Optimal Complexity Bounds for Positive LTL Games

• Authors: Jerzy Marcinkowski;Tomasz Truderung
• Venue: CSL '02 Proceedings of the 16th International Workshop and 11th Annual Conference of the EACSL on Computer Science Logic
• Year: 2002

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+(驴,驴, 驴) games is in PSPACE. This is a tight bound: the problem is known to be PSPACE-hard even for the much weaker logic LTL+(驴,驴). 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 B眉chi automata technique, is EXPSPACE. This means that we identify a natural fragment of LTL for which the algorithm resulting from the B眉chi 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+(驴, 驴, 驴, 驴). This solves an open problem from [AT01], where the authors use the B眉chi automata technique to show an EXPSPACE algorithm deciding more general LTL(驴, 驴, &and, 驴) games, but do not prove optimality of this upper bound