Simple Stochastic Games with Almost-Sure Energy-Parity Objectives are in NP and coNP

Abstract

We study stochastic games with energy-parity objectives, which combine quantitative rewards with a qualitative $$\omega $$ ω -regular condition: The maximizer aims to avoid running out of energy while simultaneously satisfying a parity condition. We show that the corresponding almost-sure problem, i.e., checking whether there exists a maximizer strategy that achieves the energy-parity objective with probability 1 when starting at a given energy level k, is decidable and in $$\mathsf {NP}\cap \mathsf {coNP}$$ NP ∩ coNP . The same holds for checking if such a k exists and if a given k is minimal.

Simple Stochastic Games with Almost-Sure Energy-Parity Objectives are in NP and coNP

“…This applies in particular to the Gain objective, but not to k-Bailout objectives, as these are not shift-invariant. A proof of Theorem 6 can be found in [41]. It uses a recursive argument based on the notion of reset strategies from [34].…”

“…One important observation is that it is possible to replace, without changing the outcome, the energy EN(k) condition in the Bailout(k, l) objective by the more restrictive energy-storage ST(k, l) condition. See [41] for further details.…”

“…Using the existence of MD optimal minimizer strategies (Lemma 15) and a coNP upper bound for checking almost sure Gain in MDPs established in [40], we can derive a coNP procedure. See [41] for full details.…”

“…For the the technical details of the G → G 1 → G 2 → G 3 constructions please see [41]. Figure 2 shows how these transformations may look like.…”

“…Due to page restrictions, most proofs in the main body of the paper were replaced by sketches. The detailed proofs can be found in the full version of this paper [41].…”