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 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.

scholarly journals Solution sets for equations over free groups are EDT0L languages

2016 ◽  
Vol 26 (05) ◽  
pp. 843-886 ◽  
Author(s):  
Laura Ciobanu ◽  
Volker Diekert ◽  
Murray Elder
Keyword(s):  
Directed Graph ◽  
Free Group ◽  
Diophantine Equations ◽  
Free Groups ◽  
Explicit Construction ◽  
Finitely Generated ◽  
Solution Sets ◽  
Linear Diophantine Equations ◽  
All Solutions ◽  
Reduced Words

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 Jeż, 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 [Formula: see text] here.

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 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 A Combinatorial Proof of a Formula of Biane and Chapuy

10.37236/7061 ◽  
2018 ◽  
Vol 25 (1) ◽  
Author(s):  
Sinho Chewi ◽  
Venkat Anantharam
Keyword(s):  
Directed Graph ◽  
Spanning Tree ◽  
Spanning Trees ◽  
Outgoing Edge ◽  
Alternative Proof ◽  
Combinatorial Proof ◽  
Tree Graph ◽  
Weighted Directed Graph ◽  
Strongly Connected

Let $G$ be a simple strongly connected weighted directed graph. Let $\mathcal{G}$ denote the spanning tree graph of $G$. That is, the vertices of $\mathcal{G}$ consist of the directed rooted spanning trees on $G$, and the edges of $\mathcal{G}$ consist of pairs of trees $(t_i, t_j)$ such that $t_j$ can be obtained from $t_i$ by adding the edge from the root of $t_i$ to the root of $t_j$ and deleting the outgoing edge from $t_j$. A formula for the ratio of the sum of the weights of the directed rooted spanning trees on $\mathcal{G}$ to the sum of the weights of the directed rooted spanning trees on $G$ was recently given by Biane and Chapuy. Our main contribution is an alternative proof of this formula, which is both simple and combinatorial.

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.

Exogeneous factors are not necessary in attachment, spreading and polarization of hela cells

1978 ◽  
Vol 36 (2) ◽  
pp. 310-311
Author(s):  
W. Liebrich
Keyword(s):  
Hela Cells ◽  
Calf Serum ◽  
Glass Tube ◽  
Serum Free Medium ◽  
Nacl Solution ◽  
Free Medium ◽  
Minimum Essential Medium ◽  
Serum Free ◽  
All Solutions ◽  
Glass Support

HeLa cells were grown for 2-3 days in EAGLE'S minimum essential medium with 10% calf serum (S-MEM; Seromed, München) and then incubated for 24 hours in serum free medium (MEM). After detaching the cells with a solution of 0. 14 % EDTA and 0. 07 % trypsin (Difco, 1 : 250) they were suspended in various solutions (S-MEM = control, MEM, buffered salt solutions with or without Me++ions, 0. 9 % NaCl solution) and allowed to settle on glass tube slips (Leighton-tubes). After 5, 10, 15, 20, 25, 30, 1 45, 60 minutes 2, 3, 4, 5 hours cells were prepared for scanning electron microscopy as described by Paweletz and Schroeter. The preparations were examined in a Jeol SEM (JSM-U3) at 25 KV without tilting.The suspended spherical HeLa cells are able to adhere to the glass support in all solutions. The rate of attachment, however, is faster in solutions without serum than in the control. The latter is in agreement with the findings of other authors.

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