Abstract
Let I be a finite set of integers and F be a finite set of maps of the form n↦k i n + ℓ i with integer coefficients. For an integer base k ≥ 2, we study the k-recognizability of the minimal set X of integers containing I and satisfying ϕ(X) ⊆ X for all ϕ ∈ F. In particular, solving a conjecture of Allouche, Shallit and Skordev, we show under some technical conditions that if two of the constants k i are multiplicatively independent, then X is not k-recognizable for any k ≥ 2.
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Kärki, T., Lacroix, A., Rigo, M. (2009). On the Recognizability of Self-generating Sets. In: Královič, R., Niwiński, D. (eds) Mathematical Foundations of Computer Science 2009. MFCS 2009. Lecture Notes in Computer Science, vol 5734. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-03816-7_45
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DOI: https://doi.org/10.1007/978-3-642-03816-7_45
Publisher Name: Springer, Berlin, Heidelberg
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