Optimal Reachability in Divergent Weighted Timed Games

TL;DR

This paper introduces divergent weighted timed games, a new class allowing negative weights and multiple clocks, and provides algorithms for computing their optimal values and deciding divergence efficiently.

Contribution

It generalizes previous subclasses by allowing arbitrary weights and multiple clocks, establishing decidability and polynomial-time algorithms for divergence and untimed cases.

Findings

First decidable class of weighted timed games with negative weights and multiple clocks.

Polynomial space algorithm for deciding divergence.

Polynomial time computation of game values in untimed cases.

Abstract

Weighted timed games are played by two players on a timed automaton equipped with weights: one player wants to minimise the accumulated weight while reaching a target, while the other has an opposite objective. Used in a reactive synthesis perspective, this quantitative extension of timed games allows one to measure the quality of controllers. Weighted timed games are notoriously difficult and quickly undecidable, even when restricted to non-negative weights. Decidability results exist for subclasses of one-clock games, and for a subclass with non-negative weights defined by a semantical restriction on the weights of cycles. In this work, we introduce the class of divergent weighted timed games as a generalisation of this semantical restriction to arbitrary weights. We show how to compute their optimal value, yielding the first decidable class of weighted timed games with negative weights and an arbitrary number of clocks. In addition, we prove that divergence can be decided in polynomial space. Last, we prove that for untimed games, this restriction yields a class of games for which the value can be computed in polynomial time.