Relativized logspace and generalized quantifiers over finite ordered structures

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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 familyQof generalized quantifiers expressing a complexity classC, what is the expressive power of first order logic FO(Q) extended by the quantifiers inQ? From previously studied examples, one would expect that FO(Q) capturesLC, i.e., logarithmic space relativized to an oracle inC. We show that this is not always true. However, after studying the problem from a general point of view, we derive sufficient conditions onCsuch that FO(Q) capturesLC. These conditions are fulfilled by a large number of relevant complexity classes, in particular, for example, byNP. As an application of this result, it follows that first order logic extended by Henkin quantifiers capturesLNP. This answers a question raised by Blass and Gurevich [Ann. Pure Appl. Logic, vol. 32, 1986]. Furthermore we show that for many familiesQof generalized quantifiers (including the family of Henkin quantifiers), each FO(Q)-formula can be replaced by an equivalent FO(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].

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