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Mathematical systems theory

, Volume 2, Issue 1, pp 57–81 | Cite as

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

  • J. W. Thatcher
  • J. B. Wright
Article

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.

Keywords

Computational Mathematic Generalize Theory Decision Problem Important Concept Finite Automaton 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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References

  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.Google Scholar
  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.Google Scholar
  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].)Google Scholar
  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.Google Scholar
  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.Google Scholar
  19. [19]
    J. W. Thatcher andJ. B. Wright, Generalized finite automata. Abstract 65T-469,Notices Amer. Math. Soc. 12 (1965), 820.Google Scholar

Copyright information

© Springer-Verlag New York Inc. 1968

Authors and Affiliations

  • J. W. Thatcher
    • 1
  • J. B. Wright
    • 1
  1. 1.IBM Watson Research CenterUSA

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