New Progress with a Forgotten Logical Game

Abstract

We describe our rediscovery of an intriguing logical game, introduced by Immerman in 1981, but then, until now, never again mentioned in the literature. The game is played on two sets A and B of structures. These multi-structural games generalize Ehrenfeucht-Fraisse games. Whereas Ehrenfeucht-Fraisse games capture the quantifier rank of a first-order sentence, multi-structural games capture the number of quantifiers, in the sense that Spoiler wins the r-round game if and only if there is a first-order sentence Φ with at most r quantifiers, where every structure in A satisfies Φ and no structure in B satisfies Φ. We use these games to give a complete characterization of the number of quantifiers required to distinguish linear orders of different sizes, and develop machinery for analyzing structures beyond linear orders.