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Michel Rigo - One of the best experts on this subject based on the ideXlab platform.
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Automatic sequences based on Parry or Bertrand Numeration systems.
arXiv: Formal Languages and Automata Theory, 2018Co-Authors: Adeline Massuir, Jarkko Peltomäki, Michel RigoAbstract:We study the factor complexity and closure properties of automatic sequences based on Parry or Bertrand Numeration systems. These automatic sequences can be viewed as generalizations of the more typical $k$-automatic sequences and Pisot-automatic sequences. We show that, like $k$-automatic sequences, Parry-automatic sequences have sublinear factor complexity while there exist Bertrand-automatic sequences with superlinear factor complexity. We prove that the set of Parry-automatic sequences with respect to a fixed Parry Numeration system is not closed under taking images by uniform substitutions or periodic deletion of letters. These closure properties hold for $k$-automatic sequences and Pisot-automatic sequences, so our result shows that these properties are lost when generalizing to Parry Numeration systems and beyond. Moreover, we show that a multidimensional sequence is $U$-automatic with respect to a positional Numeration system $U$ with regular language of Numeration if and only if its $U$-kernel is finite.
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On Cobham's theorem
2011Co-Authors: Fabien Durand, Michel RigoAbstract:In this chapter we essentially focus on the representation of non-negative integers in a given Numeration system. The main role of such a system --- like the usual integer base $k$ Numeration system --- is to replace numbers or more generally sets of numbers by their corresponding representations, {\em i.e.}, by words or by languages. First we consider integer base Numeration systems to present the main concepts but rapidly we will introduce non-standard systems and their relationships with substitutions.
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Representing real numbers in a generalized Numeration system
Journal of Computer and System Sciences, 2011Co-Authors: ímilie Charlier, Marion Le Gonidec, Michel RigoAbstract:We show how to represent an interval of real numbers in an abstract Numeration system built on a language that is not necessarily regular. As an application, we consider representations of real numbers using the Dyck language. We also show that our framework can be applied to the rational base Numeration systems.
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A DECISION PROBLEM FOR ULTIMATELY PERIODIC SETS IN NONSTANDARD Numeration SYSTEMS
International Journal of Algebra and Computation, 2009Co-Authors: Jason P. Bell, Aviezri S. Fraenkel, Emilie Charlier, Michel RigoAbstract:Consider a nonstandard Numeration system like the one built over the Fibonacci sequence where nonnegative integers are represented by words over {0,1} without two consecutive 1. Given a set X of integers such that the language of their greedy representations in this system is accepted by a finite automaton, we consider the problem of deciding whether or not X is a finite union of arithmetic progressions. We obtain a decision procedure for this problem, under some hypothesis about the considered Numeration system. In a second part, we obtain an analogous decision result for a particular class of abstract Numeration systems built on an infinite regular language.
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A Decision Problem for Ultimately Periodic Sets in Non-standard Numeration Systems
arXiv: Formal Languages and Automata Theory, 2009Co-Authors: Jason P. Bell, Aviezri S. Fraenkel, Emilie Charlier, Michel RigoAbstract:Consider a non-standard Numeration system like the one built over the Fibonacci sequence where nonnegative integers are represented by words over $\{0,1\}$ without two consecutive 1. Given a set $X$ of integers such that the language of their greedy representations in this system is accepted by a finite automaton, we consider the problem of deciding whether or not $X$ is a finite union of arithmetic progressions. We obtain a decision procedure for this problem, under some hypothesis about the considered Numeration system. In a second part, we obtain an analogous decision result for a particular class of abstract Numeration systems built on an infinite regular language.
Emilie Charlier - One of the best experts on this subject based on the ideXlab platform.
