Games on Trees and Syntactical Complexity of Formulas

Ehrenfeucht-Fraïssé 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-Fraïssé 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-Fraïssé 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 Σn and Πn 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-Fraïssé game.