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

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.