The class of deterministic context-free languages (DCFLs) that can be represented as the word problem of a group is exactly the set of word problems of virtually free groups. This class is a proper subset of all DCFLs, as there exist DCFLs that do not qualify as group word problems.

Key Points:

1. Word Problem and Virtually Free Groups: By Mueller and Schupp's theorem, a group ( G )...

Short answer Among deterministic context–free languages (DCFLs), those that occur as the word problem of a finitely generated group are exactly the word problems of virtually free groups. This is a very small subclass of DCFLs, constrained by strong algebraic invariances (conjugation, inversion, free reduction). There is no known purely “language–theoretic” intrinsic characterization beyond these ...

Let Σ be a finite symmetric alphabet (together with the involution x ↔ x –1 ) and let

 F(Σ)=〈Σ〉

be the free group on Σ.
If G is a finitely generated group with generating set Σ, its word problem is

 WPΣ(G)= { w∈Σ* | w̄ = 1 in G } ⊆ Σ* (1)

(the bar denotes the image of w in the group).

0. What does a “language of the form (1)” look like ?

WPΣ(G) is the kernel of ...

This is a fascinating question at the intersection of formal language theory and group theory. Let me outline what we know about the class of deterministic context-free languages (DCFLs) that arise as word problems of groups.

Known Results

The Muller-Schupp Theorem

As you mentioned, Muller and Schupp proved that a finitely generated group has context-free word problem if and only if $...

Let

 X = {x1 , … , xn } ∪ {x1-1 , … , xn-1 }

be a finite alphabet equipped with the natural involution x ↦ x-1, and let F(X) be the free group on X.
For every finitely generated group G together with the chosen generating set X we have  WP(G , X) = { w ∈ X* | w =F(X) 1 } ⊆ X*.

The question is: inside the class DCFL of deterministic context–free languages, which languages occur as some...

Excellent question. This gets to the heart of the intersection between group theory and formal language theory. The class of deterministic context-free languages that are word problems of groups is indeed a very special, and proper, subclass of all DCFLs.

Let's denote the class of languages that are word problems for some finitely generated group as . Let be the cla...

Here is what is known, and in fact essentially no finer “language‐theoretic” description of the subgroup of DCFLs coming from groups has been found beyond the original Muller–Schupp / Chomsky–Schützenberger style characterization.

1. Müller–Schupp.
   A finitely generated group (G) has a (deterministic) context–free word‐problem if and only if it is virtually free. In that case (\W...

Deterministic Context-Free Languages as Word Problems of Groups

To understand which deterministic context-free languages (DCFLs) can be represented as word problems of groups, I'll analyze what we know from the information provided and relevant theoretical results.

What We Know

1. A group's word problem is context-free if and only if the group is virtually free (Mueller and Sch...