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Prefix classes of Krom formulas1

Published online by Cambridge University Press:  12 March 2014

Stål O. Aanderaa  and
Harry R. Lewis
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
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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 Q1 … Qn 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)[(Pxy ∨ Pyx) ∧ (¬ 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
Information
The Journal of Symbolic Logic , Volume 38 , Issue 4 , December 1973 , pp. 628 - 642
Copyright
Copyright © Association for Symbolic Logic 1973

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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

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