Decision problems for patterns
Journal of Computer and System Sciences
Some combinatorial properties of Sturmian words
Theoretical Computer Science
Complexity of Makanin's algorithm
Journal of the ACM (JACM)
Handbook of formal languages, vol. 1
Automata on Infinite Words, Ecole de Printemps d'Informatique Théorique,
Efficient Solving of the Word Equations in One Variable
MFCS '94 Proceedings of the 19th International Symposium on Mathematical Foundations of Computer Science 1994
The Expressibility of Languages and Relations by Word Equations
ICALP '97 Proceedings of the 24th International Colloquium on Automata, Languages and Programming
Word Equations with Two Variables
IWWERT '91 Proceedings of the Second International Workshop on Word Equations and Related Topics
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We consider languages expressed by word equations in two variables and give a complete characterization for their complexity functions, that is, the functions that give the number of words of a given length. Specifically, we prove that there are only five types of complexities: constant, linear, exponential, and two in between constant and linear. For the latter two, we give precise characterizations in terms of the number of solutions of Diophantine equations of certain types. There are several consequences of our study. First, we show that the linear upper bound on the non-exponential complexities by Karhumäki et al., cf. [KMP], is optimal. Second, we derive that both of the sets of all finite Sturmian words and of all finite Standard words are expressible by word equations. Third, we characterize the languages of non-exponential complexity which are expressible by two-variable word equations as finite unions of several simple parametric formulae and solutions of a twovariable word equation with a finite graph. Fourth, we find optimal upper bounds on the solutions of (solvable) two-variable word equations, namely, linear bound for one variable and quadratric for the other. From this, we obtain an O(n6) algorithm for testing the solvability of twovariable word equation.