On the Complexity of ATL and ATL* Module Checking

TL;DR

This paper analyzes the computational complexity of module checking for ATL and ATL*, revealing that ATL module checking is EXPTIME-complete, while ATL* module checking is 3EXPTIME-complete, thus highlighting the increased complexity of ATL*.

Contribution

It establishes asymptotically optimal bounds on the complexity of ATL and ATL* module checking using automata-theoretic methods.

Findings

ATL module checking is EXPTIME-complete.

ATL* module checking is 3EXPTIME-complete.

Complexity increases significantly from ATL to ATL*.

Abstract

Module checking has been introduced in late 1990s to verify open systems, i.e., systems whose behavior depends on the continuous interaction with the environment. Classically, module checking has been investigated with respect to specifications given as CTL and CTL* formulas. Recently, it has been shown that CTL (resp., CTL*) module checking offers a distinctly different perspective from the better-known problem of ATL (resp., ATL*) model checking. In particular, ATL (resp., ATL*) module checking strictly enhances the expressiveness of both CTL (resp., CTL*) module checking and ATL (resp. ATL*) model checking. In this paper, we provide asymptotically optimal bounds on the computational cost of module checking against ATL and ATL*, whose upper bounds are based on an automata-theoretic approach. We show that module-checking for ATL is EXPTIME-complete, which is the same complexity of module checking against CTL. On the other hand, ATL* module checking turns out to be 3EXPTIME-complete, hence exponentially harder than CTL* module checking.