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Word equations with two variables
[chapter]

1993
*
Lecture Notes in Computer Science
*

This is an

doi:10.1007/3-540-56730-5_30
fatcat:bpkcl4xxpjhfhol7d4r76677oy
*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

1997
*
Mathematics Education Research Journal
*

way, using only one

doi:10.1007/bf03217307
fatcat:fejp33gjwbfrxacge3xoaibdzy
*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

2018
*
International Colloquium on Automata, Languages and Programming
*

We solve

doi:10.4230/lipics.icalp.2018.136
dblp:conf/icalp/NowotkaS18
fatcat:k6v6qweeyve5pavutocwi5hmcq
*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. ...##
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Word Equations with Length Constraints: What's Decidable?
[chapter]

2013
*
Lecture Notes in Computer Science
*

A corollary of this undecidability result is that this set is undecidable even

doi:10.1007/978-3-642-39611-3_21
fatcat:olludyqgdzhmhexsuzgs4n3qqq
*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). ...##
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(Un)Decidability Results for Word Equations with Length and Regular Expression Constraints
[article]

2013
*
arXiv
*
pre-print

A corollary of this undecidability result is that this set is undecidable even

arXiv:1306.6054v1
fatcat:53atrdlhi5hr5di7ialhzxcvha
*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). ...##
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On Word Equations in One Variable
[chapter]

2002
*
Lecture Notes in Computer Science
*

Additionally it is proven here that the set of solutions of any one-

doi:10.1007/3-540-45687-2_17
fatcat:bistlnrvtnanrpr6rvgs3d37aq
*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*[9]. ...##
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Upper Bounds on the Length of Minimal Solutions to Certain Quadratic Word Equations

2019
*
International Symposium on Mathematical Foundations of Computer Science
*

It is a long standing conjecture that the problem of deciding whether a quadratic

doi:10.4230/lipics.mfcs.2019.44
dblp:conf/mfcs/DayMN19
fatcat:dvqezcm4j5h23h46gbvrq2pc34
*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 ...##
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Two-Variable Word Equations
[chapter]

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. ...

##
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Varieties of idempotent slim groupoids

2007
*
Czechoslovak Mathematical Journal
*

So, every minimal

doi:10.1007/s10587-007-0124-y
fatcat:az4luj7255bmhhzeyuqzsfecam
*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 ...##
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Two-variable word equations

2000
*
RAIRO - Theoretical Informatics and Applications
*

and solutions of a

doi:10.1051/ita:2000126
fatcat:3ylgnin4zbawve5todtmzfrfli
*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?

2012
*
2012 14th International Symposium on Symbolic and Numeric Algorithms for Scientific Computing
*

Although some languages make distinctions among ideas such

doi:10.1109/synasc.2012.79
dblp:conf/synasc/MarcusW12
fatcat:udly5q5n6zh5therioczrls6bu
*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 ...##
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A Decision Procedure for String Logic with Equations, Regular Membership and Length Constraints
[article]

2016
*
arXiv
*
pre-print

In particular, we propose a semi-decision procedure for arbitrary string formulae

arXiv:1610.01331v2
fatcat:fjzftqgztfanhmubp5rgj7g6km
*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 ...##
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An optimal bound on the solution sets of one-variable word equations and its consequences
[article]

2018
*
arXiv
*
pre-print

We solve

arXiv:1805.09535v1
fatcat:ey2urcngr5cwjntnschmo66h2a
*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 ...##
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On the Length of the Minimum Solution of Word Equations in One Variable
[chapter]

2003
*
Lecture Notes in Computer Science
*

We show the tight upperbound of the length of the minimum solution of a

doi:10.1007/978-3-540-45138-9_13
fatcat:vxuxuem3b5dvjgexduornc4b7a
*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*. ...##
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Effective Search-Space Pruning for Solvers of String Equations, Regular Expressions and Length Constraints
[chapter]

2015
*
Lecture Notes in Computer Science
*

We present

doi:10.1007/978-3-319-21690-4_14
fatcat:czrmj6lvk5hkhapow2ojz7v2fy
*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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