Games on Trees and Syntactical Complexity of Formulas

Abstract

Ehrenfeucht-Fraisse games have been introduced as a means of characterizing the relation of elementary equivalence between structures, or relational database instances in first order logic (FO), or equivalently Relational Calculus. In∼the usual Ehrenfeucht-Fraisse games the rules are determined by a linear ordering of a fixed lenght or, equivalently, by a special kind of tree - a chain of a fixed length -, where each point of that ordering or node of that tree corresponds to a quantification operation. Here we consider Ehrenfeucht-Fraisse games whose rules are determined by arbitrary trees such that their nodes correspond either to quantification operations (“q-nodes”) or to connective operations (“c-nodes”). By playing games on trees, we can refine the class of sentences which are considered in a given game, since a tree represents a particular class of sentences. We define and study several variations of tree games, for first and second order logic (SO). We give a sufficient condition for FO and SO equivalence restricted to formulae with up to n connectives, and hence also a sufficient condition for the non expressibility of a given query in those logics with formulae whose number of logical connectives is less than a given integer. We also give a sufficient condition to prove simultaneous lower bounds in both the number of connectives and in the quantifier types needed to express a given property in FO. If we consider only quantifier types, we get a characterization of the relation of preservation of sentences in the fragment of FO with the given set of quantifier types. We also study tree games for Σ n and Π n formulae. To illustrate the use of our games we use them to prove lower bounds in the connective size for several FO queries, like size of a database, size of a clique in a graph, size of a unary relation, transitive property in a graph, and degree of a node in a graph. Regarding SO, we prove lower bounds for quantifier rank for the parity query.Finally, we give a precise characterization of the logic whose elementary equivalence is characterized by a given tree game, as well as several equivalent characterizations of the existence of a winning strategy for Duplicator in the classical Ehrenfeucht-Fraisse game.