Word Equations Research Articles

Languages Generated by Conjunctive Query Fragments of FC[REG

Abstract$${\textsf{FC}}$$ FC is a finite model variant on the theory of concatenation, $${\textsf{FC}[\textsf{REG}]}$$ FC [ REG ] extends $${\textsf{FC}}$$ FC with regular constraints. This paper considers the languages generated by their conjunctive query fragments, and . We compare the expressive power of $${\textsf {FC[REG]-CQ}}$$ FC [ REG ] - CQ to that of various related language generators, such as regular expressions, patterns, and typed patterns. We then consider decision problems for $${\textsf {FC-CQ}}$$ FC - CQ and $${\textsf {FC[REG]-CQ}}$$ FC [ REG ] - CQ , and show that certain static analysis problems (such as equivalence and regularity) are undecidable. While this paper defines $${\textsf {FC-CQ}}$$ FC - CQ based on the logic $${\textsf{FC}}$$ FC , it can equally be understood as synchronized intersections of pattern languages, or as systems of restricted word equations.

On the Solution Sets of Three-Variable Word Equations

AbstractIt is known that the set of solutions of any constant-free three-variable word equation can be represented using parametric words, and the number of numerical parameters and the level of nesting in these parametric words is at most logarithmic with respect to the length of the equation. We show that this result can be significantly improved in the case of unbalanced equations, that is, equations where at least one variable has a different number of occurrences on the left-hand side and on the right-hand side. More specifically, it is sufficient to have two numerical parameters and one level of nesting in this case. We also discuss the possibility of proving a similar result for balanced equations in the future.

A Closer Look at the Expressive Power of Logics Based on Word Equations

Word equations are equations α≐β\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\alpha \\doteq \\beta $$\\end{document} where α\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\alpha $$\\end{document} and β\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\beta $$\\end{document} are words consisting of letters from some alphabet Σ\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\Sigma $$\\end{document} and variables from a set X. Recently, there has been substantial interest in the context of string solving in logics combining word equations with other kinds of constraints on words such as (regular) language membership (regular constraints) and arithmetic over string lengths (length constraints). We consider the expressive power of such logics by looking at the set of all values a single variable might take as part of a satisfying assignment for a given formula. Hence, each formula-variable pair defines a formal language, and each logic defines a class of formal languages. We consider logics arising from combining word equations with either length constraints, regular constraints, or both. We also consider word equations with visibly pushdown language membership constraints as a generalisation of the combination of regular and length constraints. We show that word equations with visibly pushdown membership constraints are sufficient to express all recursively enumerable languages and hence satisfiability is undecidable in this case. We then establish a strict hierarchy involving the other combinations. We also provide a complete characterisation of when a thin regular language is expressible by word equations (alone) and some further partial results for regular languages in the general case.

Group Equations With Abelian Predicates

Abstract 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.

On the Expressive Power of String Constraints

We investigate properties of strings which are expressible by canonical types of string constraints. Specifically, we consider a landscape of 20 logical theories, whose syntax is built around combinations of four common elements of string constraints: language membership (e.g. for regular languages), concatenation, equality between string terms, and equality between string-lengths. For a variable x and formula f from a given theory, we consider the set of values for which x may be substituted as part of a satisfying assignment, or in other words, the property f expresses through x. Since we consider string-based logics, this set is a formal language. We firstly consider the relative expressive power of different combinations of string constraints by comparing the classes of languages expressible in the corresponding theories, and are able to establish a mostly complete picture in this regard. Secondly, we consider the question of deciding whether the language or property expressed by a variable/formula in one theory can be expressed in another theory. We establish several negative results which are relevant to preprocessing and normalisation of string constraints in practice. Some of our results have strong connections to important open problems regarding word equations and the theory of string solving.

Об уравнениях в свободных моноидах и полугруппах с ограничениями на решения

We study algorithmic problems for equations in free monoids and semigroups (equations in words and lengths) with additional restrictions on the solutions. It is proved that it is impossible to construct an algorithm that solves an arbitrary system of equations in words and lengths in a free monoid (free semigroup) of rank 2 with an additional constraint on the solution in the form that one of its components belongs to the language of balanced words or the equality of the projections of two components of the solution into a distinguished free generator to determine whether it has a solution. A similar result is obtained for systems of inequalities in words.

