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Prefix classes of Krom formulas1

  • Stål O. Aanderaa (a1) and Harry R. Lewis (a2)

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In this paper we consider decision problems for subclasses of Kr, the class of those formulas of pure quantification theory whose matrices are conjunctions of binary disjunctions of signed atomic formulas. If each of Q 1, …, Qn is an ∀ or an ∃, then let Q 1Qn be the class of those closed prenex formulas with prefixes of the form (Q 1 x 1)… (Qnxn ). Our results may then be stated as follows:

Theorem 1. The decision problem for satisfiability is solvable for the class ∀∃∀ ∩ Kr.

Theorem 2. The classes ∀∃∀∀ ∩ Kr and ∀∀∃∀ ∩ Kr are reduction classes for satisfiability.

Maslov [11] showed that the class ∃…∃∀…∀∃…∃ ∩ Kr is solvable, while the first author [1, Corollary 4] showed ∃∀∃∀ ∩ Kr and ∀∃∃∀ ∩ Kr to be reduction classes. Thus the only prefix subclass of Kr for which the decision problem remains open is ∀∃∀∃…∃∩ Kr.

The class ∀∃∀ ∩ Kr, though solvable, contains formulas whose only models are infinite (e.g., (∀x)(∃u)(∀y)[(PxyPyx) ∧ (¬ Pxy ∨ ¬Pyu)], which can be satisfied over the integers by taking P to be ≥). This is not the case for Maslov's class ∃…∃∀…∀∃…∃ ∩ Kr, which contains no formula whose only models are infinite ([2] [5]).

Theorem 1 was announced in [1, Theorem 4], but the proof sketched there is defective: Lemma 4 (p. 17) is incorrectly stated. Theorem 2 was announced in [9].

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1

This paper was prepared while the first author was visiting the IBM Thomas J. Watson Research Center, Yorktown Heights, New York. The second author was supported in part by the Center for Research in Computing Technology, Division of Engineering and Applied Physics, Harvard University, and by a Fellowship from the International Business Machines Corporation. The authors are grateful to Burton Dreben, Warren Goldfarb, and the referee for their many helpful suggestions.

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References

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[1] 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.
[2] Aanderaa, S. O. and Goldfarb, W. D., Finite controllability of the Maslov class, Notices of the American Mathematical Society, vol. 20 (1973), p. A447.
[3] Börger, E., Eine entscheidbare Klasse von Kromformeln, Zeitschrift für Mathematische Logik und Grundlagen der Mathematik (to appear).
[4] Church, A., Introduction to mathematical logic, Vol. I, Princeton University Press, Princeton, N.J., 1956.
[5] Dreben, B. and Goldfarb, W. D., A systematic treatment of the decision problem (forthcoming).
[6] Ginsburg, S., The mathematical theory of context-free languages, McGraw-Hill, N.Y., 1966.
[7] Ginsburg, S. and Spanier, E. H., Bounded ALGOL-like languages, Transactions of the American Mathematical Society, vol. 113 (1964), pp. 333368.
[8] 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.
[9] Lewis, H. R., Two reduction classes of Krom formulae, Notices of the American Mathematical Society, vol. 19 (1972), p. A715.
[10] 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.
[11] Maslov, S. Ju., An inverse method of establishing deducibilities in the classical predicate calculus, Soviet Mathematics Doklady, vol. 5 (1964), pp. 14201424.
[12] Parikh, R. J., On context-free languages, Journal of the Association for Computing Machinery, vol. 13 (1966), pp. 570581.
[13] Presburger, M., Über die Vollständigkeit eines gewissen Systems der Arithmetik ganzer Zahlen, in welchem die Addition als einzige Operation hervortritt, Sprawozdanie z I Kongresu matematyków krajów słowiańskich, Warsaw, 1930, pp. 92–101, 395.
[14] Wang, H., Dominoes and the AEA case of the decision problem, Proceedings of a Symposium on the Mathematical Theory of Automata (New York, 1962), Polytechnic Press, Brooklyn, N.Y., 1963, pp. 2355.

Prefix classes of Krom formulas1

  • Stål O. Aanderaa (a1) and Harry R. Lewis (a2)

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