The Chomsky-Sch\"utzenberger Theorem for Quantitative Context-Free Languages

TL;DR

This paper extends the Chomsky-Schützenberger theorem to a broad class of weighted context-free languages, encompassing various algebraic structures and demonstrating their equivalence to weighted pushdown automata.

Contribution

It generalizes the fundamental theorem for quantitative context-free languages beyond semirings, including averages and lattices, and establishes their equivalence to weighted pushdown automata.

Findings

Derived the Chomsky-Schützenberger theorem for general weight structures.

Proved the expressive equivalence to weighted pushdown automata.

Investigated conditions for languages to have finitely many values.

Abstract

Weighted automata model quantitative aspects of systems like the consumption of resources during executions. Traditionally, the weights are assumed to form the algebraic structure of a semiring, but recently also other weight computations like average have been considered. Here, we investigate quantitative context-free languages over very general weight structures incorporating all semirings, average computations, lattices, and more. In our main result, we derive the fundamental Chomsky-Sch\"utzenberger theorem for such quantitative context-free languages, showing that each arises as the image of a Dyck language and a regular language under a suitable morphism. Moreover, we show that quantitative context-free language are expressively equivalent to a model of weighted pushdown automata. This generalizes results previously known only for semirings. We also investigate when quantitative context-free languages assume only finitely many values.