On delay and regret determinization of max-plus automata

Abstract

Decidability of the determinization problem for weighted automata over the semiring (Z∪{−∞}, max, +), WA for short, is a long-standing open question. We propose two ways of approaching it by constraining the search space of deterministic WA: k-delay and r-regret. A WA N is k-delay determinizable if there exists a deterministic automaton D that defines the same function as N and for all words α in the language of N , the accepting run of D on α is always at most k-away from a maximal accepting run of N on α. That is, along all prefixes of the same length, the absolute difference between the running sums of weights of the two runs is at most k. A WA N is r-regret determinizable if for all words α in its language, its non-determinism can be resolved on the fly to construct a run of N such that the absolute difference between its value and the value assigned to α by N is at most r.We show that a WA is determinizable if and only if it is k-delay determinizable for some k. Hence deciding the existence of some k is as difficult as the general determinization problem. When k and r are given as input, the k-delay and r-regret determinization problems are shown to be EXPtimecomplete. We also show that determining whether a WA is r-regret determinizable for some r is in EXPtime.

Token Games and History-Deterministic Quantitative-Automata

“…In the quantitative setting, deciding history-determinism coincides with bestvalue partial domain synthesis [14], 0-regret synthesis [19] and, for some value functions with 0-regret determinization [13,7]. There are procedures to decide history determinism (which is sometimes called good-for-gameness due to erroneously assuming them equivalent) of Sum, Avg, and DSum automata on finite words, as follows.…”

Finite Sequentiality of Finitely Ambiguous Max-Plus Tree Automata

“…Maxplus automata are weighted automata [1][2][3][4][5] over the max-plus semiring. In the form of min-plus automata, they were originally introduced by Imre Simon as a means to show the decidability of the finite power property [6,7] and they enjoy a continuing interest [8][9][10][11][12][13][14]. They have found applications in many different contexts, for example to determine the star height of a language [15], to prove the termination of certain string rewriting systems [16], and to model discrete event systems [17].…”

Synthesis from Weighted Specifications with Partial Domains over Finite Words

“…Following the game metaphor explained before, those quantitative synthesis problems can be formulated as two-player games in which Adam (environment) and Eve (system) alternatively pick symbols in Σ and Σ Ó respectively. Additionally, Adam has the power to Ptime [3] Ptime [3] NP ∩ coNP strict approximate EXPtime-c [26] D…”

“…Finally, approximate synthesis corresponds to a problem known as r-regret determinization of non-deterministic weighted automata. For sum-automata, it is known to be Ex-pTime-complete [26]. For average-automata, there is no immediate reduction to the sum case, because the sum value computed by an r-regret determinizer can be arbitrarily faraway from the best sum, while its averaged value remains close to the best average.…”