Looking at mean-payoff and total-payoff through windows

Abstract

Abstract. We consider two-player games played on weighted directed graphs with mean-payoff and total-payoff objectives, two classical quantitative objectives. While for single-dimensional games the complexity and memory bounds for both objectives coincide, we show that in contrast to multi-dimensional mean-payoff games that are known to be coNP-complete, multi-dimensional total-payoff games are undecidable. We introduce conservative approximations of these objectives, where the payoff is considered over a local finite window sliding along a play, instead of the whole play. For single dimension, we show that (i) if the window size is polynomial, deciding the winner takes polynomial time, and (ii) the existence of a bounded window can be decided in NP ∩ coNP, and is at least as hard as solving mean-payoff games. For multiple dimensions, we show that (i) the problem with fixed window size is EXPTIME-complete, and (ii) there is no primitive-recursive algorithm to decide the existence of a bounded window.

Computer Aided Synthesis: A Game-Theoretic Approach

“…In [68], the author studies objectives equal to Boolean combinations of inequalities f k (ρ) ∼ µ k , with ∼ ∈ {≤, ≥} and f k ∈ {MP, MP}: deciding whether player 1 has a winning strategy in (G, v 0 ) becomes undecidable. However, this problem remains decidable and is EXPTIME-complete for CNF/DNF Boolean combinations of functions taken among {Sup, Inf, LimSup, LimInf, WMP} [13], where WMP is an interesting window variant of mean-payoff introduced in [20]. The threshold problem is P-complete (resp.…”

“…EXPTIME-complete) for a single WMP objective (resp. an intersection of WMP objectives) [20]. Recall that it is in NP ∩ co-NP for a single function MP or MP (see Theorem 22).…”

“…A sketch of proof is given in the appendix. Problem 7 is PSPACE-complete for the colored variant of Muller objectives[26] 20. Recall our comment after Theorem 22.…”

Weight monitoring with linear temporal logic

“…The (direct) fixed window properties studied in [14] Temporal logic with prefix accumulation. The concept of prefixaccumulation assertions as introduced by Boker et al [8] for weighted Kripke structures and branching-time and linear-time temporal logics is very much in the spirit of the logic LTL avg [ , : All].…”

“…They can be used to formalize, e.g., constraints on the load of a processor in the past few clock cycles or the chart-development of a stock within one day. The fixed window properties studied in [14] for non-probabilistic game structures are expressible in our logic with window DFA. More flexibility is provided by the class of acyclic DFA.…”