🤖 AI Summary
This work investigates the logical characterization and synthesizability of ω-regular positional properties in reactive synthesis games. Leveraging language cluster theory, game semantics, ω-automata, and model checking techniques for alternating-time temporal logic, it establishes that all such properties are expressible in linear temporal logic (LTL) and provides necessary and sufficient conditions for positionality. The main contributions include demonstrating the equivalence between ω-regular positionality and LTL expressibility, proving the nonexistence of any positional class that is both closed under Boolean operations and contains all prefix-independent properties, identifying several subclasses with favorable synthesis behavior, and isolating fragments of alternating-time temporal logic that admit efficient model checking while precisely delineating their expressive power.
📝 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 $ω$-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 $ω$-regular property is positional, and identify well-behaved subclasses of $ω$-regular positional properties. Using varieties of languages, we prove that no class of $ω$-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.