (1998) 95-114. | MR | Zbl

[4] M. Dam and D. Gurov, μ-calculus with explicit points and approximations. J. Logic Comput. 12 (2002) 43-57. Previously appeared in Fixed Points in Computer Science, FICS (2000). | MR | Zbl

[5] E.A. Emerson and C.L. Lei, Modalities for model checking: branching time strikes back. Sci. Comput. Program. 8 (1987) 275-306. | MR | Zbl

[6] L. Fredlund, A Framework for Reasoning about Erlang Code. Ph.D. thesis, Royal Institute of Technology, Stockholm, Sweden (2001).

[7] D. Kozen, Results on the propositional μ-calculus. Theor. Comput. Sci. 27 (1983) 333-354. | MR | Zbl

[8] D. Niwiński and I. Walukiewicz, Games for the μ-calculus. Theor. Comput. Sci. 163 (1997) 99-116. | Zbl

[9] D. Park, Finiteness is mu-ineffable. Theor. Comput. Sci. 3 (1976) 173-181. | MR | Zbl

[10] S. Safra, On the complexity of ω-automata, in 29th IEEE Symposium on Foundations of Computer Science (1988) 319-327.

[11] U. Schöpp, Formal verification of processes. Master's thesis, University of Edinburgh (2001)

[12] U. Schöpp and A. Simpson, Verifying temporal properties using explicit approximants: Completeness for context-free processes, in Foundations of Software Science and Computational Structures (FoSSaCS 02), Grenoble, France. Springer, Lecture Notes in Comput. Sci. 2303 (2002) 372-386. | MR | Zbl

[13] C. Sprenger and M. Dam, On the structure of inductive reasoning: Circular and tree-shaped proofs in the μ-calculus, Foundations of Software Science and Computational Structures (FoSSaCS 03), Warsaw, Poland, April 7-11 2003. A. Gordon, Springer, Lecture Notes in Comput. Sci. 2620 (2003) 425-440. | Zbl

[14] C. Stirling and D. Walker, Local model checking in the modal μ-calculus. Theor. Comput. Sci. 89 (1991) 161-177. | MR | Zbl

[15] W. Thomas, Automata on infinite objects. J. van Leeuwen, Elsevier Science Publishers, Amsterdam, Handb. Theor. Comput. Sci. B (1990) 133-191. | MR | Zbl

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