An Until Hierarchy and Other Applications of an Ehrenfeucht–Fraı̈ssé Game for Temporal Logic

Abstract

We prove there is a strict hierarchy of expressive power according to the Until depth of linear temporal logic (LTL) formulas: for each k, there is a natural property, based on quantitative fairness, that is not expressible with k nestings of Until operators, regardless of the number of applications of other operators, but is expressible by a formula with Until depth k+1. Our proof uses a new Ehrenfeucht–Fraı̈ssé (EF) game designed specifically for LTL. These properties can all be expressed in first-order logic with quantifier depth and size O(log k), and we use them to observe some interesting relationships between LTL and first-order expressibility. We note that our Until hierarchy proof for LTL carries over to the branching time logics, CTL and CTL*. We then use the EF game in a novel way to effectively characterize (1) the LTL properties expressible without Until, as well as (2) those expressible without both Until and Next. By playing the game “on finite automata,” we prove that the automata recognizing languages expressible in each of the two fragments have distinctive structural properties. The characterization for the first fragment was originally proved by Cohen, Perrin, and Pin using sophisticated semigroup-theoretic techniques. They asked whether such a characterization exists for the second fragment. The technique we develop is general and can potentially be applied in other contexts.