Hostname: page-component-848d4c4894-nmvwc Total loading time: 0 Render date: 2024-06-16T16:59:44.137Z Has data issue: false hasContentIssue false
  • English
  • Français

The finite controllability of the Maslov case

Published online by Cambridge University Press:  12 March 2014

Stål Aanderaa  and
Warren D. Goldfarb
Stål Aanderaa
Affiliation:
T. J. Watson Research Center, I.B.M., Yorktown Heights, New York 10598 Institute of Mathematics, Universtty of Oslo, Oslo 3, Norway
Warren D. Goldfarb
Affiliation:
Society of Fellows, Harvard University, Cambridge, Massachusetts 02138
Get access Rights & Permissions [Opens in a new window]

Extract

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 A1 ⋁ A2, 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].

Type
Research Article
Information
The Journal of Symbolic Logic , Volume 39 , Issue 3 , September 1974 , pp. 509 - 518
Copyright
Copyright © Association for Symbolic Logic 1974

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.)

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