Hostname: page-component-cd9895bd7-mkpzs Total loading time: 0 Render date: 2024-12-25T19:12:20.965Z Has data issue: false hasContentIssue false

Prefix classes of Krom formulas1

Published online by Cambridge University Press:  12 March 2014

Stål O. Aanderaa
Affiliation:
Institute of Mathematics, University of Oslo, Blindern, Oslo 3, Norway
Harry R. Lewis
Affiliation:
Aiken Computation Laboratory, Harvard University, Cambridge, Massachusetts 02138

Extract

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 Q1, …, Qn is an ∀ or an ∃, then let Q1Qn be the class of those closed prenex formulas with prefixes of the form (Q1x1)… (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].

Type
Research Article
Copyright
Copyright © Association for Symbolic Logic 1973

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

Footnotes

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.

References

REFERENCES

[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.Google Scholar
[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.Google Scholar
[3] Börger, E., Eine entscheidbare Klasse von Kromformeln, Zeitschrift für Mathematische Logik und Grundlagen der Mathematik (to appear).Google Scholar
[4] Church, A., Introduction to mathematical logic, Vol. I, Princeton University Press, Princeton, N.J., 1956.Google Scholar
[5] Dreben, B. and Goldfarb, W. D., A systematic treatment of the decision problem (forthcoming).Google Scholar
[6] Ginsburg, S., The mathematical theory of context-free languages, McGraw-Hill, N.Y., 1966.Google Scholar
[7] Ginsburg, S. and Spanier, E. H., Bounded ALGOL-like languages, Transactions of the American Mathematical Society, vol. 113 (1964), pp. 333368.Google Scholar
[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.CrossRefGoogle Scholar
[9] Lewis, H. R., Two reduction classes of Krom formulae, Notices of the American Mathematical Society, vol. 19 (1972), p. A715.Google Scholar
[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.Google Scholar
[11] Maslov, S. Ju., An inverse method of establishing deducibilities in the classical predicate calculus, Soviet Mathematics Doklady, vol. 5 (1964), pp. 14201424.Google Scholar
[12] Parikh, R. J., On context-free languages, Journal of the Association for Computing Machinery, vol. 13 (1966), pp. 570581.CrossRefGoogle Scholar
[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.Google Scholar
[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.Google Scholar