Revisiting Decidability and Optimum Reachability for Multi-Priced Timed Automata

Abstract

Abstract. We investigate the optimum reachability problem for MultiPriced Timed Automata (MPTA) that admit both positive and negative costs on edges and locations, thus bridging the gap between the results of Bouyer et al. (2007) and of Larsen and Rasmussen (2008). Our contributions are the following: (1) We show that even the location reachability problem is undecidable for MPTA equipped with both positive and negative costs, provided the costs are subject to a bounded budget, in the sense that paths of the underlying Multi-Priced Transition System (MPTS) that operationally exceed the budget are considered as not being viable. This undecidability result follows from an encoding of StopWatch Automata using such MPTA, and applies to MPTA with as few as two cost variables, and even when no costs are incurred upon taking edges. (2) We then restrict the MPTA such that each viable quasi-cyclic path of the underlying MPTS incurs a minimum absolute cost. Under such a condition, the location reachability problem is shown to be decidable and the optimum cost is shown to be computable for MPTA with positive and negative costs and a bounded budget. These results follow from a reduction of the optimum reachability problem to the solution of a linear constraint system representing the path conditions over a finite number of viable paths of bounded length.

Costs and rewards in priced timed automata

“…Multi Priced Timed Automata (MPTA) [5,7,8,10,17,18,19] extend priced timed automata [2, 3,4,6,16] with multiple observers that capture the accumulation of costs and rewards along a computation. This extension allows to model multi-objective optimization problems beyond the scope of timed automata [1].…”

“…This has not been addressed in prior work on Pareto analysis of MPTA [17], which considers only costs or only rewards. Other works on MPTA either do not address Pareto analysis [5,8,10,18,19], or have only discrete costs updated on edges [22], or are confined to a single clock [7].…”

Costs and Rewards in Priced Timed Automata

“…Multi Priced Timed Automata (MPTA) [5,7,8,10,17,18,19] extend priced timed automata [2,3,4,6,16] with multiple observers that capture the accumulation of costs and rewards along a computation. This extension allows to model multi-objective optimization problems beyond the scope of timed automata [1].…”