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Conservative reduction classes of Krom formulas1

  • Stål O. Aanderaa (a1), Egon Börger (a2) and Harry R. Lewis (a3)

Abstract

A Krom formula of pure quantification theory is a formula in conjunctive normal form such that each conjunct is a disjunction of at most two atomic formulas or negations of atomic formulas. Every class of Krom formulas that is determined by the form of their quantifier prefixes and which is known to have an unsolvable decision problem for satisfiability is here shown to be a conservative reduction class. Therefore both the general satisfiability problem, and the problem of satisfiability in finite models, can be effectively reduced from arbitrary formulas to Krom formulas of these several prefix types.

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1

The authors thank the “Stiftung Volkswagenwerk” for its financial support of the conference on logic and complexity theory held in Aachen in September 1979, during which this collaborative work was initiated, and also the Institut für Mathematische Logik und Grundlagenforschung of the University of Münster for its hospitality during the week after the conference. We are also grateful to Warren Goldfarb for his corrections to an earlier draft of this paper.

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References

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[A]Aanderaa, S. O., On the decision problem for formulas in which all disjunctions are binary, Proceedings of the Second Scandinavian Logic Symposium, North-Holland, Amsterdam, 1971, pp. 118.
[ABG]Aanderaa, S. O., Börger, E. and Gurevich, Y., Prefix classes of Kromformulae with identity, Archiv für Mathematische Logik und Grundlagenforschung (to appear).
[AL]Aanderaa, S. O. and Lewis, H. R., Prefix classes of Krom formulas, this Journal, vol. 38 (1973), pp. 628642.
[B1]Börger, E., Reduktionstypen in Krom- und Hornformeln, Inauguraldissertation, Westfälische Wilhelms-Universität, Münster, 1971.
[B2]Börger, E., Reduktionstypen der klassischen Prädikatenlogik, unpublished lecture notes, Westfälische Wilhelms-Universität, Münster, 1972.
[DG]Dreben, B. and Goldfarb, W. D., The decision problem: Solvable classes of quantificational formulas, Addision-Wesley, Reading, Massachusetts, 1979.
[Gu]Gurevich, Y., Semi-conservative reduction, Archiv für Mathematische Logik, vol. 18 (1976), pp. 2325.
[K1]Krom, M. R., The decision problem for a class of first-order formulas in which all disjunctions are binary, Zeitschrift für Mathematische Logik und Grundlagen der Mathematik, vol. 13 (1967), pp. 1520.
[K2]Krom, M. R., The decision problem for formulas in prenex conjunctive normal form with binary disjunctions, this Journal, vol. 35 (1970), pp. 210216.
[L]Lewis, H. R., Unsolvable classes of quantificational formulas, Addison-Wesley, Reading, Massachusetts, 1979.
[LG]Lewis, H. R. and Goldfarb, W. D., The decision problem for formulas with a small number of atomic subformulas, this Journal, vol. 38 (1973), pp. 471480.
[Ma]Maslov, S. Yu., Obratniy metod ystanovleniya vyvodimosti v klassicheskom ischislenii predikatov, Doklady Akademii Nauk SSSR, vol. 159 (1964), pp. 1720; English translation in Soviet Mathematics-Doklady, vol. 5 (1964), pp. 1420–1424.
[Mi]Minsky, M., Computation: finite and infinite machines, Prentice-Hall, Englewood Cliffs, New Jersey, 1967.
[T1]Trakhtenbrot, B. A., Nevozmozhnost' algorifma dlya problemy razreshimosti na konechnykh klassakh, Doklady Akademii Nauk SSSR, vol. 70 (1950), pp. 569572; translation in American Mathematical Society Translations (2), vol. 23 (1963), pp. 1–5.
[T2]Trakhtenbrot, B. A., O rekursivnoi otdyelimosti, Doklady Akademii Nauk SSSR, vol. 88 (1953), pp. 953955.

Conservative reduction classes of Krom formulas1

  • Stål O. Aanderaa (a1), Egon Börger (a2) and Harry R. Lewis (a3)

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