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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 the same 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. In particular,...
For a non-negative integer , we say that a language is
if the number of words of length in
is of order . We give a precise characterization of the
-poly-slender context-free languages. The well-known characterization
of the -poly-slender regular languages is an immediate consequence
of ours.
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