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An introduction to abstract Numeration systems
2012Co-Authors: Emilie CharlierAbstract:Numeration systems An abstract Numeration system (ANS) is a triple S = (L,Σ,
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A DECISION PROBLEM FOR ULTIMATELY PERIODIC SETS IN NONSTANDARD Numeration SYSTEMS
International Journal of Algebra and Computation, 2009Co-Authors: Jason P. Bell, Aviezri S. Fraenkel, Emilie Charlier, Michel RigoAbstract:Consider a nonstandard Numeration system like the one built over the Fibonacci sequence where nonnegative integers are represented by words over {0,1} without two consecutive 1. Given a set X of integers such that the language of their greedy representations in this system is accepted by a finite automaton, we consider the problem of deciding whether or not X is a finite union of arithmetic progressions. We obtain a decision procedure for this problem, under some hypothesis about the considered Numeration system. In a second part, we obtain an analogous decision result for a particular class of abstract Numeration systems built on an infinite regular language.
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A Decision Problem for Ultimately Periodic Sets in Non-standard Numeration Systems
arXiv: Formal Languages and Automata Theory, 2009Co-Authors: Jason P. Bell, Aviezri S. Fraenkel, Emilie Charlier, Michel RigoAbstract:Consider a non-standard Numeration system like the one built over the Fibonacci sequence where nonnegative integers are represented by words over $\{0,1\}$ without two consecutive 1. Given a set $X$ of integers such that the language of their greedy representations in this system is accepted by a finite automaton, we consider the problem of deciding whether or not $X$ is a finite union of arithmetic progressions. We obtain a decision procedure for this problem, under some hypothesis about the considered Numeration system. In a second part, we obtain an analogous decision result for a particular class of abstract Numeration systems built on an infinite regular language.
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a decision problem for ultimately periodic sets in non standard Numeration systems
Mathematical Foundations of Computer Science, 2008Co-Authors: Emilie Charlier, Michel RigoAbstract:Consider a non-standard Numeration system like the one built over the Fibonacci sequence where nonnegative integers are represented by words over {0,1} without two consecutive 1. Given a set Xof integers such that the language of their greedy representations in this system is accepted by a finite automaton, we consider the problem of deciding whether or not Xis a finite union of arithmetic progressions. We obtain a decision procedure under some hypothesis about the considered Numeration system. In a second part, we obtain an analogous decision result for a particular class of abstract Numeration systems built on an infinite regular language.
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Abstract Numeration systems on bounded languages and multiplication by a constant
Integers : Electronic Journal of Combinatorial Number Theory, 2008Co-Authors: Emilie Charlier, Michel Rigo, Wolfgang SteinerAbstract:A set of integers is $S$-recognizable in an abstract Numeration system $S$ 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 $\lambda\ge2$ does not preserve $S$-recognizability, meaning that there always exists a $S$-recognizable set $X$ such that $\lambda X$ is not $S$-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.
Rigo Michel - One of the best experts on this subject based on the ideXlab platform.
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Revisiting regular sequences in light of rational base Numeration systems
2021Co-Authors: Rigo Michel, Stipulanti ManonAbstract:Regular sequences generalize the extensively studied automatic sequences. Let $S$ be an abstract Numeration system. When the Numeration language $L$ is prefix-closed and regular, a sequence is said to be $S$-regular if the module generated by its $S$-kernel is finitely generated. In this paper, we give a new characterization of such sequences in terms of the underlying Numeration tree $T(L)$ whose nodes are words of $L$. We may decorate these nodes by the sequence of interest following a breadth-first eNumeration. For a prefix-closed regular language $L$, we prove that a sequence is $S$-regular if and only if the tree $T(L)$ decorated by the sequence is linear, i.e., the decoration of a node depends linearly on the decorations of a fixed number of ancestors. Next, we introduce and study regular sequences in a rational base Numeration system, whose Numeration language is known to be highly non-regular. We motivate and comment our definition that a sequence is $\frac{p}{q}$-regular if the underlying Numeration tree decorated by the sequence is linear. We give the first few properties of such sequences, we provide a few examples of them, and we propose a method for guessing $\frac{p}{q}$-regularity. Then we discuss the relationship between $\frac{p}{q}$-automatic sequences and $\frac{p}{q}$-regular sequences. We finally present a graph directed linear representation of a $\frac{p}{q}$-regular sequence. Our study permits us to highlight the places where the regularity of the Numeration language plays a predominant role.Comment: 31 pages, 12 figure
