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Studying Word Equations by a Method of Weighted Frequencies

Saarela A.

Fundamenta Informaticae

2018

Vol. 162, nr 2/3

223--235

We briefly survey some results and open problems on word equations, especially on those equations where the right-hand side is a power of a variable. We discuss a method that was recently used to prove one of the results, and we prove improved versions of some lemmas that are related to the method and can be used as tools when studying word equations. We use the method and the tools to give new, simple proofs for several old results.

Equivalence Relations Defined by Numbers of Occurrences of Factors

Saarela A.

Fundamenta Informaticae

2016

Vol. 145, nr 3

385--397

We study the question of what can be said about a word based on the numbers of occurrences of certain factors in it. We do this by defining a family of equivalence relations that generalize the so called k-abelian equivalence. The characterizations and answers we obtain are linear algebraic. We also use these equivalence relations to help us in solving some problems related to repetitions and palindromes, and to point out that some previous results about Sturmian words and k-abelian equivalence hold in a more general form.

The Unique Decipherability in the Monoid of Regular Languages is Undecidable

Karhumäki J., Saarela A.

Fundamenta Informaticae

2011

Vol. 110, nr 1-4

197-200

We show by a simple reduction that the unique decipherability problem in the language monoid of regular languages over a non-unary alphabet is undecidable.