Abstract
Our main purpose is to present an algorithm which decides whether or not a condition 𝕮(X, Y) stated in sequential calculus admits a finite automata solution, and produces one if it exists. This solves a problem stated in [4] and contains, as a very special case, the answer to Case 4 left open in [6]. In an equally appealing form the result can be restated in the terminology of [7], [10], [15]: Every ω-game definable in sequential calculus is determined. Moreover the player who has a winning strategy, in fact, has a winning finite-state strategy, that is one which can effectively be played in a strong sense. The main proof, that of the central Theorem 1, will be presented at the end. We begin with a discussion of its consequences.
This research was sponsored by the National Science Foundation Grant No. GJ-120. The main result was announced in [13].
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
Bibliography
J. R. Buchi, Decision methods in the theory of ordinals, Bull. Amer. Math. Soc. 71 (1965), 767–770.
J. R. Buchi, C. E. Elgot and J. B. Wright, The nonexistence of certain algorithms of finite automata theory, Notices Amer. Math. Soc. 5 (1958), 98.
J. R. Buchi, Weak second-order arithmetic and finite automata, Z. Math. Logik Grundlagen Math. 6 (1960), 66–92.
J. R. Buchi, On a decision method in restricted second order arithmetic, Proc. Internat. Congr. Logic, Method, and Philos. Sci. 1960, Stanford Univ. Press, Stanford, Calif., 1962.
A. Church, Application of recursive arithmetic to the problem of circuit synthesis, Summaries of Talks Presented at the Summer Institute for Symbolic Logic, Cornell Univ. 1957, 2nd ed; Princeton, N. J., 1960, 3–50.
A. Church, Logic arithmetic and automata, Proc. Internat. Congr. Math. 1963, Almqvist and Wiksells, Uppsala, 1963.
M. Davis, “Infinite games of perfect information” in Advances in game theory, Princeton Univ. Press, Princeton, N. J., 1964, 85–101.
C. C. Elgot, Decision problems of finite automata design and related arithmetics, Trans. Amer. Math. Soc. 98 (1961), 21–51.
J. Friedman, Some results in Church’s restricted recursive arithmetic, J. Symbolic Logic 22 (1957), 337–342.
D. Gale and F. M. Stewart, “Infinite games with perfect information” in Contributions to the theory of games, Vol. II, Princeton Univ. Press, Princeton, N. J., 1953, 245–266.
S. C. Kleene, Arithmetical predicates and function quantifiers, Trans. Amer. Math. Soc. 79 (1955), 312–340.
S. C. Kleene, “Representation of events in nerve nets and finite automata” in Automata studies, Princeton Univ. Press. Princeton, N. J., 1956, 3–41.
L. H. Landweber, Finite state games—A solvability algorithm for restricted second-order arithmetic, Notices Amer. Math. Soc. 14 (1967), 129–130.
R. McNaughton, Testing and generating infinite sequences by a finite automaton, Information and Control 9 (1966), 521–530.
J. Mycielski, S. Swierczkowski and A. Zieba, On infinite positional games, Bull. Acad. Polon. Sci. Cl. III 4 (1956), 485–488.
M. Rabin and D. Scott, Finite automata and their decision problems, IBM J. Res. Develop. 3(1959), 114–125.
M. Rabin, “Effective computability of winning strategies” in Contributions to the theory of games, Vol. III, Princeton Univ. Press, Princeton, N. J., 1957, 147–157.
B. A. Trachtenbrot, Synthesis of logic networks whose operators are described by means of single place predicate calculus, Dokl. Akad. Nauk SSSR 118 (1958), 646–649.
B. A. Trachtenbrot, Finite automata and the logic of single place predicates, Dokl. Akad. Nauk SSSR 140 (1961), 320–323 = Soviet Math. Dokl. 2 (1961), 623–626.
J. R. Buchi and L. G. Landweber, Definability in the monadic second-order theory of successor, J. Symbolic Logic 34 (1969).
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 1990 Springer-Verlag New York Inc.
About this chapter
Cite this chapter
Buchi, J.R., Landweber, L.H. (1990). Solving Sequential Conditions by Finite-State Strategies. In: Mac Lane, S., Siefkes, D. (eds) The Collected Works of J. Richard Büchi. Springer, New York, NY. https://doi.org/10.1007/978-1-4613-8928-6_29
Download citation
DOI: https://doi.org/10.1007/978-1-4613-8928-6_29
Publisher Name: Springer, New York, NY
Print ISBN: 978-1-4613-8930-9
Online ISBN: 978-1-4613-8928-6
eBook Packages: Springer Book Archive