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The finite controllability of the Maslov case

  • Stål Aanderaa (a1) (a2) and Warren D. Goldfarb (a3)

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In this paper we show the finite controllability of the Maslov class of formulas of pure quantification theory (specified immediately below). That is, we show that every formula in the class has a finite model if it has a model at all. A signed atomic formula is an atomic formula or the negation of one; a binary disjunction is a disjunction of the form A1A2, where A1 and A2 are signed atomic formulas; and a formula is Krom if it is a conjunction of binary disjunctions. Finally, a prenex formula is Maslov if its prefix is ∃···∃∀···∀∃···∃ and its matrix is Krom.

A number of decidability results have been obtained for formulas classified along these lines. It is a consequence of Theorems 1.7 and 2.5 of [4] that the following are reduction classes (for satisfiability): the class of Skolem formulas, that is, prenex formulas with prefixes ∀···∀∃···∃, whose matrices are conjunctions one conjunct of which is a ternary disjunction and the rest of which are binary disjunctions; and the class of Skolem formulas containing identity whose matrices are Krom. Moreover, the following results (for pure quantification theory, that is, without identity) are derived in [1] and [2]: the classes of prenex formulas with Krom matrices and prefixes ∃∀∃∀, or prefixes ∀∃∃∀, or prefixes ∀∃∀∀ are all reduction classes, while formulas with Krom matrices and prefixes ∀∃∀ comprise a decidable class. The latter class, however, is not finitely controllable, for it contains formulas satisfiable only over infinite universes. The Maslov class was shown decidable by Maslov in [11].

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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. 4–14.
[2]Aanderaa, S. O. and Lewis, H., Prefix classes of Krom formulas, this Journal, vol. 38 (1973), pp. 628–642.
[3]Ackermann, W., Solvable cases of the decision problem, North-Holland, Amsterdam, 1954.
[4]Chang, C. C. and Keisler, H. J., An improved prenex normal form, this Journal, vol. 27 (1962), pp. 317–326.
[5]Dreben, B., Solvable Surányi subclasses: an introduction to the Herbrand theory, Proceedings of a Harvard symposium on digital computers and their applications, 3–6 April 1961, Harvard University, Cambridge, Mass., 1962, pp. 32–47.
[6]Dreben, B. and Goldfarb, W. D., A systematic treatment of the decision problem (to appear).
[7]Erdös, P., On a problem m graph theory, Mathematical Gazette, vol. 47 (1963), pp. 220–223.
[8]Gödel, K., Ein Spezialfall des Entscheidungsproblems der theoretischen Logik, Ergebnisse eines mathematischen Kolloquiums, vol. 2 (1932), pp. 27–28.
[9]Gödel, K., Zum Entscheidungsproblem des logischen Funktionenkalküls, Monatshefte für Mathematik und Physik, vol. 40 (1933), pp. 433–443.
[10]Kalmár, L., Über die Erfüllbarkeit derjenigen Zāhlausdrücke, welche in der Normalform zwei benachbarte Allzeichen enthalten, Mathematische Annalen, vol. 108 (1933), pp. 466–484.
[11]Maslov, S. Ju., An inverse method of establishing deducibilities in the classical predicate calculus, Soviet Mathematics—Doklady, vol. 5 (1964), pp. 1420–1424.
[12]Schütte, K., Untersuchungen zum Entscheidungsproblem der mathematischen Logik, Mathematische Annalen, vol. 109 (1934), pp. 572–603.
[13]Schütte, K., Über die Erfüllbarkeit einer Klasse von logischen Formeln, Mathematische Annalen, vol. 110(1934), pp. 161–194.

The finite controllability of the Maslov case

  • Stål Aanderaa (a1) (a2) and Warren D. Goldfarb (a3)

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