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In this short survey we describe recent advances on word equations with non-rational constraints in groups and monoids, highlighting the important role that formal languages play in this area., Comment: 12 pages, to appear in the proceedings of DLT 2024. arXiv admin note: text overlap with arXiv:2204.13946

Indexed languages are a classical notion in formal language theory. As the language equivalent of second-order pushdown automata, they have received considerable attention in higher-order model checking. Unfortunately, counting properties are notoriously difficult to decide for indexed languages: So far, all results about non-regular counting properties show undecidability. In this paper, we initiate the study of slice closures of (Parikh images of) indexed languages. A slice is a set of vectors of natural numbers such that membership of $u,u+v,u+w$ implies membership of $u+v+w$. Our main result is that given an indexed language $L$, one can compute a semilinear representation of the smallest slice containing $L$'s Parikh image. We present two applications. First, one can compute the set of all affine relations satisfied by the Parikh image of an indexed language. In particular, this answers affirmatively a question by Kobayashi: Is it decidable whether in a given indexed language, every word has the same number of $a$'s as $b$'s. As a second application, we show decidability of (systems of) word equations with rational constraints and a class of counting constraints: These allow us to look for solutions where a counting function (defined by an automaton) is not zero. For example, one can decide whether a word equation with rational constraints has a solution where the number of occurrences of $a$ differs between variables $X$ and $Y$., Comment: 12 pages, accepted for publication at LICS 2024

In this paper we study the satisfiability and solutions of group equations when combinatorial, algebraic and language-theoretic constraints are imposed on the solutions. We show that the solutions to equations with length, lexicographic order, abelianisation or context-free constraints added, can be effectively produced in finitely generated virtually abelian groups. Crucially, we translate each of the constraints above into a rational set in an effective way, and so reduce each problem to solving equations with rational constraints, which is decidable and well understood in virtually abelian groups. A byproduct of our results is that the growth series of a virtually abelian group, with respect to any generating set and any weight, is effectively computable. This series is known to be rational by a result of Benson, but his proof is non-constructive., Comment: 28 pages

The survey provides an overview of the work done in the last 10 years to characterise solutions to equations in groups in terms of formal languages. We begin with the work of Ciobanu, Diekert and Elder, who showed that solutions to systems of equations in free groups in terms of reduced words are expressible as EDT0L languages. We provide a sketch of their algorithm, and describe how the free group results extend to hyperbolic groups. The characterisation of solutions as EDT0L languages is very robust, and many group constructions preserve this, as shown by Levine. The most recent progress in the area has been made for groups without negative curvature, such as virtually abelian, the integral Heisenberg group, or the soluble Baumslag-Solitar groups, where the approaches to describing the solutions are different from the negative curvature groups. In virtually abelian groups the solutions sets are in fact rational, and one can obtain them as $m$-regular sets. In the Heisenberg group producing the solutions to a single equation reduces to understanding the solutions to quadratic Diophantine equations and uses number theoretic techniques. In the Baumslag-Solitar groups the methods are combinatorial, and focus on the interplay of normal forms to solve particular classes of equations. In conclusion, EDT0L languages give an effective and simple combinatorial characterisation of sets of seemingly high complexity in many important classes of groups., Comment: 26 pages

In this paper we generalise and unify the results and methods used by Benson, Liardet, Evetts, and Evetts & Levine, to show that rational sets in a virtually abelian group G have rational (relative) growth series with respect to any generating set for G. We prove equivalences between the structures used in the literature, and establish the rationality of important classes of sets in G: definable sets, algebraic sets, conjugacy representatives and coset representatives (of any fixed subgroup), among others. Furthermore, we show that any rational set, when written as words over the generating set of G, has several EDT0L representations., Comment: 24 pages, 3 figures. To appear in L'Enseignement Mathematique

In this paper we begin the systematic study of group equations with abelian predicates in the main classes of groups where solving equations is possible. We extend the line of work on word equations with length constraints, and more generally, on extensions of the existential theory of semigroups, to the world of groups. We use interpretability by equations to establish model-theoretic and algebraic conditions which are sufficient to get undecidability. We apply our results to (non-abelian) right-angled Artin groups, and show that the problem of solving equations with abelian predicates is undecidable for these. We obtain the same result for hyperbolic groups whose abelianisation has torsion-free rank at least two. By contrast, we prove that in groups with finite abelianisation, the problem can be reduced to solving equations with recognisable constraints, and so this is decidable in right-angled Coxeter groups, or more generally, graph products of finite groups, as well as hyperbolic groups with finite abelianisation.

