Dedicated to Jean Berstel for his 60th birthday with our very best wishes.
AbstractWe study the palindrome complexity of infinite sequences on finite alphabets, i.e., the number of palindromic factors (blocks) of given length occurring in a given sequence. We survey the known results and obtain new results for some sequences, in particular for Rote sequences and for fixed points of primitive morphisms of constant length belonging to "class P" of Hof-Knill-Simon. We also give an upper bound for the palindrome complexity of a sequence in terms of its (block-)complexity.
We slightly improve the result of Klarner, Birget and Satterfield, showing that the freeness of finitely presented multiplicative semigroups of 3×3 matrices over ℕ is undecidable even for triangular matrices. This is achieved by proving a new variant of Post correspondence problem. We also consider the freeness problem for 2×2 matrices. On the one hand, we show that it cannot be proved undecidable using the above methods which work in higher dimensions, and, on the other hand, we give some evidence that its decidability, if so, is also a challenging problem.
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