Costs and Rewards in Priced Timed Automata

TL;DR

This paper analyzes the complexity of Pareto reachability in multi-priced timed automata, showing undecidability in general but decidability with restrictions, and introduces approximate solutions and connections to Diophantine equations.

Contribution

It establishes the decidability boundaries of the Pareto Domination Problem in multi-priced timed automata and links the problem to Diophantine equations.

Findings

Undecidable in general for more than three observers.

Decidable for up to three observers.

PSPACE-complete when all observers are costs or rewards.

Abstract

We consider Pareto analysis of reachable states of multi-priced timed automata (MPTA): timed automata equipped with multiple observers that keep track of costs (to be minimised) and rewards (to be maximised) along a computation. Each observer has a constant non-negative derivative which may depend on the location of the MPTA. We study the Pareto Domination Problem, which asks whether it is possible to reach a target location via a run in which the accumulated costs and rewards Pareto dominate a given objective vector. We show that this problem is undecidable in general, but decidable for MPTA with at most three observers. For MPTA whose observers are all costs or all rewards, we show that the Pareto Domination Problem is PSPACE-complete. We also consider an epsilon-approximate Pareto Domination Problem that is decidable without restricting the number and types of observers. We develop connections between MPTA and Diophantine equations. Undecidability of the Pareto Domination Problem is shown by reduction from Hilbert's 10th Problem, while decidability for three observers is shown by a translation to a fragment of arithmetic involving quadratic forms.