Costs and rewards in priced timed automata

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 ε-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 10 th Problem, while decidability for three observers is shown by a translation to a fragment of arithmetic involving quadratic forms.

Optimal Infinite Temporal Planning: Cyclic Plans for Priced Timed Automata

“…If the rest of the cycle has cost/reward 1/0, then the better ratio is 3/3, and if it has 0/2, then the better ratio is 1/3. While this could be dealt with multi-objective search (Fränzle et al 2022;Larsen and Rasmussen 2008), maintaining the entire Pareto front for each clock valuation is unnecessary because we are interested on a single objective: the minimum ratio. Gondran and Minoux (1995) show how to overcome this problem by combining the cost and reward into a single weight, and then incrementally finding better solutions (according to the current weight) until the optimal is found.…”