scholarly journals Solutions to twisted word equations and equations in virtually free groups

2020 ◽  
Vol 30 (04) ◽  
pp. 731-819
Author(s):  
Volker Diekert ◽  
Murray Elder
Keyword(s):  
Free Groups ◽  
Expressive Power ◽  
Solution Set ◽  
Complexity Bound ◽  
Free Monoids ◽  
Word Equation ◽  
All Solutions ◽  
Standard Normal ◽  
Word Equations ◽  
Solving Equations

It is well known that the problem solving equations in virtually free groups can be reduced to the problem of solving twisted word equations with regular constraints over free monoids with involution. In this paper, we prove that the set of all solutions of a twisted word equation is an EDT0L language whose specification can be computed in PSPACE . Within the same complexity bound we can decide whether the solution set is empty, finite, or infinite. In the second part of the paper we apply the results for twisted equations to obtain in PSPACE an EDT0L description of the solution set of equations with rational constraints for finitely generated virtually free groups in standard normal forms with respect to a natural set of generators. If the rational constraints are given by a homomorphism into a fixed (or “small enough”) finite monoid, then our algorithms can be implemented in [Formula: see text], that is, in quasi-quadratic nondeterministic space. Our results generalize the work by Lohrey and Sénizergues (ICALP 2006) and Dahmani and Guirardel (J. of Topology 2010) with respect to both complexity and expressive power. Neither paper gave any concrete complexity bound and the results in these papers are stated for subsets of solutions only, whereas our results concern all solutions.

WORD EQUATIONS WITH ONE UNKNOWN

2011 ◽  
Vol 22 (02) ◽  
pp. 345-375 ◽  
Author(s):  
MARKKU LAINE ◽  
WOJCIECH PLANDOWSKI
Keyword(s):  
Infinite Number ◽  
Linear Time ◽  
Time Algorithm ◽  
Solution Set ◽  
Infinitely Many Solutions ◽  
Linear Time Algorithm ◽  
Number Of Solutions ◽  
Word Equation ◽  
Word Equations

We consider properties of the solution set of a word equation with one unknown. We prove that the solution set of a word equation possessing infinite number of solutions is of the form (pq)*p where pq is primitive. Next, we prove that a word equation with at most four occurrences of the unknown possesses either infinitely many solutions or at most two solutions. We show that there are equations with at most four occurrences of the unknown possessing exactly two solutions. Finally, we prove that a word equation with at most 2k occurrences of the unknown possesses either infinitely many solutions or at most 8 log k + O(1) solutions. Hence, if we consider a class εk of equations with at most 2k occurrences of the unknown, then each equation in this class possesses either infinitely many solutions or O( log k) number of solutions. Our considerations allow to construct the first alphabet independent linear time algorithm for computing the solution set of an equation in a nontrivial class of equations.

scholarly journals On the structure of solution-sets to regular word equations

2021 ◽  
Author(s):  
Joel D. Day ◽  
Florin Manea
Keyword(s):  
Directed Graph ◽  
Regular Word ◽  
Combinatorial Proof ◽  
Solution Sets ◽  
Word Equation ◽  
All Solutions ◽  
Word Equations ◽  
Rewriting Rules

AbstractFor quadratic word equations, there exists an algorithm based on rewriting rules which generates a directed graph describing all solutions to the equation. For regular word equations – those for which each variable occurs at most once on each side of the equation – we investigate the properties of this graph, such as bounds on its diameter, size, and DAG-width, as well as providing some insights into symmetries in its structure. As a consequence, we obtain a combinatorial proof that the problem of deciding whether a regular word equation has a solution is in NP.

scholarly journals Relations between counting functions on free groups and free monoids

2018 ◽  
Vol 12 (4) ◽  
pp. 1485-1521
Author(s):  
Tobias Hartnick ◽  
Alexey Talambutsa
Keyword(s):  
Free Groups ◽  
Free Monoids ◽  
Counting Functions

scholarly journals Word equations in a uniquely divisible group

2010 ◽  
Vol DMTCS Proceedings vol. AN,... (Proceedings) ◽  
Author(s):  
Christopher J. Hillar ◽  
Lionel Levine ◽  
Darren Rhea
Keyword(s):  
Positive Integer ◽  
Complete Classification ◽  
Irreducible Factor ◽  
Nous Obtenons ◽  
Nous Pensons ◽  
Word Equation ◽  
Nous Montrons ◽  
Finite Word ◽  
Word Equations ◽  
International Audience

International audience We study equations in groups $G$ with unique $m$-th roots for each positive integer $m$. A word equation in two letters is an expression of the form$ w(X,A) = B$, where $w$ is a finite word in the alphabet ${X,A}$. We think of $A,B ∈G$ as fixed coefficients, and $X ∈G$ as the unknown. Certain word equations, such as $XAXAX=B$, have solutions in terms of radicals: $X = A^-1/2(A^1/2BA^1/2)^1/3A^-1/2$, while others such as $X^2 A X = B$ do not. We obtain the first known infinite families of word equations not solvable by radicals, and conjecture a complete classification. To a word w we associate a polynomial $P_w ∈ℤ[x,y]$ in two commuting variables, which factors whenever $w$ is a composition of smaller words. We prove that if $P_w(x^2,y^2)$ has an absolutely irreducible factor in $ℤ[x,y]$, then the equation $w(X,A)=B$ is not solvable in terms of radicals. Nous étudions des équations dans les groupes $G$ avec les $m$-th racines uniques pour chaque nombre entier positif m. Une équation de mot dans deux lettres est une expression de la forme $w(X, A) = B$, où $w$ est un mot fini dans l'alphabet ${X, A}$. Nous pensons $A, B ∈G$ en tant que coefficients fixes, et $X ∈G$ en tant que inconnu. Certaines équations de mot, telles que $XAXAX=B$, ont des solutions en termes de radicaux: $X = A^-1/2(A^1/2BA^1/2)^1/3A^-1/2$, alors que d'autres tel que $X^2 A X = B$ ne font pas. Nous obtenons les familles infinies d'abord connues des équations de mot non solubles par des radicaux, et conjecturons une classification complété. Á un mot $w$ nous associons un polynôme $P_w ∈ℤ[x, y]$ dans deux variables de permutation, qui factorise toutes les fois que $w$ est une composition de plus petits mots. Nous montrons que si $P_w(x^2, y^2)$ a un facteur absolument irréductible dans $ℤ[x, y]$, alors l'équation $w(X, A)=B$ n'est pas soluble en termes de radicaux.

