Infinite Runs in Weighted Timed Automata with Energy Constraints

Abstract

Abstract.We study the problems of existence and construction of infinite schedules for finite weighted automata and one-clock weighted timed automata, subject to boundary constraints on the accumulated weight. More specifically, we consider automata equipped with positive and negative weights on transitions and locations, corresponding to the production and consumption of some resource (e.g. energy). We ask the question whether there exists an infinite path for which the accumulated weight for any finite prefix satisfies certain constraints (e.g. remains between 0 and some given upper-bound). We also consider a game version of the above, where certain transitions may be uncontrollable.

Lower-bound-constrained runs in weighted timed automata

“…Still, the general problem of existence of infinite lower-bound runs for weighted timed automata has remained unsettled since [10]. In this paper we close this open problem showing that it is undecidable for weighted timed automata with four or more clocks.…”

“…This is a weighted graph whose size is at most |L|. Using a modified Bellman-Ford algorithm [10], we can compute the least c for which profit(( , r), c), if any. This algorithm runs in polynomial time, and returns a natural number (because discrete weights are integers).…”

“…In [10], a new class of resource-allocation problems was introduced, namely that of constructing infinite schedules subject to boundary constraints on the accumulation of resources.…”

“…Inside the feasible area, the optimal final credit is e f = e 0 − t 0 on the right of the solid line, while it is e f = 2e 0 + 2t 0 − 5 on the left. For weighted timed automata with a single clock and a single weight variable, the existence of a lower-bound constrained infinite run has been shown decidable in polynomial time [10] with the restriction that no discrete updates of the accumulated weight occur on transitions. In [8], it is shown that the problem remains decidable if this restriction is lifted and even if the accumulated weight grows not only linearly but also exponentially.…”

“…two clocks and two weight variables [17], one clock and two weight variables [13], and two clocks and one weight variable [15]. Also, the interval-constrained problem is undecidable for weighted timed automata with one clock and one weight variable in the game setting [10].…”