Weighted Automata and Weighted Logics

Abstract

Abstract. Weighted automata are used to describe quantitative properties in various areas such as probabilistic systems, image compression, speech-to-text processing. The behaviour of such an automaton is a mapping, called a formal power series, assigning to each word a weight in some semiring. We generalize Büchi's and Elgot's fundamental theorems to this quantitative setting. We introduce a weighted version of MSO logic and prove that, for commutative semirings, the behaviours of weighted automata are precisely the formal power series definable with our weighted logic. We also consider weighted first-order logic and show that aperiodic series coincide with the first-order definable ones, if the semiring is locally finite, commutative and has some aperiodicity property.

Definable Transductions and Weighted Logics for Texts

“…A formula ϕ ∈ MSO(¡ , σ) is restricted if it does not contain universal set quantification and whenever ϕ has subformula ∀x.ψ, then ψ is a definable step function. The following theorem extends the result of Droste and Gastin [5] to trees in T Σ (∆). The domain of a tree is a finite, nonempty, prefix-closed subset of * and it has relations for the node labeling and relations E i (x, y) saying that y is the i-th child of x. Theorem 3.6 (Droste & Vogler [7]).…”

“…We now define weighted MSO logic as introduced in [5]. Formulae of MSO(¡ , σ) are built from the atomic formulae k (for k ∈ ¡ ), x = y, R i (x 1 .…”

“…In particular, the fundamental result of Büchi on the coincidence of recognizable and definable languages has been extended to texts [15]. Recently, Droste and Gastin [5] introduced weighted logics over words and showed a Büchi-type characterization for weighted automata over words. They enrich the language of monadic second order logic with values from a semiring in order to add quantitative expressiveness.…”

Language and Automata Theory and Applications

“…For a comprehensive survey, see the Handbook of Weighted Automata [27]. Recognizable word functions over commutative semirings S were characterized using logic through the formalism of Weighted Monadic Second Order Logic (WMSOL), [26], and the formalism of MSOLEVAL 1 , [44].…”

Energy and Mean-Payoff Games with Imperfect Information

“…This problem has tight connection (through polynomial-time reductions) with important theoretical questions about logics and games, such as the µ-calculus model-checking, and solving parity games [13,14,17,18]. Second, quantitative objectives in general are gaining interest in the specification and design of reactive systems [8,5,11], where the weights represent resource usage (e.g., energy consumption or network usage); the problem of controller synthesis with resource constraints requires the solution of quantitative games [20,6,1,3]. Finally, mean-payoff games are log-space equivalent to energy games (EG) where the objective of Player 1 is to maintain the sum of the weight (called the energy level) positive, given a fixed initial credit of energy.…”