Weighted Automata and Regular Expressions Over Valuation Monoids

Abstract

Quantitative aspects of systems like consumption of resources, output of goods, or reliability can be modeled by weighted automata. Recently, objectives like the average cost or the longtime peak power consumption of a system have been modeled by weighted automata which are not semiring weighted anymore. Instead, operations like limit superior, limit average, or discounting are used to determine the behavior of these automata. Here, we introduce a new class of weight structures subsuming a range of these models as well as semirings. Our main result shows that such weighted automata and Kleene-type regular expressions are expressively equivalent both for finite and infinite words.

Weighted automata and weighted MSO logics for average and long-time behaviors

“…A characterization of this new kind of weighted automata by regular expressions was obtained in [14,33]. Moreover, results similar to the ones of this paper were also obtained for finite traces [26] and finite trees [11].…”

“…Weighted Büchi automata were considered in several weighted settings, cf. [14,16,22,30,33]. For ω-valuation monoids several types of behavior of weighted automata on infinite words can be defined [33].…”

Weighted Recognizability over Infinite Alphabets

“…Furthermore, it should be very interesting to study the weighted variable automata theory over more general weight structures, contributing to real world applications, like for instance over valuation monoids [15].…”

Weighted finite automata over hemirings

“…Using our previous results, we obtain a Kleene characterization of automata over such weight structures, which extends Schützenberger's original result. Here our assumptions are slightly stronger than the ones of [11], but they apply to all natural examples including Chatterjee, Doyen, and Henzinger's quantitative automata, realizing our goal mentioned above.…”

“…Due to the average operation, the theory of semiring-weighted automata is not (or at least not directly) applicable. Nevertheless, Droste and Meinecke [11] could characterize the behaviors of such quantitative automata with average weight computations by rational series using valuation functions and direct constructions of weighted automata.…”