Context-free Word Problem Research Articles

On the word problem for weakly compressible monoids

We study the language-theoretic properties of the word problem, in the sense of Duncan & Gilman, of weakly compressible monoids, as defined by Adian & Oganesian. We show that if C is a reversal-closed super-, as defined by Greibach, then M has word problem in C if and only if its compressed left monoid L(M) has word problem in C. As a special case, we may take C to be the class of context-free or indexed languages. As a corollary, we find many new classes of monoids with decidable rational subset membership problem. Finally, we show that it is decidable whether a one-relation monoid containing a non-trivial idempotent has context-free word problem. This answers a generalization of a question first asked by Zhang in 1992.

Non-finitely generated maximal subgroups of context-free monoids

A finitely generated group or monoid is said to be context-free if it has context-free word problem. In this note, we give an example of a context-free monoid, none of whose maximal subgroups are finitely generated. This answers a question of Brough, Cain & Pfeiffer on whether the group of units of a context-free monoid is always finitely generated, and highlights some of the contrasts between context-free monoids and context-free groups. Finally, we ask whether the group of units of a context-free monoid is always coherent.

On the word problem for special monoids

A monoid is called special if it admits a presentation in which all defining relations are of the form w = 1. Every group is special, but not every monoid is special. In this article, we describe the language-theoretic properties of the word problem, in the sense of Duncan and Gilman, for special monoids in terms of their group of units. We prove that a special monoid has context-free word problem if and only if its group of units is virtually free, giving a full generalisation of the Muller-Schupp theorem. This fully answers, for the class of special monoids, a question posed by Duncan and Gilman (Math Proc Camb Philos Soc 136:513–524, 2004). We describe the congruence classes of words in a special monoid, and prove that these have the same language-theoretic properties as the word problem. This answers a question first posed by Zhang (Math Proc Camb Philos Soc 112:495–505, 1992). As a corollary, we prove that it is decidable (in polynomial time) whether a special one-relation monoid has context-free word problem. This completely answers another question from 1992, also posed by Zhang.

Closure properties in the class of multiple context-free groups

AbstractWe show that the class of groups withk-multiple context-free word problem is closed under graphs of groups with finite edge groups.

A dynamical definition of f.g. virtually free groups

We show that the class of finitely generated virtually free groups is precisely the class of finitely generated demonstrable subgroups for Thompson’s group [Formula: see text]. The class of demonstrable groups for [Formula: see text] consists of all groups which can embed into [Formula: see text] with a natural dynamical behavior in their induced actions on the Cantor space [Formula: see text]. There are also connections with formal language theory, as the class of groups with context-free word problem is also the class of finitely generated virtually free groups, while Thompson’s group [Formula: see text] is a candidate as a universal [Formula: see text] group by Lehnert’s conjecture, corresponding to the class of groups with context free co-word problem (as introduced by Holt, Rees, Röver, and Thomas). Our main results answers a question of Berns-Zieve, Fry, Gillings, Hoganson, and Matthews, and separately of Bleak and Salazar-Díaz, and it fits into the larger exploration of the class of [Formula: see text] groups as it shows that all four of the known closure properties of the class of [Formula: see text] groups hold for the set of finitely generated subgroups of [Formula: see text].

A NOTE ON CONTEXT-SENSITIVE LANGUAGES AND WORD PROBLEMS

Anisimov and Seifert show that a group has a regular word problem ifand only if it is finite. Muller and Schupp (together with Dunwoody's accessibility result) show that a group has context free word problem if and only if it is virtually free. In this note, we exhibit a class of groups where the word problem is as close as possible to being a context sensitive language. This class includes the automatic groups and is closed under passing to finitely generated subgroups. Consequently, it is quite large, including many groups which are not finitely presented.

Context-free languages and random walks on groups

The Green function of an arbitrary, finitely supported random walk on a discrete group with context-free word problem is algebraic. It is shown how this theorem can be deduced from basic results of formal language theory. Context-free groups are precisely the finite extensions of free groups.