Eigenvalues and Transduction of Morphic Sequences

Abstract

We study finite state transduction of automatic and morphic sequences. Dekking [4] proved that morphic sequences are closed under transduction and in particular morphic images. We present a simple proof of this fact, and use the construction in the proof to show that non-erasing transductions preserve a condition called α-substitutivity. Roughly, a sequence is α-substitutive if the sequence can be obtained as the limit of iterating a substitution with dominant eigenvalue α. Our results culminate in the following fact: for multiplicatively independent real numbers α and β, if v is a α-substitutive sequence and w is an β-substitutive sequence, then v and w have no common non-erasing transducts except for the ultimately periodic sequences. We rely on Cobham's theorem for substitutions, a recent result of Durand [5]. * This is an extended version of our paper [9] presented at Developments in Language Theory 2014. This extended version contains examples and additional remarks. Related workDurand [5] proved that if w is an α-substitutive sequence and h is a non-erasing morphism, then h(w) is α k -substitutive for some k ∈ N. We strengthen this result in two directions. First, we show that k may be taken to be 1; hence h(w) is α k -substitutive for every k ∈ N.Second, we show that Durand's result also holds for non-erasing transductions. PreliminariesWe recall some of the main concepts that we use in the paper. For a thorough introduction to morphic sequences, automatic sequences and finite state transducers, we refer to [1,8].We are concerned with infinite sequences Σ ω over a finite alphabet Σ. We write Σ * for the set of finite words, Σ + for the finite, non-empty words, Σ ω for the infinite words, and Σ ∞ = Σ * ∪ Σ ω for all finite or infinite words over Σ.

Asymptotic properties of free monoid morphisms

“…Roughly speaking, a transducer is a finite-state machine, i.e., a deterministic finite automaton where transitions are labeled with input letters and (possibly empty) output words, used to replace an infinite word by another one, where the nth output depends on the first n symbols of the original word [STEM14]. For instance, Dekking proved that morphic words are closed under transduction [Dek94].…”

Transducer degrees: atoms, infima and suprema

“…That is, at least one of the output words along the edges must be empty. A non-erasing transducer cannot do the transformation since it preserves α-substitutivity[37].…”