33 research outputs found
Syndeticity and independent substitutions
We associate in a canonical way a substitution to any abstract numeration
system built on a regular language. In relationship with the growth order of
the letters, we define the notion of two independent substitutions. Our main
result is the following. If a sequence is generated by two independent
substitutions, at least one being of exponential growth, then the factors of
appearing infinitely often in appear with bounded gaps. As an
application, we derive an analogue of Cobham's theorem for two independent
substitutions (or abstract numeration systems) one with polynomial growth, the
other being exponential
Self-similar tiling systems, topological factors and stretching factors
In this paper we prove that if two self-similar tiling systems, with
respective stretching factors and , have a common factor
which is a non periodic tiling system, then and are
multiplicatively dependent
Numeration Systems: a Link between Number Theory and Formal Language Theory
We survey facts mostly emerging from the seminal results of Alan Cobham
obtained in the late sixties and early seventies. We do not attempt to be
exhaustive but try instead to give some personal interpretations and some
research directions. We discuss the notion of numeration systems, recognizable
sets of integers and automatic sequences. We briefly sketch some results about
transcendence related to the representation of real numbers. We conclude with
some applications to combinatorial game theory and verification of
infinite-state systems and present a list of open problems.Comment: 21 pages, 3 figures, invited talk DLT'201
Cobham-Semenov theorem and \NN^d-subshifts
We give a new proof of the Cobham's first theorem using ideas from symbolic
dynamics and of the Cobham-Semenov theorem (in the primitive case) using ideas
from tiling dynamics.Comment: 24 page
On the Sets of Real Numbers Recognized by Finite Automata in Multiple Bases
This article studies the expressive power of finite automata recognizing sets
of real numbers encoded in positional notation. We consider Muller automata as
well as the restricted class of weak deterministic automata, used as symbolic
set representations in actual applications. In previous work, it has been
established that the sets of numbers that are recognizable by weak
deterministic automata in two bases that do not share the same set of prime
factors are exactly those that are definable in the first order additive theory
of real and integer numbers. This result extends Cobham's theorem, which
characterizes the sets of integer numbers that are recognizable by finite
automata in multiple bases.
In this article, we first generalize this result to multiplicatively
independent bases, which brings it closer to the original statement of Cobham's
theorem. Then, we study the sets of reals recognizable by Muller automata in
two bases. We show with a counterexample that, in this setting, Cobham's
theorem does not generalize to multiplicatively independent bases. Finally, we
prove that the sets of reals that are recognizable by Muller automata in two
bases that do not share the same set of prime factors are exactly those
definable in the first order additive theory of real and integer numbers. These
sets are thus also recognizable by weak deterministic automata. This result
leads to a precise characterization of the sets of real numbers that are
recognizable in multiple bases, and provides a theoretical justification to the
use of weak automata as symbolic representations of sets.Comment: 17 page
A problem around Mahler functions
Let be a field of characteristic zero and and be two
multiplicatively independent positive integers. We prove the following result
that was conjectured by Loxton and van der Poorten during the Eighties: a power
series satisfies both a - and a -Mahler type functional
equation if and only if it is a rational function.Comment: 52 page
Deciding Properties of Automatic Sequences
In this thesis, we show that several natural questions about automatic sequences can be expressed as logical predicates and then decided mechanically. We extend known results in this area to broader classes of sequences (e.g., paperfolding words), introduce new operations that extend the space of possible queries, and show how to process the results.
We begin with the fundamental concepts and problems related to automatic sequences, and the corresponding numeration systems. Building on that foundation, we discuss the general logical framework that formalizes the questions we can mechanically answer. We start with a first-order logical theory, and then extend it with additional predicates and operations. Then we explain a slightly different technique that works on a monadic second- order theory, but show that it is ultimately subsumed by an extension of the first-order theory.
Next, we give two applications: critical exponent and paperfolding words. In the critical exponent example, we mechanically construct an automaton that describes a set of rational numbers related to a given automatic sequence. Then we give a polynomial-time algorithm to compute the supremum of this rational set, allowing us to compute the critical exponent and many similar quantities. In the paperfolding example, we extend our mechanical procedure to the paperfolding words, an uncountably infinite collection of infinite words.
In the following chapter, we address abelian and additive problems on automatic sequences. We give an example of a natural predicate which is provably inexpressible in our first-order theory, and discuss alternate methods for solving abelian and additive problems on automatic sequences.
We close with a chapter of open problems, drawn from the earlier chapters
Numeration systems on a regular language: Arithmetic operations, Recognizability and Formal power series
Generalizations of numeration systems in which N is recognizable by a finite
automaton are obtained by describing a lexicographically ordered infinite
regular language L over a finite alphabet A. For these systems, we obtain a
characterization of recognizable sets of integers in terms of rational formal
series. We also show that, if the complexity of L is Theta (n^q) (resp. if L is
the complement of a polynomial language), then multiplication by an integer k
preserves recognizability only if k=t^{q+1} (resp. if k is not a power of the
cardinality of A) for some integer t. Finally, we obtain sufficient conditions
for the notions of recognizability and U-recognizability to be equivalent,
where U is some positional numeration system related to a sequence of integers.Comment: 34 pages; corrected typos, two sections concerning exponential case
and relation with positional systems adde
Automatic Sequences and Zip-Specifications
We consider infinite sequences of symbols, also known as streams, and the
decidability question for equality of streams defined in a restricted format.
This restricted format consists of prefixing a symbol at the head of a stream,
of the stream function `zip', and recursion variables. Here `zip' interleaves
the elements of two streams in alternating order, starting with the first
stream. For example, the Thue-Morse sequence is obtained by the
`zip-specification' {M = 0 : X, X = 1 : zip(X,Y), Y = 0 : zip(Y,X)}. Our
analysis of such systems employs both term rewriting and coalgebraic
techniques. We establish decidability for these zip-specifications, employing
bisimilarity of observation graphs based on a suitably chosen cobasis. The
importance of zip-specifications resides in their intimate connection with
automatic sequences. We establish a new and simple characterization of
automatic sequences. Thus we obtain for the binary zip that a stream is
2-automatic iff its observation graph using the cobasis (hd,even,odd) is
finite. The generalization to zip-k specifications and their relation to
k-automaticity is straightforward. In fact, zip-specifications can be perceived
as a term rewriting syntax for automatic sequences. Our study of
zip-specifications is placed in an even wider perspective by employing the
observation graphs in a dynamic logic setting, leading to an alternative
characterization of automatic sequences. We further obtain a natural extension
of the class of automatic sequences, obtained by `zip-mix' specifications that
use zips of different arities in one specification. We also show that
equivalence is undecidable for a simple extension of the zip-mix format with
projections like even and odd. However, it remains open whether zip-mix
specifications have a decidable equivalence problem