Deterministic generators and games for LTL fragments

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 Buchi game on a graph with n vertices and longest distance d. Then, for the LTL fragment with only 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 provide a translation to deterministic generators that is still of exponential size but also with exponential longest distance, show both 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 bounds to be optimal and prove the corresponding games to be EXPSPACE. Finally, we show tightness of the double-exponential bound on the size as well as the longest distance for deterministic generators for LTL, even in the absence of “next” and “until” modalities