Linear Time Computable Problems and First-Order Descriptions (1996)
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BibTeX
@MISC{Seese96lineartime,
author = {Detlef Seese},
title = {Linear Time Computable Problems and First-Order Descriptions},
year = {1996}
}
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Abstract
this article is a proof that each FO problem can be solved in linear time if only relational structures of bounded degree are considered. The basic idea of the proof is a localization technique based on a method that was originally developed by Hanf (Hanf 1965) to show that the elementary theories of two structures are equal under certain conditions, i.e., that two structures agree on all first-order sentences. Fagin, Stockmeyer and Vardi (Fagin et al. 1993) developed a variant of this technique, which is applicable in descriptive complexity theory to classes of finite relational structures of uniformly bounded degree. Variants of this result can also be found in Gaifman (1982) (see also Thomas (1991)). The essential content of this result, which is also called the Hanf-Sphere Lemma, is that two relational structures of bounded degree satisfy the same first-order sentences of a certain quantifier-rank if both contain, up to a certain number m, the same number of isomorphism types of substructures of a bounded radius r. In addition, a technique of model interpretability from Rabin (1965) (see also Arnborg et al. (1991), Seese (1992), Compton and Henson (1987) and Baudisch et al. (1982)) is adapted to descriptive complexity classes, and proved to be useful for reducing the case of an arbitrary class of relational structures to a class of structures consisting only of the domain and one binary irreflexive and symmetric relation, i.e., the class of simple graphs. It is shown that the class of simple graphs is lintime-universal with respect to first-order logic, which shows that many problems on descriptive complexity classes, described in languages extending first-order logic for arbitrary structures, can be reduced to problems on simple graphs. This paper is organized as f...