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Automatic sequences: from rational bases to trees
2021Co-Authors: Rigo Michel, Stipulanti ManonAbstract:The $n$th term of an automatic sequence is the output of a deterministic finite automaton fed with the representation of $n$ 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 $r$-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.Comment: 25 pages, 15 figure
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The carry propagation of the successor function
2020Co-Authors: Berthé Valérie, Rigo Michel, Frougny Christiane, Sakarovitch JacquesAbstract:Given any Numeration system, we call carry propagation at a number N the number of digits that are changed when going from the representation of N to the one of N+1, and amortized carry propagation the limit of the mean of the carry propagations at the first N integers, when N tends to infinity, if this limit exists.In the case of the usual base p Numeration system, it can be shown that the limit indeed exists and is equal to p/(p −1). We recover a similar value for those Numeration systems we consider and for which the limit exists.We address the problem of the existence of the amortized carry propagation in non-standard Numeration systems of various kinds: abstract Numeration systems, rational base Numeration systems, greedy Numeration systems and beta-Numeration. We tackle the problem with three different types of techniques: combinatorial, algebraic, and ergodic. Fo r each kind of Numeration systems that we consider, the relevant method allows for establishing sufficient conditions for the existence of the carry propagation and examples show that these conditions are close to being necessary conditions.Peer reviewe
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The carry propagation of the successor function
'Elsevier BV', 2020Co-Authors: Berthé Valérie, Rigo Michel, Frougny Christiane, Sakarovitch JacquesAbstract:peer reviewedaudience: researcher, professionalGiven any Numeration system, we call carry propagation at a number N the number of digits that are changed when going from the representation of N to the one of N+1, and amortized carry propagation the limit of the mean of the carry propagations at the first N integers, when N tends to infinity, if this limit exists.In the case of the usual base p Numeration system, it can be shown that the limit indeed exists and is equal to p/(p −1). We recover a similar value for those Numeration systems we consider and for which the limit exists.We address the problem of the existence of the amortized carry propagation in non-standard Numeration systems of various kinds: abstract Numeration systems, rational base Numeration systems, greedy Numeration systems and beta-Numeration. We tackle the problem with three different types of techniques: combinatorial, algebraic, and ergodic. Fo r each kind of Numeration systems that we consider, the relevant method allows for establishing sufficient conditions for the existence of the carry propagation and examples show that these conditions are close to being necessary conditions
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The carry propagation of the successor function
2020Co-Authors: Berthé Valérie, Rigo Michel, Frougny Christiane, Sakarovitch JacquesAbstract:Given any Numeration system, we call carry propagation at a number $N$ the number of digits that are changed when going from the representation of $N$ to the one of $N+1$, and amortized carry propagation the limit of the mean of the carry propagations at the first $N$ integers, when $N$ tends to infinity, if this limit exists. In the case of the usual base $p$ Numeration system, it can be shown that the limit indeed exists and is equal to $p/(p-1)$. We recover a similar value for those Numeration systems we consider and for which the limit exists. We address the problem of the existence of the amortized carry propagation in non-standard Numeration systems of various kinds: abstract Numeration systems, rational base Numeration systems, greedy Numeration systems and beta-Numeration. We tackle the problem by three different types of techniques: combinatorial, algebraic, and ergodic. For each kind of Numeration systems that we consider, the relevant method allows for establishing sufficient conditions for the existence of the carry propagation and examples show that these conditions are close to being necessary conditions.Comment: 55 page
S C Ingham - One of the best experts on this subject based on the ideXlab platform.
