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

Abstract

Weighted automata model quantitative aspects of systems like memory or power consumption. Recently, Chatterjee, Doyen, and Henzinger introduced a new kind of weighted automata which compute objectives like the average cost or the long-time peak power consumption. In these automata, operations like average, limit superior, limit inferior, limit average, or discounting are used to assign values to finite or infinite words. In general, these weighted automata are not semiring weighted anymore. Here, we establish a connection between such new kinds of weighted automata and weighted logics. We show that suitable weighted MSO logics and these new weighted automata are expressively equivalent, both for finite and infinite words. The constructions employed are effective, leading to decidability results for the weighted logic formulas considered.

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

“…As in [14], for timed pv-monoid with additional properties there are larger fragments of weighted relative-distance logic which are still expressively equivalent to WTA. For the simplicity of presentation, we restricted ourselves to idempotent timed pv-monoids.…”

“…For untimed words, weighted MSO logic over semirings was defined in [10]. A weighted MSO logic over (untimed) product valuation monoids was considered in [14]. We will use a similar approach to define the syntax and the semantics of our weighted relative distance logic.…”

“…We will use a similar approach to define the syntax and the semantics of our weighted relative distance logic. In [14], valuation monoids were augmented with a product operation and a unit element to define the semantics of weighted formulas. Here, we proceed in a similar way and consider timed product valuation monoids.…”

“…(ϕ 1 , ϕ 2 ). Our approach also extends the idea of [14] to define the semantics of formulas with a first-order universal quantifier using the valuation function.…”

“…The classical Büchi-Elgot theorem [9] was extended to both weighted [10,11,14] and timed settings [26,27]. In [24,25], a semiring-weighted extension of Wilke's relative distance logic [26,27] was considered.…”