Construction of Some Nonautomatic Sequences by Cellular Automata
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.
KeywordsCellular automata Automatic sequences Nonautomatic sequences Computability Polynomials Fibonacci word
The authors thank N. Fatès and E. Jeandel for fruitful discussions and the referees for valuable comments. The work has been supported by the ANR-FWF bilateral project MuDeRa “Multiplicativity: Determinism and Randomness” (France-Austria), ANR-14-CE34-0009.