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Increased D-values for Salmonella enteritidis resulting from the use of anaerobic eNumeration methods
Food Microbiology, 1993Co-Authors: I.j. Xavier, S C InghamAbstract:Abstract D-value (decimal reduction time) was determined for Salmonella enteritides ATCC # 4931 at 52, 54, 56 and 58°C in skim milk and casein peptone soymeal peptone broth supplemented with 0·6% yeast extract (CASO-YE). Four combinations of heating conditions and survivor eNumeration method were tested for each medium: (1) pre-reduced heating medium with anaerobic eNumeration (PAn); (2) non-reduced heating medium with anaerobic eNumeration (NAn); (3) non-reduced heating medium with a fluid thioglycollate overlay eNumeration method (NOverlay); (4) non-reduced medium with aerobic eNumeration (NAer). The D-values obtained using PAn, NAn, and NOverlay were significantly (P
Sakarovitch Jacques - One of the best experts on this subject based on the ideXlab platform.
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The carry propagation of the successor function
2020Co-Authors: Berthé Valérie, Rigo Michel, Frougny Christiane, Sakarovitch JacquesAbstract:Given any Numeration system, we call carry propagation at a number N the number of digits that are changed when going from the representation of N to the one of N+1, and amortized carry propagation the limit of the mean of the carry propagations at the first N integers, when N tends to infinity, if this limit exists.In the case of the usual base p Numeration system, it can be shown that the limit indeed exists and is equal to p/(p −1). We recover a similar value for those Numeration systems we consider and for which the limit exists.We address the problem of the existence of the amortized carry propagation in non-standard Numeration systems of various kinds: abstract Numeration systems, rational base Numeration systems, greedy Numeration systems and beta-Numeration. We tackle the problem with three different types of techniques: combinatorial, algebraic, and ergodic. Fo r each kind of Numeration systems that we consider, the relevant method allows for establishing sufficient conditions for the existence of the carry propagation and examples show that these conditions are close to being necessary conditions.Peer reviewe
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The carry propagation of the successor function
'Elsevier BV', 2020Co-Authors: Berthé Valérie, Rigo Michel, Frougny Christiane, Sakarovitch JacquesAbstract:peer reviewedaudience: researcher, professionalGiven any Numeration system, we call carry propagation at a number N the number of digits that are changed when going from the representation of N to the one of N+1, and amortized carry propagation the limit of the mean of the carry propagations at the first N integers, when N tends to infinity, if this limit exists.In the case of the usual base p Numeration system, it can be shown that the limit indeed exists and is equal to p/(p −1). We recover a similar value for those Numeration systems we consider and for which the limit exists.We address the problem of the existence of the amortized carry propagation in non-standard Numeration systems of various kinds: abstract Numeration systems, rational base Numeration systems, greedy Numeration systems and beta-Numeration. We tackle the problem with three different types of techniques: combinatorial, algebraic, and ergodic. Fo r each kind of Numeration systems that we consider, the relevant method allows for establishing sufficient conditions for the existence of the carry propagation and examples show that these conditions are close to being necessary conditions
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The carry propagation of the successor function
2020Co-Authors: Berthé Valérie, Rigo Michel, Frougny Christiane, Sakarovitch JacquesAbstract:Given any Numeration system, we call carry propagation at a number $N$ the number of digits that are changed when going from the representation of $N$ to the one of $N+1$, and amortized carry propagation the limit of the mean of the carry propagations at the first $N$ integers, when $N$ tends to infinity, if this limit exists. In the case of the usual base $p$ Numeration system, it can be shown that the limit indeed exists and is equal to $p/(p-1)$. We recover a similar value for those Numeration systems we consider and for which the limit exists. We address the problem of the existence of the amortized carry propagation in non-standard Numeration systems of various kinds: abstract Numeration systems, rational base Numeration systems, greedy Numeration systems and beta-Numeration. We tackle the problem by three different types of techniques: combinatorial, algebraic, and ergodic. For each kind of Numeration systems that we consider, the relevant method allows for establishing sufficient conditions for the existence of the carry propagation and examples show that these conditions are close to being necessary conditions.Comment: 55 page
