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### Word equations with two variables [chapter]

Witold Charatonik, Leszek Pacholski
1993 Lecture Notes in Computer Science
This is an equation with two variables λ, µ, over alphabet ∪ {v} of coefficients.  ...  Let¯ denote the projection of the word onto the alphabet . Let 1 2 = 1 2 be an equation with two variables such that ν( 1 ) = ν( 1 ) and c = |¯ 1 | − |¯ 1 | ≥ 0.  ...

### The Effect of using two variables when there are two unknowns in solving algebraic word problems

Susann M. Mathews
1997 Mathematics Education Research Journal
way, using only one variable to translate from prose to an algebraic equation.  ...  This article reports an experiment in which Algebra I students learned to translate word problems with· two unknowns from the prose representation to symbolic representation using two variables (one to  ...  After solving word problems for most of the year, they are finally taught to solve two equations in two variables.  ...

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

Dirk Nowotka, Aleksi Saarela, Michael Wagner
2018 International Colloquium on Automata, Languages and Programming
We solve two long-standing open problems on word equations. Firstly, we prove that a onevariable word equation with constants has either at most three or an infinite number of solutions.  ...  Secondly, we consider independent systems of three-variable word equations without constants. If such a system has a nonperiodic solution, then this system of equations is at most of size 17.  ...  with a nonperiodic solution is somewhere between two and 17.  ...

### Word Equations with Length Constraints: What's Decidable? [chapter]

Vijay Ganesh, Mia Minnes, Armando Solar-Lezama, Martin Rinard
2013 Lecture Notes in Computer Science
A corollary of this undecidability result is that this set is undecidable even with sentences with at most two occurrences of a string variable.  ...  We prove several decidability and undecidability results for the satisfiability and validity problems for languages that can express solutions of word equations with length constraints.  ...  Example 1 Consider the L 0 e formula which is a word equation X = aYbZa with three variables (X, Y, Z) and two string constants (a, b).  ...

### (Un)Decidability Results for Word Equations with Length and Regular Expression Constraints [article]

Vijay Ganesh, Mia Minnes, Armando Solar-Lezama, Martin Rinard
2013 arXiv   pre-print
A corollary of this undecidability result is that this set is undecidable even with sentences with at most two occurrences of a string variable.  ...  We prove several decidability and undecidability results for the satisfiability and validity problems for languages that can express solutions to word equations with length constraints.  ...  Example 1 Consider the L 0 e formula which is a word equation X = aYbZa with three variables (X, Y, Z) and two string constants (a, b).  ...

### On Word Equations in One Variable [chapter]

Robert Dabrowski, Wojtek Plandowski
2002 Lecture Notes in Computer Science
Additionally it is proven here that the set of solutions of any one-variable word equation is either of the form F or of the form F ∪ (uv) + u where F is a set of O(log n) words and u, v are some words  ...  For a word equation E of length n in one variable x occurring # x times in E a resolution algorithm of O(n + # x log n) time complexity is presented here.  ...  There exists also an efficient O(n log 2 n) time algorithm for a certain class of special equations with two variables .  ...

### Upper Bounds on the Length of Minimal Solutions to Certain Quadratic Word Equations

Joel D. Day, Florin Manea, Dirk Nowotka, Michael Wagner
2019 International Symposium on Mathematical Foundations of Computer Science
It is a long standing conjecture that the problem of deciding whether a quadratic word equation has a solution is in NP.  ...  We show that both conjectures hold for some natural subclasses of quadratic equations, namely the classes of regular-reversed, k-ordered, and variable-sparse quadratic equations.  ...  For regular equations with two variables, the chain-words will be over an alphabet of size two, meaning we get an immediate answer: a quick exhaustive search reveals that any word over two letters of length  ...

### Two-Variable Word Equations [chapter]

Lucian Ilie, Wojciech Plandowski
2000 Lecture Notes in Computer Science
word equation with a nite graph.  ...  Third, we nd optimal upper bounds on the solutions of (solvable) two-variable word equations, namely, linear bound for one variable and quadratric for the other.  ...  and solutions of a two-variable word equation with a nite graph.  ...

### Varieties of idempotent slim groupoids

J. Ježek
2007 Czechoslovak Mathematical Journal
So, every minimal word contains at most two occurrences of each of the k variables.  ...  Then . . x m can be arbitrarily permuted; in particular, they can be permuted in such a way that x i gets to the position with index 2, so that the word starts with two equal variables and can be shortened  ...

### Two-variable word equations

Lucian Ilie, Wojciech Plandowski
2000 RAIRO - Theoretical Informatics and Applications
and solutions of a two-variable word équation with a finite graph.  ...  Our analysis deals with languages which are defined by two-variable word équations.  ...

### What is an Equation?

Solomon Marcus, Stephen M. Watt
2012 2012 14th International Symposium on Symbolic and Numeric Algorithms for Scientific Computing
Although some languages make distinctions among ideas such equations that are always satisfied versus those that may not be, or between equations with variables and relations without, we observe that it  ...  The equation concept is one of the most fundamental in mathematics, yet the word "equation" means different things to different people.  ...  For their help with language usage, we thank Sergey  ...

### A Decision Procedure for String Logic with Equations, Regular Membership and Length Constraints [article]

Quang Loc Le
2016 arXiv   pre-print
In particular, we propose a semi-decision procedure for arbitrary string formulae with word equations, regular membership and length functions.  ...  In this paper, we consider the satisfiability problem for string logic with equations, regular membership and Presburger constraints over length functions.  ...  Conclusion and Future Work We have presented the semi-decision procedures S2 1SEA for the problem of solving satisfiability of a SEA formula with word equations, regular membership and length functions  ...

### An optimal bound on the solution sets of one-variable word equations and its consequences [article]

Dirk Nowotka, Aleksi Saarela
2018 arXiv   pre-print
We solve two long-standing open problems on word equations. Firstly, we prove that a one-variable word equation with constants has either at most three or an infinite number of solutions.  ...  Secondly, we consider independent systems of three-variable word equations without constants. If such a system has a nonperiodic solution, then this system of equations is at most of size 17.  ...  We will use it with two codes Z: The set of all minimal zero-sum words (those zero-sum words which cannot be written as a product of two shorter zero-sum words), and the set of words of a specific length  ...

### On the Length of the Minimum Solution of Word Equations in One Variable [chapter]

Kensuke Baba, Satoshi Tsuruta, Ayumi Shinohara, Masayuki Takeda
2003 Lecture Notes in Computer Science
We show the tight upperbound of the length of the minimum solution of a word equation L = R in one variable, in terms of the differences between the positions of corresponding variable occurrences in L  ...  A word equation (in one variable) is a pair of words over Σ ∪ {x} and is usually written by connecting two words with "=".  ...  An approach to the problem is to analyze word equations with a restricted number of variables.  ...

### Effective Search-Space Pruning for Solvers of String Equations, Regular Expressions and Length Constraints [chapter]

Yunhui Zheng, Vijay Ganesh, Sanu Subramanian, Omer Tripp, Julian Dolby, Xiangyu Zhang
2015 Lecture Notes in Computer Science
We present two techniques, designed for word-based SMT string solvers, to mitigate these problems: (i) sound and complete detection of overlapping variables, which is essential to avoiding common cases  ...  Such solvers typically support a theory of string equations, the length function as well as the regular-expression membership predicate.  ...  Conclusion We have described two techniques that dramatically improve the efficiency of wordbased string solvers: (i) a sound and complete procedure to detect overlapping variables, thereby automatically  ...
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