Positional Properties in Temporal Logic

Abstract

We study positional properties in the context of game-based reactive synthesis. Our motivation stems from having a usable specification logic, for which tractable synthesis is guaranteed. We demonstrate that every $\omega$-regular positional property (with respect to state- or edge-labelled game graphs), is expressible in linear-time temporal logic. Additionally, we provide some necessary and sufficient conditions for when an $\omega$-regular property is positional, and identify well-behaved subclasses of $\omega$-regular positional properties. Using varieties of languages, we prove that no class of $\omega$-regular positional properties can simultaneously contain a prefix-independent property and be closed under Boolean operations. We conclude by discussing the implications on alternating-time temporal logic, where we isolate a few different fragments with tractable model checking, and compare the associated expressivity of such fragments.