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Random models and solvable Skolem classes

Published online by Cambridge University Press:  12 March 2014

Warren Goldfarb
Warren Goldfarb*
Affiliation:
Department of Philosophy, Harvard University, Cambridge, Massachusetts 02138, E-mail: [email protected]
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Extract

A Skolem class is a class of formulas of pure quantification theory in Skolem normal form: closed, prenex formulas with prefixes ∀…∀∃…∃. (Pure quantification theory contains quantifiers, truth-functions, and predicate letters, but neither the identity sign nor function letters.) The Gödel Class, in which the number of universal quantifiers is limited to two, was shown effectively solvable (for satisfiability) sixty years ago [G1]. Various solvable Skolem classes that extend the Gödel Class can be obtained by allowing more universal quantifiers but restricting the combinations of variables that may occur together in atomic subformulas [DG, Chapter 2]. The Gödel Class and these extensions are also finitely controllable, that is, every satisfiable formula in them has a finite model. In this paper we isolate a model-theoretic property that connects the usual solvability proofs for these classes and their finite controllability. For formulas in the solvable Skolem classes, the property is necessary and sufficient for satisfiability. The solvability proofs implicitly relied on this fact. Moreover, for any formula in Skolem normal form, the property implies the existence of a finite model.

The proof of the latter implication uses the random models technique introduced in [GS] for the Gödel Class and modified and applied in [Go] to the Maslov Class. The proof thus substantiates the claim made in [Go] that random models can be adapted to the Skolem classes of [DG, Chapter 2]. As a whole, the results of this paper provide a more general, systematic approach to finite controllability than previous methods.

Type
Research Article
Information
The Journal of Symbolic Logic , Volume 58 , Issue 3 , September 1993 , pp. 908 - 914
Copyright
Copyright © Association for Symbolic Logic 1993

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References

REFERENCES

[DG]Dreben, B. and Goldfarb, W., The decision problem: Solvable classes of quantificational formulas, Addison-Wesley, Reading, Massachusetts, 1979.Google Scholar
[Gl]Gödel, K., Ein Spezialfall des Entscheidungsproblems der theoretischen Logik, Ergebnisse eines mathematischen Kolloquiums der Universität Wien, vol. 2 (1932), pp. 27–28.Google Scholar
[G2]Gödel, K., Zum Entscheidungsproblem des logischen Funktionenkalküls, Monatshefte für Mathematik und Physik, vol. 40 (1933), pp. 433–443.CrossRefGoogle Scholar
[Go]Goldfarb, W., Random models and the Moslov class, this Journal, vol. 54 (1989), pp. 460–465.Google Scholar
[GS]Gurevich, Y. and Shelah, S., Random models and the Gödel case of the decision problem, this Journal, vol. 48 (1983), pp. 1120–1124.Google Scholar
[K]Kalmár, L., Über die Erfüllbarkeit derjenigen Zählausdrücke, welche in der Normalform zwei benachbarte Allzeichen enthalten, Mathematische Annalen,, vol. 108 (1933), pp. 466–484.CrossRefGoogle Scholar
[S]Schütte, K., Untersuchungen zum Entscheidungsproblem der mathematischen Logik, Mathema-tisch Annalen, vol. 109 (1934), pp. 572–603.CrossRefGoogle Scholar