[Paper Review] Deciding Word Problems of Semigroups using Finite State Automata

Max Neunhöffer, Markus Pfeiffer+1 more|arXiv (Cornell University)|Jun 8, 2012

semigroups and automata theory18 references4 citations

TL;DR

This paper introduces rwp-semigroups—semigroups with rational word problems decidable via finite state automata—enabling constant-memory, quadratic-time algorithms for the word problem. The key contribution is proving that rwp-semigroups are closed under finitely generated subsemigroups, direct products, and extensions by finite ideals, and that finiteness, group, monoid, and freeness properties are decidable within this class.

ABSTRACT

We explore a natural class of semigroups that have word problem decidable by finite state automata. Among the main results are invariance of this property under change of generators, invariance under basic algebraic constructions and algebraic properties of these semigroups.

Motivation & Objective

• To define and characterize a class of semigroups—rwp-semigroups—where the word problem is decidable using finite state automata.
• To establish closure properties of rwp-semigroups under common semigroup constructions such as direct products, subsemigroups, and adding zeros or identities.
• To prove decidability of structural properties like finiteness, grouphood, monoid structure, and freeness within rwp-semigroups.
• To explore connections between rwp-semigroups and word-hyperbolic semigroups, showing that all rwp-semigroups are word-hyperbolic.
• To investigate open problems, particularly the isomorphism problem for rwp-semigroups, and to compare rwp-semigroups with rational monoids.

Proposed method

• Define the word problem of a semigroup as a rational relation over the free monoid generated by its alphabet.
• Use finite state automata to recognize the rational relations that define the word problem, enabling constant-memory, quadratic-time decision procedures.
• Apply closure properties of rational relations and recognizable languages to prove closure of rwp-semigroups under subsemigroups, direct products, and extensions by finite ideals.
• Leverage the fact that rational relations are closed under intersection and preimage under morphisms to construct context-free languages that encode the word problem.
• Use the construction of a context-free language $ M = \{u\#_1 v\#_2 w^{\mathrm{rev}} \mid (u\pi)(v\pi) = w\pi\} $ to show word-hyperbolicity of rwp-semigroups.
• Reduce decision problems (freeness, grouphood, etc.) to known decidable problems in word-hyperbolic semigroups via effective construction of word-hyperbolic structures from rational word problems.

Experimental results

Research questions

• RQ1 Can the word problem of a semigroup be decided efficiently when it is defined by a rational relation over a finite generating set?
• RQ2 Which semigroup constructions preserve the rational word problem property, and are rwp-semigroups closed under these operations?
• RQ3 Is it decidable whether an rwp-semigroup is finite, a group, a monoid, or free?
• RQ4 Do rwp-semigroups coincide with rational monoids as defined by Sakarovitch?
• RQ5 Is the isomorphism problem undecidable for rwp-semigroups, and if so, can it be reduced to known undecidability results?

Key findings

• The word problem for rwp-semigroups is decidable in constant memory and quadratic time using finite state automata.
• rwp-semigroups are closed under taking finitely generated subsemigroups, direct products, adding zeros or identities, and extension by finite ideals.
• Every rwp-semigroup is word-hyperbolic, as the associated language $ M $ is context-free and constructible from the rational word problem.
• It is decidable whether an rwp-semigroup is finite, a group, a monoid, or free, with grouphood decidable via finiteness testing and exhaustive verification.
• The isomorphism problem for rwp-semigroups remains open, but is a strong candidate for undecidability.
• The Green’s relations of rwp-semigroups are intersections of rational relations, suggesting a deeper structural connection to rationality in semigroup theory.