Metric geometric means with arbitrary weights of positive definite matrices involving semi-tensor products

<abstract><p>We extend the notion of classical metric geometric mean (MGM) for positive definite matrices of the same dimension to those of arbitrary dimensions, so that usual matrix products are replaced by semi-tensor products. When the weights are arbitrary real numbers, the weighted MGMs possess not only nice properties as in the classical case, but also affine change of parameters, exponential law, and cancellability. Moreover, when the weights belong to the unit interval, the weighted MGM has remarkable properties, namely, monotonicity and continuity from above. Then we apply a continuity argument to extend the weighted MGM to positive semidefinite matrices, here the weights belong to the unit interval. It turns out that this matrix mean posses rich algebraic, order, and analytic properties, such as, monotonicity, continuity from above, congruent invariance, permutation invariance, affine change of parameters, and exponential law. Furthermore, we investigate certain equations concerning weighted MGMs of positive definite matrices. It turns out that such equations are always uniquely solvable with explicit solutions. The notion of MGMs can be applied to solve certain symmetric word equations in two letters.</p></abstract>

Design and Implementation of a Final Project Plagiarism Detection System Using Cosine Similarity Method

Plagiarism is an act of taking ideas, taking research results, acquiring research results, and summarizing writing without mentioning the source, either intentionally or unintentionally. The cosine similarity method can be used to calculate the score of similarities between documents by comparing existing documents on the database with uploaded ones to gain a similarity percentage. External plagiarism detection (EPD) is used to compare the contents of articles. The design and implementation of the system will be carried out in several stages in the hope that the system can work optimally and detect text similarities accurately. This research objective is to check the plagiarism rate of the document using the cosine similarity method as a method of calculating word equations called Kipcheck. The purpose of checking the level of plagiarism is to ensure that the documents created have a minimum level of fraud to avoid academic sanctions. Kipcheck uses three application system tests; maximum word calculation that can be processed; comparing application and manual calculation; and testing the consistency of application calculation results based on two different schemes.

An Optimal Bound on the Solution Sets of One-Variable Word Equations and its Consequences

Related DatabasesWeb of Science You must be logged in with an active subscription to view this.Article DataHistorySubmitted: 2 January 2020Accepted: 17 September 2021Published online: 10 January 2022Keywordscombinatorics on words, word equations, systems of equationsAMS Subject Headings68R15Publication DataISSN (print): 0097-5397ISSN (online): 1095-7111Publisher: Society for Industrial and Applied MathematicsCODEN: smjcat

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.

On the structure of solution-sets to regular word equations

For 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.

Program Specialization as a Tool for Solving Word Equations

The paper focuses on the automatic generating of the witnesses for the word equation satisfiability problem by means of specializing an interpreter which tests whether a composition of variable substitutions of a given word equation system produces its solution. We specialize such an interpreter w.r.t. the equation system, while the substitutions are unknown. We show that several variants of such interpreters, when specialized using the basic unfold/fold specialization methods, are able to construct the whole solution sets for some classes of the word equations whose left- and right-hand sides share variables. We prove that the specialization process wrt the constructed interpreters gives a simple syntactic criterion of the satisfiability of the equations considered, and show that the suggested approach can solve some equations not solvable by Z3str3 and CVC4, the widely-used SMT-solvers.

Word equations in non-deterministic linear space

Satisfiability of word equations problem is: Given two sequences consisting of letters and variables decide whether there is a substitution for the variables that turns this equation into true equality. The exact computational complexity of this problem remains unknown, with the best lower and upper bounds being, respectively, NP and PSPACE. Recently, the novel technique of recompression was applied to this problem, simplifying the known proofs and lowering the space complexity to (non-deterministic) O(nlog⁡n). In this paper we show that satisfiability of word equations is in non-deterministic linear space, thus the language of satisfiable word equations is context-sensitive. We use the known recompression-based algorithm and additionally employ Huffman coding for letters. The proof, however, uses analysis of how the fragments of the equation depend on each other as well as a new strategy for non-deterministic choices of the algorithm.

On equations and first-order theory of one-relator monoids

We investigate systems of equations and the first-order theory of one-relator monoids. We describe a family F of one-relator monoids of the form 〈A|w=1〉 where for each monoid M in F, the longstanding open problem of decidability of word equations with length constraints reduces to the Diophantine problem (i.e. decidability of systems of equations) in M. We achieve this result by finding an interpretation in M of a free monoid, using only systems of equations together with length relations. It follows that each monoid in F has undecidable positive AE-theory, hence in particular it has undecidable first-order theory. The family F includes many one-relator monoids with torsion 〈A|wn=1〉 (n>1). In contrast, all one-relator groups with torsion are hyperbolic, and all hyperbolic groups are known to have decidable Diophantine problem. We further describe a different class of one-relator monoids with decidable Diophantine problem.

