Abstract
This paper presents translations forth and back between formulas of the linear time μ-calculus and finite automata with a weak parity acceptance condition. This yields a normal form for these formulas, in fact showing that the linear time alternation hierarchy collapses at level 0 and not just at level 1 as known so far. The translation from formulas to automata can be optimised yielding automata whose size is only exponential in the alternation depth of the formula.
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Barringer, H., Kuiper, R., Pnueli, A.: A really abstract concurrent model and its temporal logic. In: Proc. 13th Annual ACM Symp. on Principles of Programming Languages, pp. 173–183. ACM, New York (1986)
Békić, H.: In: Bekic, H. (ed.) Programming Languages and their Definition. LNCS, vol. 177. Springer, Heidelberg (1984)
Biere, A., Cimatti, A., Clarke, E.M., Zhu, Y.: Symbolic model checking without BDDs. In: Cleaveland, W.R. (ed.) TACAS 1999. LNCS, vol. 1579, p. 193. Springer, Heidelberg (1999)
Bradfield, J.C.: The modal μ-calculus alternation hierarchy is strict. In: Sassone, V., Montanari, U. (eds.) CONCUR 1996. LNCS, vol. 1119, pp. 233–246. Springer, Heidelberg (1996)
Büchi, J.R.: On a decision method in restricted second order arithmetic. In: Proc. Congress on Logic, Method, and Philosophy of Science, pp. 1–12. Stanford University Press, Stanford (1962)
Chandra, A.K., Kozen, D.C., Stockmeyer, L.J.: Alternation. Journal of the ACM 28(1), 114–133 (1981)
Dziembowski, S., Jurdziński, M., Walukiewicz, I.: How much memory is needed to win infinite games? In: Proc. 12th Symp. on Logic in Computer Science, LICS 1997, Warsaw, Poland, pp. 99–110. IEEE, Los Alamitos (1997)
Emerson, E.A.: Model checking and the μ-calculus. In: Immerman, N., Kolaitis, P.G. (eds.) Descriptive Complexity and Finite Models, ch. 6. DIMACS: Series in Discrete Mathematics and Theoretical Computer Science, vol. 31, AMS (1997)
Emerson, E.A., Jutla, C.S.: Tree automata, μ-calculus and determinacy. In: Proc. 32nd Symp. on Foundations of Computer Science, San Juan, Puerto Rico, pp. 368–377. IEEE, Los Alamitos (1991)
Janin, D., Walukiewicz, I.: On the expressive completeness of the propositional μ-calculus with respect to monadic second order logic. In: Sassone, V., Montanari, U. (eds.) CONCUR 1996. LNCS, vol. 1119, pp. 263–277. Springer, Heidelberg (1996)
Kaivola, R.: Using Automata to Characterise Fixed Point Temporal Logics. PhD thesis, LFCS, Division of Informatics, The University of Edinburgh, Tech. Rep. ECS-LFCS-97-356 (1997)
Kozen, D.: Results on the propositional μ-calculus. TCS 27, 333–354 (1983)
Kupferman, O., Vardi, M.Y.: Weak alternating automata are not that weak. ACM Transactions on Computational Logic 2(3), 408–429 (2001)
Kupferman, O., Vardi, M.Y., Wolper, P.: An automata-theoretic approach to branching-time model checking. Journal of the ACM 47(2), 312–360 (2000)
Löding, C., Thomas, W.: Alternating automata and logics over infinite words. In: Watanabe, O., Hagiya, M., Ito, T., van Leeuwen, J., Mosses, P.D. (eds.) TCS 2000. LNCS, vol. 1872, pp. 521–535. Springer, Heidelberg (2000)
Muller, D., Schupp, P.: Alternating automata on infinite objects: determinacy and rabin’s theorem. In: Perrin, D., Nivat, M. (eds.) Automata on Infinite Words. LNCS, vol. 192, pp. 100–107. Springer, Heidelberg (1985)
Muller, D.E., Saoudi, A., Schupp, P.E.: Weak alternating automata give a simple explanation of why most temporal and dynamic logics are decidable in exponential time. In: Proc. 3rd Symp. on Logic in Computer Science, LICS 1988, Edinburgh, Scotland, pp. 422–427. IEEE, Los Alamitos (1988)
Rabin, M.O.: Decidability of second-order theories and automata on infinite trees. Trans. of Amer. Math. Soc. 141, 1–35 (1969)
Stirling, C.: Local model checking games. In: Lee, I., Smolka, S.A. (eds.) CONCUR 1995. LNCS, vol. 962, pp. 1–11. Springer, Heidelberg (1995)
Vardi, M.Y.: A temporal fixpoint calculus. In: Proc. Conf. on Principles of Programming Languages, POPL 1988, pp. 250–259. ACM Press, New York (1988)
Vardi, M.Y.: An Automata-Theoretic Approach to Linear Temporal Logic. In: Moller, F., Birtwistle, G. (eds.) Logics for Concurrency. LNCS, vol. 1043. Springer, Heidelberg (1996)
Walukiewicz, I.: Completeness of Kozen’s axiomatization of the propositional μ- calculus. In: Proc. 10th Symp. on Logic in Computer Science, LICS 1995, pp. 14–24. IEEE, Los Alamitos (1995)
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Lange, M. (2005). Weak Automata for the Linear Time μ-Calculus. In: Cousot, R. (eds) Verification, Model Checking, and Abstract Interpretation. VMCAI 2005. Lecture Notes in Computer Science, vol 3385. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-540-30579-8_18
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DOI: https://doi.org/10.1007/978-3-540-30579-8_18
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