49 research outputs found
Abstract numeration systems on bounded languages and multiplication by a constant
A set of integers is -recognizable in an abstract numeration system if
the language made up of the representations of its elements is accepted by a
finite automaton. For abstract numeration systems built over bounded languages
with at least three letters, we show that multiplication by an integer
does not preserve -recognizability, meaning that there always
exists a -recognizable set such that is not
-recognizable. The main tool is a bijection between the representation of an
integer over a bounded language and its decomposition as a sum of binomial
coefficients with certain properties, the so-called combinatorial numeration
system
The minimal automaton recognizing mN in a linear numeration system
We study the structure of automata accepting the greedy representations of N in a wide class of numeration systems. We describe the conditions under which such automata can have more than one strongly connected component and the form of any such additional components. Our characterization applies, in particular, to any automaton arising from a Bertrand numeration system. Furthermore, we show that for any automaton A arising from a system with a dominant root β > 1, there is a morphism mapping A onto the automaton arising from the Bertrand system associated with the number β. Under some mild assumptions, we also study the state complexity of the trim minimal automaton accepting the greedy representations of the multiples of m ≥ 2 for a wide class of linear numeration systems. As an example, the number of states of the trim minimal automaton accepting the greedy representations of mN in the Fibonacci system is exactly 2m2
Ostrowski numeration systems, addition and finite automata
We present an elementary three pass algorithm for computing addition in
Ostrowski numeration systems. When is quadratic, addition in the Ostrowski
numeration system based on is recognizable by a finite automaton. We deduce
that a subset of is definable in
, where is the function that maps a natural number
to the smallest denominator of a convergent of that appears in the
Ostrowski representation based on of with a non-zero coefficient, if
and only if the set of Ostrowski representations of elements of is
recognizable by a finite automaton. The decidability of the theory of
follows
Structural properties of bounded languages with respect to multiplication by a constant
peer reviewedWe consider the preservation of recognizability of a set of integers after multiplication by a constant for numeration systems built over a bounded language. As a corollary we show that any nonnegative integer can be written as a sum of binomial coefficients with some prescribed properties
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
Generalization of automatic sequences for numeration systems on a regular language
Let L be an infinite regular language on a totally ordered alphabet (A,<).
Feeding a finite deterministic automaton (with output) with the words of L
enumerated lexicographically with respect to < leads to an infinite sequence
over the output alphabet of the automaton. This process generalizes the concept
of k-automatic sequence for abstract numeration systems on a regular language
(instead of systems in base k). Here, I study the first properties of these
sequences and their relations with numeration systems.Comment: 10 pages, 3 figure
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