[Paper Review] Some Combinatorial Operators in Language Theory

Jean-Gabriel Luque, Ludovic Mignot+1 more|arXiv (Cornell University)|May 15, 2012

semigroups and automata theory9 references3 citations

TL;DR

This paper introduces an operadic framework to study multitilde operators in formal language theory, providing an algebraic foundation for their composition and action on languages. By modeling multitildes as elements of a free operad, the authors establish connections to boolean vectors, partial orders, and combinatorial structures, enabling enumeration of inequivalent operators and a new representation of finite and regular languages via operator actions.

ABSTRACT

Multitildes are regular operators that were introduced by Caron et al. in order to increase the number of Glushkov automata. In this paper, we study the family of the multitilde operators from an algebraic point of view using the notion of operad. This leads to a combinatorial description of already known results as well as new results on compositions, actions and enumerations.

Motivation & Objective

• To provide an algebraic, operadic characterization of multitilde operators used in regular expression theory.
• To formalize the composition and action of multitilde operators on languages using operad theory.
• To establish a combinatorial enumeration of inequivalent multitilde operators via posets and relations.
• To extend the framework to include star operations and represent regular languages as actions of operators on letter tuples.
• To lay the groundwork for a new combinatorial approach to formal language theory using operads.

Proposed method

• Model the multitilde operators as elements of a free operad, using partial composition operations to encode their algebraic structure.
• Introduce an operad of boolean vectors to describe the action of multitildes on languages via characteristic functions.
• Construct a quotient operad from the multitilde operad to model equivalence of operator actions.
• Define a poset-based operad (POSet) to formalize the equivalence of multitilde actions and ensure uniqueness of representation.
• Use the structure of free lists and prefix closures to analyze the action of multitildes on k-tuples of languages.
• Extend the operad to include a star operator, defining its action as Kleene star and quotienting by the identity $\star \circ \star \equiv \star$.

Experimental results

Research questions

• RQ1 How can the composition of multitilde operators be formalized algebraically using operad theory?
• RQ2 What is the action of multitilde operators on languages, and how can it be encoded via boolean vectors?
• RQ3 How can two multitilde operators be considered equivalent, and what combinatorial structure captures this equivalence?
• RQ4 Can the number of inequivalent multitilde operators be enumerated, and what is the underlying combinatorics?
• RQ5 How can regular languages be represented as actions of extended multitilde operators on letter tuples?

Key findings

• The multitilde operators form a free operad, allowing a systematic study of their composition and algebraic properties.
• The action of multitildes on languages is isomorphic to an action of a boolean vector operad, providing a linear algebraic interpretation.
• The poset-based operad POSet≤ provides a complete and optimal representation of multitilde actions, distinguishing all inequivalent operators.
• Two multitildes have the same action on all k-tuples of languages if and only if they are equivalent under the POSet≤ structure.
• The number of inequivalent multitilde operators on k languages is equal to the number of distinct posets in POSet≤, enabling full enumeration.
• Every regular language can be expressed as the action of an element in the quotient operad $T^\star$ on a k-tuple of letters, with the star operator satisfying $\star \circ \star \equiv \star$.