scholarly journals On finitely generated submonoids of virtually free groups

2018 ◽  
Vol 10 (2) ◽  
pp. 63-82
Author(s):  
Pedro V. Silva ◽  
Alexander Zakharov
Keyword(s):  
Word Problem ◽  
Free Group ◽  
Free Groups ◽  
Isomorphism Problem ◽  
Geometric Characterization ◽  
Finitely Generated ◽  
Free Monoids ◽  
Virtually Free Group

AbstractWe prove that it is decidable whether or not a finitely generated submonoid of a virtually free group is graded, introduce a new geometric characterization of graded submonoids in virtually free groups as quasi-geodesic submonoids, and show that their word problem is rational (as a relation). We also solve the isomorphism problem for this class of monoids, generalizing earlier results for submonoids of free monoids. We also prove that the classes of graded monoids, regular monoids and Kleene monoids coincide for submonoids of free groups.

scholarly journals Quadratic Word Equations with Length Constraints, Counter Systems, and Presburger Arithmetic with Divisibility

2021 ◽  
Vol Volume 17, Issue 4 ◽  
Author(s):  
Anthony W. Lin ◽  
Rupak Majumdar
Keyword(s):  
Theoretical Foundation ◽  
Constraint Solving ◽  
Simple Loop ◽  
Decidability Result ◽  
Presburger Arithmetic ◽  
Existential Theory ◽  
Word Equation ◽  
Word Equations ◽  
Left And Right ◽  
Np Complete

Word equations are a crucial element in the theoretical foundation of constraint solving over strings. A word equation relates two words over string variables and constants. Its solution amounts to a function mapping variables to constant strings that equate the left and right hand sides of the equation. While the problem of solving word equations is decidable, the decidability of the problem of solving a word equation with a length constraint (i.e., a constraint relating the lengths of words in the word equation) has remained a long-standing open problem. We focus on the subclass of quadratic word equations, i.e., in which each variable occurs at most twice. We first show that the length abstractions of solutions to quadratic word equations are in general not Presburger-definable. We then describe a class of counter systems with Presburger transition relations which capture the length abstraction of a quadratic word equation with regular constraints. We provide an encoding of the effect of a simple loop of the counter systems in the existential theory of Presburger Arithmetic with divisibility (PAD). Since PAD is decidable (NP-hard and is in NEXP), we obtain a decision procedure for quadratic words equations with length constraints for which the associated counter system is flat (i.e., all nodes belong to at most one cycle). In particular, we show a decidability result (in fact, also an NP algorithm with a PAD oracle) for a recently proposed NP-complete fragment of word equations called regular-oriented word equations, when augmented with length constraints. We extend this decidability result (in fact, with a complexity upper bound of PSPACE with a PAD oracle) in the presence of regular constraints.

scholarly journals SLP compression for solutions of equations with constraints in free and hyperbolic groups

2015 ◽  
Vol 25 (01n02) ◽  
pp. 81-111
Author(s):  
Volker Diekert ◽  
Olga Kharlampovich ◽  
Atefeh Mohajeri Moghaddam
Keyword(s):  
Free Groups ◽  
Hyperbolic Group ◽  
Hyperbolic Groups ◽  
Boolean Formula ◽  
System Of Equations ◽  
Exponential Bound ◽  
Word Equation ◽  
Relatively Hyperbolic Group ◽  
Solutions Of Equations ◽  
Relatively Hyperbolic

The paper is a part of an ongoing program which aims to show that the problem of satisfiability of a system of equations in a free group (hyperbolic or even toral relatively hyperbolic group) is NP-complete. For that, we study compression of solutions with straight-line programs (SLPs) as suggested originally by Plandowski and Rytter in the context of a single word equation. We review some basic results on SLPs and give full proofs in order to keep this fundamental part of the program self-contained. Next we study systems of equations with constraints in free groups and more generally in free products of abelian groups. We show how to compress minimal solutions with extended Parikh-constraints. This type of constraints allows to express semi-linear conditions as e.g. alphabetic information. The result relies on some combinatorial analysis and has not been shown elsewhere. We show similar compression results for Boolean formula of equations over a torsion-free δ-hyperbolic group. The situation is much more delicate than in free groups. As byproduct we improve the estimation of the "capacity" constant used by Rips and Sela in their paper "Canonical representatives and equations in hyperbolic groups" from a double-exponential bound in δ to some single-exponential bound. The final section shows compression results for toral relatively hyperbolic groups using the work of Dahmani: We show that given a system of equations over a fixed toral relatively hyperbolic group, for every solution of length N there is an SLP for another solution such that the size of the SLP is bounded by some polynomial p(s + log N) where s is the size of the system.

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