The rabin index and chain automata, with applications to automata and games
In this paper we relate the Rabin Index of an ω-language to the complexity of translation amongst automata, strategies for two-person regular games, and the complexity of controller-synthesis and verification of finite state systems, via a new construction to transform Rabin automata to Chain automata. The Rabin Index is the minimum number of pairs required to realize the language as a deterministic Rabin automaton (DRA), and is a measure of the inherent complexity of the ω-language. Chain automata are a special kind of Rabin automata where the sets comprising the acceptance condition form a chain. Our main construction translates a DRA with n states and h pairs to a deterministic chain automaton (DCA) with n.h k states, where k is the Rabin Index of the language. Using this construction, we can transform a DRA into a minimum-pair DRA or a minimum-pair deterministic Streett automaton (DSA), each with n.h k states. Using a simple correspondence between tree automata (TA) and games, we extend the constructions to translate between nondeterministic Rabin and Streett TA while simultaneously reducing the number of pairs; for the class of “trim” deterministic Rabin TA our construction gives a minimum-index deterministic Chain TA, or a minimum-pair DRTA or DSTA, each with n.h k states, where k is RI of the tree-language.
Using these results, we obtain upper bounds on the memory required to implement strategies in infinite games. In particular, the amount of memory required in a game presented as a DRA, or DSA, is bounded by nh k , where k is the RI of the game language.
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- 1.O. Carton. Chain Automata. In IFIP 13th World Computer Congress, pages 451–458, August 1994.Google Scholar
- 2.E. S. Chang, Z. Manna, and A. Pnueli. The Safety-Progress Classification. In F. L. Bauer, W. Bauer, and H. Schwichtenberg, editors, Logic and Algebra of Specification, pages 143–202, 1993.Google Scholar
- 3.E. A. Emerson. Automata, tableaux, and temporal logics. In Logics of Programs, LNCS, pages 79–88. Springer-Verlag, 1985.Google Scholar
- 4.E. A. Emerson and C. S. Jutla. The complexity of tree automata and logics of programs. In Proc. of the Symp. on Foundations of Computer Science, pages 328–337, October 1988.Google Scholar
- 5.E. A. Emerson and C. S. Jutla. Trees automata, Mu-calculus and determinacy. In Proc. of the Symp. on Foundations of Computer Science, pages 368–377, October 1991.Google Scholar
- 6.E. A. Emerson, C. S. Jutla, and A. P. Sistla. On model-checking for fragments of mu-calculus. In Computer Aided Verification, volume 697 of LNCS, pages 385–396. Springer-Verlag, 1994.Google Scholar
- 7.Y. Gurevich and L. Harrington. Trees, automata, and games. In Proc. of the ACM Symposium on the Theory of Computing, pages 60–65, May 1982.Google Scholar
- 8.C. S. Jutla. Personal communication, February 1995.Google Scholar
- 9.M. Kaminski. A Classification of ω-regular languages. Theoretical Computer Science, 36:217–229, 1985.Google Scholar
- 10.S. C. Krishnan, A. Puri, and R. K. Brayton. Deterministic ω-automata vis-a-vis Deterministic Buchi Automata. In Algorithms and Computation, volume 834 of LNCS, pages 378–386. Springer-Verlag, 1994.Google Scholar
- 11.S. C. Krishnan, A. Puri, and R. K. Brayton. Structural Complexity of ω-automata. In Symposium on Theoretical Aspects of Computer Science, volume 900 of LNCS, pages 143–156. Springer-Verlag, 1995.Google Scholar
- 12.R. P. Kurshan. Computer-aided Verification of Coordinating Processes: the Automata-theoretic approach. Princeton University Press, 1994.Google Scholar
- 13.O. Maler, A. Pnueli, and J. Sifakis. On the synthesis of Discrete Controllers for Timed Systems. In Symposium on Theoretical Aspects of Computer Science, volume 900 of LNCS, pages 229–242. Springer-Verlag, 1995.Google Scholar
- 14.R. McNaughton. Infinite gmaes played on finite graphs. Annals of Pure and Applied Logic, 65:149–184, 1993.Google Scholar
- 15.A. Pnueli and R. Rosner. On the synthesis of a reactive module. In Proc. of the ACM Symposium on Principles of Programming Languages, pages 179–180, 1989.Google Scholar
- 16.M. O. Rabin. Automata on Infinite Objects and Church's Problem, volume 13 of Regional Conf. Series in Mathematics. American Mathematical Society, Providence, Rhode Island, 1972.Google Scholar
- 17.S. Safra and M. Y. Vardi. On ω-Automata and Temporal Logic. In Proc. of the ACM Symposium on the Theory of Computing, pages 127–137, May 1989.Google Scholar
- 18.Shmuel Safra. Complexity of Automata on Infinite Objects. PhD thesis, The Weizmann Institute of Science, Rehovot, Israel, March 1989.Google Scholar
- 19.W. Thomas. Automata on Infinite Objects. In J. van Leeuwen, editor, Formal Models and Semantics, volume B of Handbook of Theoretical Computer Science, pages 133–191. Elsevier Science, 1990.Google Scholar
- 20.W. Thomas. On the synthesis of strategies in infinite games. In Symposium on Theoretical Aspects of Computer Science, volume 900 of LNCS, pages 1–13. Springer-Verlag, 1995.Google Scholar