Abstract
Many of the important concepts and results of conventional finite automata theory are developed for a generalization in which finite algebras take the place of finite automata. The standard closure theorems are proved for the class of sets “recognizable” by finite algebras, and a generalization of Kleene's regularity theory is presented. The theorems of the generalized theory are then applied to obtain a positive solution to a decision problem of second-order logic.
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References
G. Birkhoff, On the structure of abstract algebras.Proc. Cambridge Phil. Soc. 31 (1938), 433–454.
G. Birkhoff,Lattice Theory, Amer. Math. Soc. Colloq. Publ., Vol. 25, New York, 1948.
J. R. Büchi andC. C. Elgot, Decision problems of weak second-order arithmetics and finite automata. Abstract 553-112,Notices Amer. Math. Soc. 5 (1958), 834.
J. R. Büchi, “Weak second-order arithmetic and finite automata”. University of Michigan, Logic of Computers Group Technical Report, September 1959;Z. Math. Logik Grundlagen Math. 6 (1960), 66–92.
J. R. Büchi andJ. B. Wright, “Mathematical Theory of Automata”. Notes on material presented by J. R. Büchi and J. B. Wright, Communication Sciences 403, Fall 1960, The University of Michigan.
J. R. Büchi, Mathematische Theorie des Verhaltens endlicher Automaten.Z. Angewandte Math. und Mech. 42 (1962), 9–16.
I. M. Copi, C. C. Elgot andJ. B. Wright, Realization of events by logical nets.J. Assoc. Comp. Mach. 5 (1958) 181–196. (Also in Moore [13].)
J. E. Doner, Decidability of the weak second-order theory of two successors. Abstract 65T-468,Notices Amer. Math. Soc. 12 (1965), 819; erratum,ibid. 13 (1966), 513.
C. C. Elgot, “Decision problems of finite automaton design and related arithmetics”. University of Michigan, Department of Mathematics and Logic of Computers Group Technical Report, June 1959;Trans. Amer. Math. Soc. 98 (1961), 21–51.
S. Feferman andR. L. Vaught, The first-order properties of products of algebraic systems.Fund. Math. 47 (1959), 57–103.
S. C. Kleene, Representation of events in nerve nets and finite automata.Automata Studies pp. 3–42, Annals of Math. Studies, No. 34, Princeton University Press, Princeton, N. J., 1956.
I. T. Medvedev, On a class of events representable in a finite automaton. Supplement to the Russian translation ofAutomata Studies (C. Shannon and J. McCarthy, eds.), published by the Publishing Agency for Foreign Literature, Moscow, 1956. Translated by Jacques J. Schorr-Kon for Lincoln Laboratory Report 34-73, June 1958. (Also in Moore [13].)
E. F. Moore,Sequential Machines, Selected Papers. Addison-Wesley, Reading, Mass., 1964.
M. O. Rabin andDana Scott, Finite automata and their decision problems.IBM J. Res. Develop. 3 (1959), 114–125. (Also in Moore [13].)
R. W. Ritchie, Classes of predictably computable functions.Trans. Amer. Math. Soc. 106 (1963), 139–173.
Raphael M. Robinson, Restricted set-theoretic definitions in arithmetic.Proc. Amer. Math. Soc. 9 (1958), 238–242.
J. W. Thatcher, “Notes on Mathematical Automata Theory”. University of Michigan Technical Note, December, 1963.
J. W. Thatcher, “Decision Problems and Definability for Generalized Arithmetic”. Doctoral Dissertation, The University of Michigan; IBM Research Report RC 1316, November, 1964.
J. W. Thatcher andJ. B. Wright, Generalized finite automata. Abstract 65T-469,Notices Amer. Math. Soc. 12 (1965), 820.
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Thatcher, J.W., Wright, J.B. Generalized finite automata theory with an application to a decision problem of second-order logic. Math. Systems Theory 2, 57–81 (1968). https://doi.org/10.1007/BF01691346
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DOI: https://doi.org/10.1007/BF01691346