The Post Correspondence Problem is a classical decision problem about equalisers of free monoid homomorphisms. We prove connections between several variations of this classical problem, but in the setting of free groups and free group homomorphisms. Among other results, and working under certain injectivity assumptions, we prove that computing the rank of the equaliser of a pair of free group homomorphisms can be applied to computing a basis of this equaliser, and also to solve the "generalised" Post Correspondence Problem for free groups., Comment: 14 pages

A marked free monoid morphism is a morphism for which the image of each generator starts with a different letter, and immersions are the analogous maps in free groups. We show that the (simultaneous) PCP is decidable for immersions of free groups, and provide an algorithm to compute bases for the sets, called equalisers, on which the immersions take the same values. We also answer a question of Stallings about the rank of the equaliser. Analogous results are proven for marked morphisms of free monoids., Comment: 16 pages, final version incorporating referees comments

We show that the full set of solutions to systems of equations and inequations in a hyperbolic group, as shortlex geodesic words (or any regular set of quasigeodesic normal forms), is an EDT0L language whose specification can be computed in NSPACE$(n^2\log n)$ for the torsion-free case and NSPACE$(n^4\log n)$ in the torsion case. Furthermore, in the presence of quasi-isometrically embeddable rational constraints, we show that the full set of solutions to systems of equations in a hyperbolic group remains EDT0L. Our work combines the geometric results of Rips, Sela, Dahmani and Guirardel on the decidability of the existential theory of hyperbolic groups with the work of computer scientists including Plandowski, Je\.z, Diekert and others on PSPACE algorithms to solve equations in free monoids and groups using compression, and involves an intricate language-theoretic analysis., Comment: 33 pages, 3 figures, 1 table. Minor edits made. An extended abstract of a preliminary version of this paper was presented at the conference ICALP 2019 arXiv:1902.07349

We show that the full set of solutions to systems of equations and inequations in a hyperbolic group, with or without torsion, as shortlex geodesic words, is an EDT0L language whose specification can be computed in $\mathsf{NSPACE}(n^2\log n)$ for the torsion-free case and $\mathsf{NSPACE}(n^4\log n)$ in the torsion case. Our work combines deep geometric results by Rips, Sela, Dahmani and Guirardel on decidability of existential theories of hyperbolic groups, work of computer scientists including Plandowski, Je\.z, Diekert and others on $\mathsf{PSPACE}$ algorithms to solve equations in free monoids and groups using compression, and an intricate language-theoretic analysis. The present work gives an essentially optimal formal language description for all solutions in all hyperbolic groups, and an explicit and surprising low space complexity to compute them., Comment: 18 pages, 2 figures

L systems generalise context-free grammars by incorporating parallel rewriting, and generate languages such as EDT0L and ET0L that are strictly contained in the class of indexed languages. In this paper we show that many of the languages naturally appearing in group theory, and that were known to be indexed or context-sensitive, are in fact ET0L and in many cases EDT0L. For instance, the language of primitives in the free group on two generators, the Bridson-Gilman normal forms for the fundamental groups of 3-manifolds or orbifolds, and the co-word problem of Grigorchuk's group can be generated by L systems. To complement the result on primitives in free groups, we show that the language of primitives, and primitive sets, in free groups of rank higher than two is context-sensitive. We also show the existence of EDT0L and ET0L languages of intermediate growth., Comment: Revised following referees suggestions. 21 pages, 2 figures

For a language $L$, we consider its cyclic closure, and more generally the language $C^k(L)$, which consists of all words obtained by partitioning words from $L$ into $k$ factors and permuting them. We prove that the classes of ET0L and EDT0L languages are closed under the operators $C^k$. This both sharpens and generalises Brandst\"adt's result that if $L$ is context-free then $C^k(L)$ is context-sensitive and not context-free in general for $k\geq 3$. We also show that the cyclic closure of an indexed language is indexed., Comment: 11 pages, 1 figure. Improved proof of the main theorem from previous version arXiv:1412.5512

We show that, given an equation over a finitely generated free group, the set of all solutions in reduced words forms an effectively constructible EDT0L language. In particular, the set of all solutions in reduced words is an indexed language in the sense of Aho. The language characterization we give, as well as further questions about the existence or finiteness of solutions, follow from our explicit construction of a finite directed graph which encodes all the solutions. Our result incorporates the recently invented recompression technique of Je\.z, and a new way to integrate solutions of linear Diophantine equations into the process. As a byproduct of our techniques, we improve the complexity from quadratic nondeterministic space in previous works to $\mathsf{NSPACE}(n\log n)$ here., Comment: 38 pages, 3 figures. A conference version of this paper was presented at ICALP 2015, Kyoto (Japan), July 4-10, 2015, see http://arxiv.org/abs/1502.03426

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 Fert\'e, Marin and S\'enizergues (ToCS 2014), which is a very restricted case. Our general result became possible due to the recent recompression technique of Je\.z. 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., Comment: 37 pages, 2 figures

We consider the cyclic closure of a language, and its generalisation to the operators $C^k$ introduced by Brandst\"adt. We prove that the cyclic closure of an indexed language is indexed, and that if $L$ is a context-free language then $C^k(L)$ is indexed., Comment: 13 pages, 5 figures