Optimal Complexity Bounds for Positive LTL Games

Abstract

We prove two tight bounds on complexity of deciding graph games with winning conditions defined by formulas from fragments of LTL.

Our first result is that deciding LTL ₊ (◊, ⋀, ∨) games is in PSPACE. This is a tight bound: the problem is known to be PSPACE-hard even for the much weaker logic LTL ₊ (◊, ⋀). We use a method based on a notion of, as we call it, persistent strategy: we prove that in games with positive winning condition the opponent has a winning strategy if and only if he has a persistent winning strategy.

The best upper bound one can prove for our problem with the Büchi automata technique, is EXPSPACE. This means that we identify a natural fragment of LTL for which the algorithm resulting from the Büchi automata tool is one exponent worse than optimal.

As our second result we show that the problem is EXPSPACE-hard if the winning condition is from the logic LTL ₊ (◊, ○, ⋀, ∨). This solves an open problem from [AT01], where the authors use the Büchi automata technique to show an EXPSPACE algorithm deciding more general LTL(◊, ○, ⋀, ∨) games, but do not prove optimality of this upper bound

Partially supported by Polish KBN grant 2 PO3A 01818.

Partially supported by Polish KBN grant 8T11C 04319.