[Paper Review] A Chomsky-Sch\\"utzenberger representation for weighted multiple context-free languages

Tobias Denkinger|arXiv (Cornell University)|Jun 13, 2016

semigroups and automata theory15 references4 citations

TL;DR

This paper establishes a Chomsky-Schützenberger (CS) representation theorem for multiple context-free languages (MCFLs) weighted over complete commutative strong bimonoids. It introduces an algebraic characterization of multiple Dyck languages via congruence relations and proves that any weighted MCFL can be expressed as the image of the intersection of a regular language and a congruence-based multiple Dyck language under an alphabetic weighted homomorphism, generalizing prior results beyond distributive semirings.

ABSTRACT

We prove a Chomsky-Sch\\"utzenberger representation theorem for multiple context-free languages weighted over complete commutative strong bimonoids.

Motivation & Objective

• To generalize the Chomsky-Schützenberger representation to weighted multiple context-free languages (MCFLs) over complete commutative strong bimonoids, extending beyond distributive semirings.
• To provide an algebraic, congruence-based definition of multiple Dyck languages that supports efficient membership checking via a decision algorithm.
• To separate the weight algebra from the underlying grammar structure, enabling modular construction of the weighted CS representation.
• To establish a formal connection between weighted MCFLs and the intersection of regular languages with congruence multiple Dyck languages through weighted homomorphisms.
• To lay the foundation for parsing algorithms for weighted MCFLs by leveraging the derived representation theorem.

Proposed method

• Defining multiple Dyck languages using congruence relations on term algebras, enabling a structural and decidable characterization of membership.
• Introducing a decision algorithm for membership in congruence multiple Dyck languages based on the underlying congruence relations.
• Separating the weight component from the grammar by decomposing weighted MCFLs into an unweighted MCFL and a weighted homomorphism.
• Constructing a weighted CS representation by composing a regular language, a congruence multiple Dyck language, and an alphabetic weighted homomorphism.
• Using the modular approach from Droste and Vogler (2013) to prove the weighted CS representation without requiring distributivity.
• Proving that the weighted homomorphism composition preserves the structure of the language via a bijection between derivation trees and strings in the intersection of the regular and multiple Dyck language.

Experimental results

Research questions

• RQ1 Can a Chomsky-Schützenberger representation be established for weighted multiple context-free languages over complete commutative strong bimonoids without requiring distributivity?
• RQ2 How can multiple Dyck languages be defined algebraically using congruence relations to support effective membership checking?
• RQ3 What is the precise relationship between weighted MCFLs and the intersection of regular languages with congruence multiple Dyck languages under weighted homomorphisms?
• RQ4 How can the weight algebra be modularly separated from the grammar structure in weighted MCFLs to enable generalization beyond semirings?
• RQ5 Can the derived representation theorem be used to construct a parsing algorithm for weighted MCFLs?

Key findings

• The paper proves that any weighted MCFL over a complete commutative strong bimonoid is representable as the image of the intersection of a regular language and a congruence multiple Dyck language under an alphabetic weighted homomorphism.
• A decision algorithm for membership in congruence multiple Dyck languages is provided, based on the underlying congruence relations, enabling efficient recognition.
• The representation theorem holds without requiring distributivity, generalizing prior results from complete commutative semirings to complete commutative strong bimonoids.
• A bijection is established between the derivation trees of a weighted MCFG and the strings in the intersection of the regular language and the multiple Dyck language of its Boolean version.
• The construction enables a modular proof strategy by first handling the unweighted case and then lifting it to the weighted setting via homomorphism composition.
• The results support the design of parsing algorithms for weighted MCFLs in the style of Hulden (2011), leveraging the derived representation.