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Preprints, Working Papers, ... Year : 2023

The categorical contours of the Chomsky-Schützenberger representation theorem

Abstract

We develop fibrational perspectives on context-free grammars and on finite state automata over categories and operads. A generalized CFG is a functor from a free colored operad (= multicategory) generated by a pointed finite species into an arbitrary base operad: this encompasses classical CFGs by taking the base to be a certain operad constructed from a free monoid, as an instance of a more general construction of an operad of spliced arrows W C for any category C. A generalized NDFA is a functor satisfying the unique lifting of factorizations and finite fiber properties, from an arbitrary bipointed category or pointed operad: this encompasses classical word automata and tree automata without ϵ-transitions, but also automata over non-free categories and operads. We show that generalized context-free and regular languages satisfy suitable generalizations of many of the usual closure properties, and in particular we give a simple conceptual proof that contextfree languages are closed under intersection with regular languages. Finally, we observe that the splicing functor W : Cat → Oper admits a left adjoint C : Oper → Cat, which we call the contour category construction since the arrows of C O have a geometric interpretation as oriented contours of operations of O. A direct consequence of the contour / splicing adjunction is that every pointed finite species induces a universal CFG generating a language of tree contour words. This leads us to a generalization of the Chomsky-Schützenberger Representation Theorem, establishing that a subset of a homset L ⊆ C(A, B) is a CFL of arrows iff it is a functorial image of the intersection of a C-chromatic tree contour language with a regular language.
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Dates and versions

hal-04399404 , version 1 (17-01-2024)

Identifiers

  • HAL Id : hal-04399404 , version 1

Cite

Paul-André Melliès, Noam Zeilberger. The categorical contours of the Chomsky-Schützenberger representation theorem. 2023. ⟨hal-04399404⟩
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