5 research outputs found
On Periodically Iterated Morphisms
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
Automatic sequences in rational base numeration systems (and even more)
The nth term of an automatic sequence is the output of a deterministic finite automaton fed with the representation of n in a suitable numeration system. Here, instead of considering automatic sequences built on a numeration system with a regular numeration language, we consider these built on languages associated with trees having periodic labeled signatures and, in particular, rational base numeration systems. We obtain two main characterizations of these sequences. The first one is concerned with r-block substitutions where r morphisms are applied periodically. In particular, we provide examples of such sequences that are not morphic. The second characterization involves the factors, or subtrees of finite height, of the tree associated with the numeration system and decorated by the terms of the sequence
Automatic sequences: from rational bases to trees
The th term of an automatic sequence is the output of a deterministic
finite automaton fed with the representation of in a suitable numeration
system. In this paper, instead of considering automatic sequences built on a
numeration system with a regular numeration language, we consider these built
on languages associated with trees having periodic labeled signatures and, in
particular, rational base numeration systems. We obtain two main
characterizations of these sequences. The first one is concerned with -block
substitutions where morphisms are applied periodically. In particular, we
provide examples of such sequences that are not morphic. The second
characterization involves the factors, or subtrees of finite height, of the
tree associated with the numeration system and decorated by the terms of the
sequence.Comment: 25 pages, 15 figure
Degrees of Infinite Words, Polynomials and Atoms
We study finite-state transducers and their power for transforming infinite words. Infinite sequences of symbols are of paramount importance in a wide range of fields, from formal languages to pure mathematics and physics. While finite automata for recognising and transforming languages are well-understood, very little is known about the power of automata to transform infinite words.The word transformation realised by finite-state transducers gives rise to a complexity comparison of words and thereby induces equivalence classes, called (transducer) degrees, and a partial order on these degrees. The ensuing hierarchy of degrees is analogous to the recursion-theoretic degrees of unsolvability, also known as Turing degrees, where the transformational devices are Turing machines. However, as a complexity measure, Turing machines are too strong: they trivialise the classification problem by identifying all computable words. Finite-state transducers give rise to a much more fine-grained, discriminating hierarchy. In contrast to Turing degrees, hardly anything is known about transducer degrees, in spite of their naturality.We use methods from linear algebra and analysis to show that there are infinitely many atoms in the transducer degrees, that is, minimal non-trivial degrees