Deterministic generators and games for Ltl fragments

Abstract

Deciding infinite two-player games on finite graphs with the winning condition specified by a linear temporal logic (Ltl) formula, is known to be 2Exptime-complete. In this paper, we identify Ltl fragments of lower complexity. Solving Ltl games typically involves a doubly exponential translation from Ltl formulas to deterministic ω-automata. First, we show that the longest distance (length of the longest simple path) of the generator is also an important parameter, by giving an O ( d log n )-space procedure to solve a Büchi game on a graph with n vertices and longest distance d . Then, for the Ltl fragment of the Boolean combinations of formulas obtained only by eventualities and conjunctions, we provide a translation to deterministic generators of exponential size and linear longest distance, show both of these bounds to be optimal, and prove the corresponding games to be Pspace-complete. Introducing next modalities in this fragment, we give a translation to deterministic generators still of exponential size but also with exponential longest distance, show both of these bounds to be optimal, and prove the corresponding games to be Exptime-complete. For the fragment resulting by further adding disjunctions, we provide a translation to deterministic generators of doubly exponential size and exponential longest distance, show both of these bounds to be optimal, and prove the corresponding games to be Expspace. We also show tightness of the double exponential bound on the size as well as the longest distance for deterministic generators of Ltl formulas without next and until modalities. Finally, we identify a class of deterministic Büchi automata corresponding to a fragment of Ltl with restricted use of always and until modalities, for which deciding games is Pspace-complete.