Contributions to the reduction theory of the decision problem

Fifth paper Ackermann prefix with three universal quantifiers
  • János Surányl


Decision Problem Reduction Theory 
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  1. 1.
    János Surányi, Contributions to the reduction theory of the decision problem, second paper, Three universal, one existential quatifiers,Acta Math. Hung.,1, (1950), pp. 261–270. We shall quote this paper as “second paper”.Google Scholar
  2. 2.
    K. Gödel, Zum Entscheidungsproblem des logischen Funktionenkalküls,Monatshefte für Math. und Phys. 40, (1933), pp. 433–443.Google Scholar
  3. 3.
    J. Pepis, Ein Verfahren der mathematischen Logik.Journal of Symbolic Logic,3 (1938), pp. 61–76 and Untersuchungen über das Entscheidungsproblem der mathematischen Logik,Fundamenta Math.,30 (1938), pp. 257–348.Google Scholar
  4. 4.
    A formula whose prefix is mentioned in this paper is always meant to be prenex.Google Scholar
  5. 5.
    Wilhelm Ackermanh, Beiträge zum Entscheidungsproblem der mathematischen Logik,Math. Annalen,112 (1936), pp. 419–432.Google Scholar
  6. 6.
    This result, with a sketch of its proof, is contained in the following paper:János Surányi, Reduction of the decision problem to formulas containing a bounded number of quantifiers only,Proc. of the X th International Congr. of Philosophy (Amsterdam 1948), fasc. II., pp. 759–762.Google Scholar
  7. 7.
    Two first order formulas are called equivalent if the satisfiability of one of them implies that of the other, and conversely.Google Scholar
  8. 8.
    Alonzo Church, A note on the Entscheidungsproblem,Journal of Symbolic Logic,1 (1936), pp. 40–41, 101–102. See also footnote8 in the second paper.Google Scholar
  9. 9.
    A. M. Turing, On computable numbers, with an application to the Entscheidungsproblem,Proceedings London Math. Society,42 (1937), pp. 230–265, especially pp. 259–263.Google Scholar
  10. 10.
    Cf.Th. Skolem, Über einige Grundlagenfragen der Mathematik,Skrifter utgitt av det Norske Videnskaps-Akademi i Oslo, Mat-naturw. Klasse,1929, no 4, pp. 1–49, especially pp. 23–29 andD. Hilbert andP. Bernays,Grundlagen der Mathematik, vol. 2 (Berlin 1939), p. 174, rule 3*.Google Scholar
  11. 11.
    Triad means ordered triad;i,j,k,n denote throughout positive integers.Google Scholar
  12. 12.
    This idea was used at the first time byLászló Kalmár andJános Surányi, On the reduction of the decision preblem, second paper, Gödel prefix, a single binary predicate,Journal of Symbolic Logic,12 (1947), pp. 65–73.Google Scholar
  13. 13.
    Lemma 1 shows that the function ω(i, j, k)=n furnishes a one-to-one correspondence between the triads (i, j, k) and the positive integersn.Google Scholar
  14. 14.
    We may suppose without loss of generality, thatA does not contain unary predicate variables. Indeed, if we attach to each unary predicate variableF occurring inA a binary one, say,F *, different from the binary predicate variables occurring originally inA and from the binary predicate variables attached to the other unary ones occurring inA, and replacing each part of the formF(x) ofA byF * (x,x), then we get a formula which is plainly equivalent toA.Google Scholar
  15. 15.
    For simplicity we omit conjunction signs (except at the end of a line when dividing formulas).Google Scholar
  16. 16.
    Of course, the numberl of the Φλ is not fixed.Google Scholar
  17. 17.
    We abbreviate conjunction of many terms by the sign π as they were products:Google Scholar
  18. 18.
    The range of the quantifiers is the setJ.Google Scholar

Copyright information

© Magyar Tudományos Akadémia 1951

Authors and Affiliations

  • János Surányl
    • 1
  1. 1.Budapest

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