Energy Games in Multiweighted Automata

Abstract

Abstract. Energy games have recently attracted a lot of attention. These are games played on finite weighted automata and concern the existence of infinite runs subject to boundary constraints on the accumulated weight, allowing e.g. only for behaviours where a resource is always available (nonnegative accumulated weight), yet does not exceed a given maximum capacity. We extend energy games to a multiweighted and parameterized setting, allowing us to model systems with multiple quantitative aspects. We present reductions between Petri nets and multiweighted automata and among different types of multiweighted automata and identify new complexity and (un)decidability results for both one-and two-player games. We also investigate the tractability of an extension of multiweighted energy games in the setting of timed automata.

Optimal Bounds for Multiweighted and Parametrised Energy Games

“…Here they study the existence of a winning strategy for Player 1 for a fixed initial value and fixed upper and lower bound and provide bounds on the complexity for the identified problems both in a timed and untimed setting. The paper [9] extends the results from [2] to the multiweighted case. The work of [7] treats multiweighted energy games with only a lower bound and show that deciding whether there exists a vector of initial values for the resources such that Player 1 can win the energy game is coNP-complete and that only finite-memory strategies are sufficient.…”

“…Another interesting question besides constructing the sets of winning vectors for a given game, is the question of membership; given a k-weighted game G and a vectorb ∈ AE k decide whetherb ∈ W (orb ∈ I orb ∈ U ). The membership problem has been addressed in [9] among others. Table 1 (also found in [9]) gives a full overview of the so far obtained decidability and complexity results for the membership problem.…”

“…Energy games have recently attracted considerable attention [1][2][3][4][5][6][7][8][9]. An energy game is played by two players on a weighted game automaton.…”

“…For multiweighted energy games with an unknown upper bound (both strict and weak) and fixed initial value we calculate the set of minimal upper bounds such that the energy game is winning. For a strict upper bound we make use of results from [3] and [9] in order to construct the set, yielding an algorithm running in 2k-exponential time. In the case of a weak upper bound we utilise an algorithm given by Valk and Jantzen in [10], that constructs the set of minimal elements of an upward-closed set (the so-called Pareto frontier), by showing that the preconditions for applying the algorithm are fulfilled.…”

Lower-bound-constrained runs in weighted timed automata

“…In contrast, the existence of interval-constrained infinite runs-where a simple energy-maximizing strategy does not suffice-have recently been proven undecidable for weighted timed automata with varying numbers of clocks and weight variables: e.g. 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].…”