MSO logics for weighted timed automata

Abstract

We aim to generalize Büchi's fundamental theorem on the coincidence of recognizable and MSO-definable languages to a weighted timed setting. For this, we investigate weighted timed automata and show how we can extend Wilke's relative distance logic with weights taken from an arbitrary semiring. We show that every formula in our logic can effectively be transformed into a weighted timed automaton, and vice versa. The results indicate the robustness of weighted timed automata and may also be used for specification purposes.

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

“…However, we also obtained a more complicated result for non-idempotent timed pv-monoids. In [24,25], for weighted relative distance logic over non-idempotent semirings, a strong restriction on the use of a first-order universal quantification was done. Surprisingly, in our result we could avoid this restriction.…”

“…Remark 6.3. In [24,25], Quaas introduced a weighted version of relative distance logic over a semiring S = (S, +, ·, 0, 1) and a family of functions F ⊆ S R ≥0 where elements of S model discrete weights and functions f ∈ F model continuous weights. If F is a one-parametric family of functions (f s ) s∈S , then our weighted logic incorporates the logic of Quaas over S and F .…”

“…In the literature, various models of WTA were considered, e.g., linearly priced timed automata [3,4,21], multi-weighted timed automata with knapsack-problem objective [22], and WTA with measures like average, reward-cost ratio [7,8] and discounting [2,18,19]. In [24,25], WTA over semirings were studied with respect to the classical automata-theoretic questions. However, various models, e.g., WTA with average and discounting measures as well as multi-weighted automata cannot be defined using semirings.…”

“…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. Here, we develop a different weighted version of relative distance logic based on our notion of timed valuation monoids.…”

Automata, Languages, and Programming

“…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. Here, we develop a different weighted version of relative distance logic based on our notion of timed valuation monoids.…”