[Paper Review] Solution sets for equations over free groups are EDT0L languages -- ICALP 2015 version

Laura Ciobanu, Volker Diekert+1 more|arXiv (Cornell University)|Feb 11, 2015

semigroups and automata theory11 references3 citations

TL;DR

This paper proves that the solution sets of word equations over finitely generated free groups—when expressed in reduced words—are EDT0L languages, a highly structured class of formal languages. Using a novel integration of Jeürs' recompression technique with linear Diophantine equation solving, the authors construct an NFA that generates all solutions in quasi-linear nondeterministic space, yielding an improved complexity bound of NSPACE(n log n) for the existential theory of free groups.

ABSTRACT

We show that, given a word equation over a finitely generated free group, the set of all solutions in reduced words forms an EDT0L language. In particular, it is an indexed language in the sense of Aho. The question of whether a description of solution sets in reduced words as an indexed language is possible has been been open for some years, apparently without much hope that a positive answer could hold. Nevertheless, our answer goes far beyond: they are EDT0L, which is a proper subclass of indexed languages. We can additionally handle the existential theory of equations with rational constraints in free products $\star_{1 \leq i \leq s}F_i$, where each $F_i$ is either a free or finite group, or a free monoid with involution. In all cases the result is the same: the set of all solutions in reduced words is EDT0L. This was known only for quadratic word equations by Ferté, Marin and Sénizergues (ToCS 2014), which is a very restricted case. Our general result became possible due to the recent recompression technique of Jeż. In this paper we use a new method to integrate solutions of linear Diophantine equations into the process and obtain more general results than in the related paper (arXiv 1405.5133). For example, we improve the complexity from quadratic nondeterministic space in (arXiv 1405.5133) to quasi-linear nondeterministic space here. This implies an improved complexity for deciding the existential theory of non-abelian free groups: NSPACE($n\log n$). The conjectured complexity is NP, however, we believe that our results are optimal with respect to space complexity, independent of the conjectured NP.

Motivation & Objective

• To resolve the long-standing open question of whether solution sets of word equations over free groups can be described as indexed languages, with a focus on EDT0L structure.
• To extend the formal language characterization of solutions beyond quadratic equations to general word equations with rational constraints.
• To improve the space complexity of solving the existential theory of non-abelian free groups from quadratic to quasi-linear nondeterministic space.
• To develop a method that integrates linear Diophantine equation solving into the recompression framework for word equations.
• To provide an effective, algorithmically constructible NFA representation of solution sets in reduced words, enabling structural and complexity-theoretic analysis.

Proposed method

• Leverages Jeürs' recompression technique to compress repeated patterns in words, enabling efficient manipulation of solution sets.
• Introduces a novel compression strategy—non-standard block compression—targeting factors of the form $c\overline{c}(c\overline{c})^\ell$ to reduce redundancy while preserving solution structure.
• Uses substitution arcs to replace repeated patterns with new symbols, maintaining the solution relation through morphism extensions.
• Applies alphabet reduction to eliminate redundant letters, simplifying the word structure while preserving solvability.
• Employs a recursive compression process combining standard block compression, non-standard block compression, and pair compression to reduce word length while controlling blowup.
• Constructs a nondeterministic finite automaton (NFA) over endomorphisms in a free monoid with a special symbol \#, such that the language generated corresponds exactly to the solution set in reduced words.

Experimental results

Research questions

• RQ1 Can the solution set of any word equation over a free group be described as an EDT0L language?
• RQ2 Is it possible to construct an effective NFA that generates all solutions in reduced words using a formal language framework?
• RQ3 Can the recompression technique be extended to handle linear Diophantine constraints arising from word equation solutions?
• RQ4 What is the optimal space complexity for deciding the existential theory of non-abelian free groups?
• RQ5 Does the solution set of a word equation with rational constraints in free products of free and finite groups remain EDT0L?

Key findings

• The solution set of any word equation over a finitely generated free group is an EDT0L language when expressed in reduced words.
• The paper provides an effective algorithm to compute an NFA that generates the solution set in quasi-linear nondeterministic space, i.e., $\mathsf{NSPACE}(n\log n)$.
• The solution set is shown to be a proper subclass of indexed languages, confirming a stronger structural result than previously known.
• The method extends to free products $\star_{1\leq i\leq s}F_i$, where each $F_i$ is a free group, finite group, or free monoid with involution, and the solution set remains EDT0L.
• The complexity of deciding the existential theory of non-abelian free groups is improved to $\mathsf{NSPACE}(n\log n)$, which is optimal with respect to space complexity independent of the $\mathsf{NP}$ conjecture.
• The recursive compression process ensures that the word length remains linear in each iteration, leading to an overall $\mathcal{O}(n)$ length bound after compression, despite intermediate blowup.