On Periodically Iterated Morphisms

TL;DR

This paper explores the computational complexity and properties of PD0L systems, demonstrating their ability to generate complex and all-encompassing computable words, and establishing key undecidability results.

Contribution

It introduces a method to embed any computable word into a PD0L word using Turing-complete encoding, and proves several undecidability results related to PD0L systems.

Findings

Existence of PD0L words with exponential subword complexity

Undecidability of certain properties of PD0L words

PD0L systems can generate all computable words

Abstract

We investigate the computational power of periodically iterated morphisms, also known as D0L systems with periodic control, PD0L systems for short. These systems give rise to a class of one-sided infinite sequences, called PD0L words. We construct a PD0L word with exponential subword complexity, thereby answering a question raised by Lepisto (1993) on the existence of such words. We solve another open problem concerning the decidability of the first-order theories of PD0L words; we show it is already undecidable whether a certain letter occurs in a PD0L word. This stands in sharp contrast to the situation for D0L words (purely morphic words), which are known to have at most quadratic subword complexity, and for which the monadic theory is decidable. The main result of our paper, leading to these answers, is that every computable word w over an alphabet Sigma can be embedded in a PD0L word u over an extended alphabet Gamma in the following two ways: (i) such that every finite prefix of w is a subword of u, and (ii) such that w is obtained from u by erasing all letters from Gamma not in Sigma. The PD0L system generating such a word u is constructed by encoding a Fractran program that computes the word w; Fractran is a programming language as powerful as Turing Machines. As a consequence of (ii), if we allow the application of finite state transducers to PD0L words, we obtain the set of all computable words. Thus the set of PD0L words is not closed under finite state transduction, whereas the set of D0L words is. It moreover follows that equality of PD0L words (given by their PD0L system) is undecidable. Finally, we show that if erasing morphisms are admitted, then the question of productivity becomes undecidable, that is, the question whether a given PD0L system defines an infinite word.