Equations and Coequations for Weighted Automata

Abstract

Abstract. We study weighted automata from both an algebraic and a coalgebraic perspective. In particular, we consider equations and coequations for weighted automata. We prove a duality result that relates sets of equations (congruences) with (certain) subsets of coequations. As a consequence, we obtain two equivalent but complementary ways to define classes of weighted automata. We show that this duality cannot be generalized to linear congruences in general but we obtain partial results when weights are from a field.

Duality of Equations and Coequations via Contravariant Adjunctions

“…is finite}, where supp(φ) = {w ∈ A * | φ(w) = 0} is the support of φ, and the functionτ is defined asτ (f )(a) = f a (see [25]). …”

“…Given an L-algebra (X, [i, α]), we have that G(X, [i, α]) = (X ∂ , c, β ) where the linear maps c : [25]). …”

“…This gives us a duality result between equations and coequations. We derive known dualities between equations and coequations for automata [7,24,25] as a special case of this abstract approach, and we generalize the duality shown in [7] to include (general) dynamical systems.…”