A Nivat Theorem for Weighted Timed Automata and Weighted Relative Distance Logic

Abstract

Abstract. Weighted timed automata (WTA) model quantitative aspects of real-time systems like continuous consumption of memory, power or financial resources. They accept quantitative timed languages where every timed word is mapped to a value, e.g., a real number. In this paper, we prove a Nivat theorem for WTA which states that recognizable quantitative timed languages are exactly those which can be obtained from recognizable boolean timed languages with the help of several simple operations. We also introduce a weighted extension of relative distance logic developed by Wilke, and we show that our weighted relative distance logic and WTA are equally expressive. The proof of this result can be derived from our Nivat theorem and Wilke's theorem for relative distance logic. Since the proof of our Nivat theorem is constructive, the translation process from logic to automata and vice versa is also constructive. This leads to decidability results for weighted relative distance logic.

Weight Assignment Logic

“…In this paper we introduced a weight assignment logic which is a simple and intuitive logical formalism for reasoning about quantitative ω-languages. Moreover, it works with arbitrary valuation functions whereas in weighted logics of [12], [14] some additional restrictions on valuation functions were added. We showed that WAL is expressively equivalent to unambiguous weighted Büchi automata.…”

“…We have also extended the results of this paper to the timed setting and obtained a logical characterization of multi-weighted timed automata (cf., e.g., [5], [21]). For the proof of this result we applied a Nivat decomposition theorem for weighted timed automata [14]. Due to space constraints we cannot present this result here.…”

“…A logical characterization of the quantitative languages of Chatterjee, Doyen and Henzinger was given in [12] (again by a restricted fragment of weighted MSO logic). In [14], a multi-weighted extension of weighted MSO logic of [12] with the multiset-based semantics was considered.Our contributions. In this paper, we introduce a new approach to logic for quantitative languages, different from [9,12,14,15].…”

“…In [14], a multi-weighted extension of weighted MSO logic of [12] with the multiset-based semantics was considered.Our contributions. In this paper, we introduce a new approach to logic for quantitative languages, different from [9,12,14,15]. We develop a so-called weight assignment logic (WAL) on infinite words, an extension of the classical MSO logic to the quantitative setting.…”

A unifying survey on weighted logics and weighted automata

“…This semantics is in the spirit of a transducer. It has already been used in similar contexts: in [16] with an operator H(ω) which relabels trees with operations taken from a multi-operator monoid, in [19] with a weight assignment logic over infinite words, in [12,19] with Nivat theorems for weighted automata over various structures. In a second phase, a concrete semantics is given, by means of an aggregation operator taking the abstract semantics and aggregating every multiset of weight structures to a single value (in a possibly different weight domain).…”