Local Normal Forms for First-Order Logic with Applications to Games and Automata

First-order (FO) logic and its extensions play an important role in many branches of (theoretical) computer science. Examples that will be considered in this paper are automata theory and descriptive complexity. Since Büchi’s and Elgot’s famous characterization of the regular string languages as the sets of models of (existential) monadic second-order (MSO) sentences, (existential) MSO logic has been used as a guideline in the search for reasonable automata models for other kinds of structures like trees or graphs. In descriptive complexity, since Fagin [Fag74] showed that the complexity class NP coincides with the sets of models of existential second-order ( 11) sentences, many complexity classes have been characterized by extensions of FO logic [Var82, Imm86, Imm87, AV89, Gr92] and there is still hope that separations of complexity classes might be possible by separating the expressive power of the respective logics. For a recent result in this direction see the paper of Libkin and Wong [LW98]. Despite its importance as an ingredient for more expressive logics, it is well-known that the expressive power of FO logic is rather limited. It can only express properties that depend on the local appearance of a structure. This intuition has been formalized in different ways by Hanf [Han65] and Gaifman [Gai82]. Hanf showed that, for every first-order formula , there is an r such that whether holds in a structure A (“A j= ”) only depends on the multiset of isomorphism types of all r-spheres in A. Here an r-sphere is a