Weak Second-Order Arithmetic and Finite Automata

  • J. Richakd Büchi


The formalism of regular expressions was introduced by S. C. Kleene [6] to obtain the following basic theorems.


Regular Expression Special Output Finite Automaton Expanded Form Propositional Variable 
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  1. [1]
    Behmann, Heinrich, Beiträge zur Algebra der Logik, insbesondere zum Entschei dungs-problem. Math. Ann. 86, 163–229 (1922).MathSciNetzbMATHCrossRefGoogle Scholar
  2. [2]
    Burks, A. W., and Wright, J. B., Theory of Logical Nets. Proc. IRE, 41, 1357–1365 (1953).MathSciNetCrossRefGoogle Scholar
  3. [3]
    Church, Alonzo, Application of Recursive Arithmetic to the Problem of Circuit Synthesis, Proceedings of the Cornell Logic Conference, Cornell University, 1957. Also see “Application of Recursive Arithmetic in the Theory of Computing and Automata”, Notes for a summer course, The University of Michigan, 1959.Google Scholar
  4. [4]
    Copi, I. M., Elgot, C. C., and Wright, J. B., Realization of Events by Logical Nets. Journal Ass. Comp. Mach. 5, pp. 181–196 (1958).MathSciNetzbMATHGoogle Scholar
  5. [5]
    Gödel, Kurt, On undecidable propositions of formal mathematical systems. Notes by S. C. Kleene and Barkley Rosser on lectures at the Institute for Advanced Study, 1934. Mimeographed, Princeton, N. J., 30 pp.Google Scholar
  6. [6]
    Kleene, S. C., Representation of Events in Nerve Nets and Finite Automata. Automata Studies, Princeton University Press, 1956, 3–41.Google Scholar
  7. [7]
    Medvedev, I. T., On a Class of Events Representable in a Finite Automaton. MIT Lincoln Laboratory Group Report, 34–73, translated from the Russian by J. Schorr-Kon, June 30, 1958.Google Scholar
  8. [8]
    Myhill, John, Finite Automata and Representation of Events. WADC Report TR 57–624, Fundamental Concepts in the Theory of Systems, October, 1957, 112–137.Google Scholar
  9. [9]
    Moore, E. F., Gedanken-Experiments on Sequential Machines. Automata Studies, Princeton, 1956, 129–153.Google Scholar
  10. [10]
    Presburger, M., Über die Vollständigkeit eines gewissen Systems der Arithmetik ganzer Zahlen, in welchem die Addition als einzige Operation hervortritt. Comptes-rendus du I Congres des Mathématiciens des Pays Slavs, Warsaw, 1930, 92–101, 395.Google Scholar
  11. [11]
    Putnam, H., Decidability and Essential Undecidability. Journ. Symb. Log. 22, 39–54. (1957)MathSciNetzbMATHCrossRefGoogle Scholar
  12. [12]
    Quine, W. V., Mathematical Logic. Harvard Univ. Press, Cambridge, 1947.zbMATHGoogle Scholar
  13. [13]
    Rabin, M., and Scott, D., Finite Automata and Their Decision Problems. IBM Journal, April, 1959, 114–125.Google Scholar
  14. [14]
    Robinson, R. M., Restricted Set-Theoretical Definitions in Arithmetic. Proc. Am. Math. Soc. 9, 238–242 (1958).zbMATHCrossRefGoogle Scholar
  15. [15]
    Skolem, Thoralf, Untersuchungen über die Axiome des Klassenkalküls und über Produktions- und Summationsprobleme, welche gewisse Klassen von Aussagen betreffen. Skrifter utgit av Videnskapsselskapet i Kristiania, I. Mat.-nat. kl. 1919, No. 3, 37 pp.Google Scholar

Copyright information

© Springer-Verlag New York Inc. 1990

Authors and Affiliations

  • J. Richakd Büchi
    • 1
  1. 1.Ann ArborUSA

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