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

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.

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Solutions to twisted word equations and equations in virtually free groups

International Journal of Algebra and Computation ◽

10.1142/s0218196720500198 ◽

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.

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WORD EQUATIONS WITH ONE UNKNOWN

International Journal of Foundations of Computer Science ◽

10.1142/s0129054111008088 ◽

2011 ◽

Vol 22 (02) ◽

pp. 345-375 ◽

Cited By ~ 9

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.

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Word equations in a uniquely divisible group

Discrete Mathematics & Theoretical Computer Science ◽

10.46298/dmtcs.2807 ◽

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.

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On the structure of solution-sets to regular word equations

Theory of Computing Systems ◽

10.1007/s00224-021-10058-5 ◽

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.

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On systems of word equations with simple loop sets

Theoretical Computer Science ◽

10.1016/j.tcs.2007.03.026 ◽

2007 ◽

Vol 380 (3) ◽

pp. 363-372 ◽

Cited By ~ 3

Author(s):

Štěpán Holub ◽

Juha Kortelainen

Keyword(s):

Simple Loop ◽

Word Equations

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Learning via queries in [+, <]

Journal of Symbolic Logic ◽

10.2307/2275176 ◽

1992 ◽

Vol 57 (1) ◽

pp. 53-81 ◽

Cited By ~ 12

Author(s):

William I. Gasarch ◽

Mark G. Pleszkoch ◽

Robert Solovay

Keyword(s):

Active Learning ◽

Query Language ◽

Independent Interest ◽

Decidability Result ◽

Presburger Arithmetic ◽

First Order ◽

Recursive Functions ◽

Bounded Number ◽

Open Question ◽

Primitive Recursive

AbstractWe prove that the set of all recursive functions cannot be inferred using first-order queries in the query language containing extra symbols [+ , <]. The proof of this theorem involves a new decidability result about Presburger arithmetic which is of independent interest. Using our machinery, we show that the set of all primitive recursive functions cannot be inferred with a bounded number of mind changes, again using queries in [+, <]. Additionally, we resolve an open question in [7] about passive versus active learning.

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WORD EQUATIONS OVER GRAPH PRODUCTS

International Journal of Algebra and Computation ◽

10.1142/s0218196708004548 ◽

2008 ◽

Vol 18 (03) ◽

pp. 493-533 ◽

Cited By ~ 10

Author(s):

VOLKER DIEKERT ◽

MARKUS LOHREY

Keyword(s):

Graph Products ◽

Graph Product ◽

Positive Theory ◽

Cancellation Condition ◽

Existential Theory ◽

Word Equations ◽

Product Of Groups

For monoids that satisfy a weak cancellation condition, it is shown that the decidability of the existential theory of word equations is preserved under graph products. Furthermore, it is shown that the positive theory of a graph product of groups can be reduced to the positive theories of those factors, which commute with all other factors, and the existential theories of the remaining factors. Both results also include suitable constraints for the variables. Larger classes of constraints lead in many cases to undecidability results.

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