49 research outputs found

    Abstract numeration systems on bounded languages and multiplication by a constant

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    A set of integers is SS-recognizable in an abstract numeration system SS 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 λ≥2\lambda\ge2 does not preserve SS-recognizability, meaning that there always exists a SS-recognizable set XX such that λX\lambda X is not SS-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

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    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

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    We present an elementary three pass algorithm for computing addition in Ostrowski numeration systems. When aa is quadratic, addition in the Ostrowski numeration system based on aa is recognizable by a finite automaton. We deduce that a subset of X⊆NnX\subseteq \mathbb{N}^n is definable in (N,+,Va)(\mathbb{N},+,V_a), where VaV_a is the function that maps a natural number xx to the smallest denominator of a convergent of aa that appears in the Ostrowski representation based on aa of xx with a non-zero coefficient, if and only if the set of Ostrowski representations of elements of XX is recognizable by a finite automaton. The decidability of the theory of (N,+,Va)(\mathbb{N},+,V_a) follows

    Structural properties of bounded languages with respect to multiplication by a constant

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    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

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    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

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    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

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    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
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