Entropy analysis of substitutive sequences revisited

A given finite sequence of letters over a finite alphabet can always be algorithmically generated, in particular by a Turing machine. This fact is at the heart of complexity theory in the sense of Kolmogorov and Chaitin. A relevant question in this context is whether, given a statistically ‘sufficiently long’ sequence, there exists a deterministic finite automaton that generates it. In this paper we propose a simple criterion, based on measuring block entropies by lumping, which is satisfied by all automatic sequences. On the basis of this, one can determine that a given sequence is not automatic and obtain interesting information when the sequence is automatic. Following previous work on the Feigenbaum sequence, we give a necessary entropy-based condition valid for all automatic sequences read by lumping. Applications of these ideas to representative examples are discussed. In particular, we establish new entropic decimation schemes for the Thue–Morse, the Rudin–Shapiro and the paperfolding sequences read by lumping.