Generalized finite automata theory with an application to a decision problem of second-order logic


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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  1. [1]

    G. Birkhoff, On the structure of abstract algebras.Proc. Cambridge Phil. Soc. 31 (1938), 433–454.

    Google Scholar 

  2. [2]

    G. Birkhoff,Lattice Theory, Amer. Math. Soc. Colloq. Publ., Vol. 25, New York, 1948.

  3. [3]

    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.

    Google Scholar 

  4. [4]

    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.

    Google Scholar 

  5. [5]

    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.

  6. [6]

    J. R. Büchi, Mathematische Theorie des Verhaltens endlicher Automaten.Z. Angewandte Math. und Mech. 42 (1962), 9–16.

    Google Scholar 

  7. [7]

    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].)

    Google Scholar 

  8. [8]

    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.

    Google Scholar 

  9. [9]

    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.

    Google Scholar 

  10. [10]

    S. Feferman andR. L. Vaught, The first-order properties of products of algebraic systems.Fund. Math. 47 (1959), 57–103.

    Google Scholar 

  11. [11]

    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.

    Google Scholar 

  12. [12]

    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].)

  13. [13]

    E. F. Moore,Sequential Machines, Selected Papers. Addison-Wesley, Reading, Mass., 1964.

    Google Scholar 

  14. [14]

    M. O. Rabin andDana Scott, Finite automata and their decision problems.IBM J. Res. Develop. 3 (1959), 114–125. (Also in Moore [13].)

    Google Scholar 

  15. [15]

    R. W. Ritchie, Classes of predictably computable functions.Trans. Amer. Math. Soc. 106 (1963), 139–173.

    Google Scholar 

  16. [16]

    Raphael M. Robinson, Restricted set-theoretic definitions in arithmetic.Proc. Amer. Math. Soc. 9 (1958), 238–242.

    Google Scholar 

  17. [17]

    J. W. Thatcher, “Notes on Mathematical Automata Theory”. University of Michigan Technical Note, December, 1963.

  18. [18]

    J. W. Thatcher, “Decision Problems and Definability for Generalized Arithmetic”. Doctoral Dissertation, The University of Michigan; IBM Research Report RC 1316, November, 1964.

  19. [19]

    J. W. Thatcher andJ. B. Wright, Generalized finite automata. Abstract 65T-469,Notices Amer. Math. Soc. 12 (1965), 820.

    Google Scholar 

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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).

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  • Computational Mathematic
  • Generalize Theory
  • Decision Problem
  • Important Concept
  • Finite Automaton