Construction of Some Nonautomatic Sequences by Cellular Automata

Irène Marcovici (1, 2) , Thomas Stoll (1) , Pierre-Adrien Tahay (1)

1 IECL - Institut Élie Cartan de Lorraine
2 MOCQUA - Designing the Future of Computational Models

It is known that if p is a prime number, the columns of linear CA are p-automatic sequences and all p-automatic sequences can be realized by some linear CA with memory. We give some constructions of (nonlinear) CA that realize certain nonautomatic sequences. First, we show through a recoding that from a construction with additional symbols, we can construct a CA using only the symbols occurring in the sequence. This answers a question posed by Rowland and Yassawi. Then, we propose a construction for the characteristic sequence of the integer polynomials, which are nonautomatic sequences by the Minsky–Papert criterion. We also provide a construction based on the indicator of Fibonacci numbers for the Fibonacci word, which is an emblematic nonautomatic sequence.

Mots clés: Automatic sequences; Nonautomatic sequences; Computability; Polynomials; Fibonacci word; Cellular automata

Domaines: Informatique [cs]

Irène Marcovici, Thomas Stoll, Pierre-Adrien Tahay. Construction of Some Nonautomatic Sequences by Cellular Automata. AUTOMATA 2018 - 24th International Workshop on Cellular Automata and Discrete Complex Systems, Jun 2018, Ghent, Belgium. pp.113-126, ⟨10.1007/978-3-319-92675-9_9⟩. ⟨hal-01824876⟩