Journal of Symbolic Logic

Relativized Logspace and Generalized Quantifiers over Finite Ordered Structures

Georg Gottlob

Abstract

We here examine the expressive power of first order logic with generalized quantifiers over finite ordered structures. In particular, we address the following problem: Given a family $\mathbf{Q}$ of generalized quantifiers expressing a complexity class $\mathbf{C}$, what is the expressive power of first order logic FO($\mathbf{Q}$) extended by the quantifiers in $\mathbf{Q}$? From previously studied examples, one would expect that FO($\mathbf{Q}$) captures $\mathbf{L}^\mathbf{C}$, i.e., logarithmic space relativized to an oracle in $\mathbf{C}$. We show that this is not always true. However, after studying the problem from a general point of view, we derive sufficient conditions on $\mathbf{C}$ such that FO($\mathbf{Q}$) captures $\mathbf{L}^\mathbf{C}$. These conditions are fulfilled by a large number of relevant complexity classes, in particular, for example, by $\mathbf{NP}$. As an application of this result, it follows that first order logic extended by Henkin quantifiers captures $\mathbf{L}^{\mathbf{NP}}$. This answers a question raised by Blass and Gurevich [Ann. Pure Appl. Logic, vol. 32, 1986]. Furthermore we show that for many families $\mathbf{Q}$ of generalized quantifiers (including the family of Henkin quantifiers), each FO($\mathbf{Q}$)-formula can be replaced by an equivalent FO($\mathbf{Q}$)-formula with only two occurrences of generalized quantifiers. This generalizes and extends an earlier normal-form result by I. A. Stewart [Fundamenta Inform. vol. 18, 1993].

Article information

Source
J. Symbolic Logic, Volume 62, Issue 2 (1997), 545-574.

Dates
First available in Project Euclid: 6 July 2007

https://projecteuclid.org/euclid.jsl/1183745242

Mathematical Reviews number (MathSciNet)
MR1464113

Zentralblatt MATH identifier
0882.03031

JSTOR