Decidability of One-Clock Weighted Timed Games with Arbitrary Weights

2022-07-04

International Conference on Concurrency Theory

AI Summary

This paper resolves the long-standing open problem of decidability for one-clock weighted timed games (WTGs) under arbitrary real-valued weights. Prior work was restricted to non-negative weights; we establish, for the first time, that the value function exists and is computable for one-clock WTGs with arbitrary (positive, negative, or zero) weights. Methodologically, we introduce a novel compositional analysis framework integrating piecewise-affine function representation, refined region abstraction, and fixed-point iteration. This framework not only proves decidability but also yields an exact exponential-time algorithm, where the complexity is exponential in the unary encoding size of the weights. Our key contribution is the removal of sign restrictions on weights, enabling rigorous construction and effective computation of the value function. This breakthrough advances the theoretical foundations of weighted real-time games and opens new avenues for quantitative verification of real-time systems with cost accumulation.

Abstract

Weighted Timed Games (WTG for short) are the most widely used model to describe controller synthesis problems involving real-time issues. Unfortunately, they are notoriously difficult, and undecidable in general. As a consequence, one-clock WTGs have attracted a lot of attention, especially because they are known to be decidable when only non-negative weights are allowed. However, when arbitrary weights are considered, despite several recent works, their decidability status was still unknown. In this paper, we solve this problem positively and show that the value function can be computed in exponential time (if weights are encoded in unary).

Problem

• Weighted Timed Games
• Arbitrary Weights
• Solvability

Innovation

• Weighted Timed Games
• Deterministic Computation
• Single-Clock WTG