Standard words and solutions of the word equation [formula omitted

We consider solutions of the word equation X12⋯Xn2=(X1⋯Xn)2 such that the squares Xi2 are minimal squares found in optimal squareful infinite words. We apply a method developed by the second author for studying word equations and prove that there are exactly two families of solutions: reversed standard words and words obtained from reversed standard words by a simple substitution scheme. A particular and remarkable consequence is that a word w is a standard word if and only if its reversal is a solution to the word equation and gcd⁡(|w|,|w|1)=1. This result can be interpreted as a yet another characterization for standard Sturmian words.We apply our results to the symbolic square root map ⋅ studied by the first author and M. A. Whiteland. We prove that if the language of a minimal subshift Ω contains infinitely many solutions to the word equation, then either Ω is Sturmian and ⋅-invariant or Ω is a so-called SL-subshift and not ⋅-invariant. This result is progress towards proving the conjecture that a minimal and ⋅-invariant subshift is necessarily Sturmian.

Equations enforcing repetitions under permutations

The notion of repetition of factors in words is central to combinatorics on words. A recent generalisation of this concept considers repetitions under permutations: given an alphabet Σ and a morphism or antimorphism f on Σ∗, whose restriction to Σ is a permutation, w is an [f]-repetition if there exists γ∈Σ∗, an integer k≥2, and the positive integers i1,…,ik such that w=fi1(γ)fi2(γ)⋯fik(γ). In this paper, we extend a series of classical repetition enforcing word equations to this general setting to obtain a series of word equations whose solutions are [f]-repetitions.

Solutions to twisted word equations and equations in virtually free groups

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.

Pseudo-solutions of word equations

We present a framework which allows a uniform approach to the recently introduced concept of pseudo-repetitions on words in the morphic case. This framework is at the same time more general and simpler. We introduce the concept of a pseudo-solution and a pseudo-rank of an equation. In particular, this allows to prove that if a classical equation forces periodicity then it also forces pseudo-periodicity. Consequently, there is no need to investigate generalizations of important equations one by one.

Deciding Context Unification

In first-order term unification, variables represent well-formed terms over a given signature, and we are to solve equations built using function symbols from the signature and such variables; this problem is well-known to be decidable (in linear time). In second-order term unification, the variables take arguments (i.e., other terms) and a substitution uses those arguments an arbitrary number of times; for instance, an equation f ( X ( c ), X ( c )) = X ( f ( c , c )) has a solution X = •, where • is a special symbol denoting the place in which the argument is substituted. Under this substitution, both sides evaluate to f ( c , c ). There are other solutions, for instance X = f (•,•), which evaluates both sides to f ( f ( c , c ), f ( c , c )); in general, a solution that evaluates both sides to full binary tree of arbitrary height is easy to construct. Second-order unification is in general undecidable. Context unification is a natural problem in between first- and second-order unification—we deal with equations over terms, the variables take arguments, but we restrict the set of solutions: The argument is used exactly once. Formally, contexts are terms with exactly one occurrence of the special symbol • and in context unification, we are given an equation over terms with variables representing contexts and ask about the satisfiability of this equation. For instance, when the aforementioned equation f ( X ( c ), X ( c )) = X ( f ( c , c )) is treated as a context unification problem, then it has exactly one solution: X = •. Other substitutions that are solutions of it as an instance of the second-unification problem, say X = f (•, •), are not valid, as • is used more than once. Context unification also generalizes satisfiability of word equations, which is decidable (in PSPACE). The decidability status of context unification remained unknown for almost two decades. In this article, we show that context unification is in PSPACE (in EXPTIME , when tree regular constraints are also allowed). Those results are obtained by extending the recently developed recompression technique, which was previously defined for strings and used to obtain a new PSPACE algorithm for satisfiability of word equations. In this article, the technique is generalized to trees, and the corresponding algorithm is generalized from word equations to context unification. The idea of recompression is to apply simple compression rules (replacing pairs of neighboring function symbols) to the solution of the context equation; to this end, we appropriately modify the equation (without the knowledge of the actual solution) so compressing the solution can be simulated by compressing parts of the equation. It is shown that if the compression operations are appropriately chosen, then the size of the instance is polynomial during the whole algorithm, thus giving a PSPACE-upper bound.

Some Sufficient conditions for a group to be abelian

A group is said to satisfy a word $w$ in the symbols $\{x, x^{-1}, y, y^{-1} \}$ provided that if the 'x' and 'y' are replaced by arbitrary elements of the group then the equation $w=1$ is satisfied. This paper studies certain equations in words, as above, which together with other conditions imply that groups which satisfy these equations and conditions must be abelian.