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Star-free sets of integers

Aldo de Luca, Antonio Restivo
1986 Theoretical Computer Science  
Recognizable sets of nonnegative integers which can be represented in suitable bases by star-free languages are considered.  ...  Acknowledgment The work for this paper was partly supported by the Italian Ministry of the education (M.P.I.). The authors wish to thank the anonymous referee for his (or her) useful suggestions.  ...  A fundamental result of Cobham [ 1] characterizes the sets of nonnegative integers which are k-recognizable for all k > 1.  ... 
doi:10.1016/0304-3975(86)90180-5 fatcat:kf4pigcsovhpxgh4l36gnn7cbq

Page 7261 of Mathematical Reviews Vol. , Issue 87m [page]

1987 Mathematical Reviews  
A set I of nonnegative integers is k-recognizable if its k-ary repre- sentation R is recognizable. If R is aperiodic (noncounting), then I is a recognizable k-periodic set.  ...  The density 7 of the set J is the integer function defined by 1(n) = card(IM{0, 1,---,n—1}) for all n>1. Karel Culik II (Columbia, S.C.)  ... 

Numeration Systems: A Link between Number Theory and Formal Language Theory [chapter]

Michel Rigo
2010 Lecture Notes in Computer Science  
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. 3 of elements in the set {2n + (0, 4/3) | n ∈ Z}.  ...  Acknowledgments I would like to thank Boris Adamczewski, Valérie Berthé, Véronique Bruyère, Eric Duchêne, Narad Rampersad for the careful reading of a first draft of this paper.  ... 
doi:10.1007/978-3-642-14455-4_6 fatcat:nl3tcpwgcfdfxffsyyvzkeher4

Numeration Systems: a Link between Number Theory and Formal Language Theory [article]

Michel Rigo
2012 arXiv   pre-print
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.  ...  Acknowledgments I would like to thank Boris Adamczewski, Valérie Berthé, Véronique Bruyère, Eric Duchêne, Narad Rampersad for the careful reading of a first draft of this paper.  ... 
arXiv:1204.5887v1 fatcat:xoju4renerhqzcr245luz7klye

A Generalization of Semenov's Theorem to Automata over Real Numbers [chapter]

Bernard Boigelot, Julien Brusten, Jérôme Leroux
2009 Lecture Notes in Computer Science  
This result can be seen as a generalization to the mixed integer and real domain of Semenov's theorem, which characterizes the sets of integer vectors recognizable by finite automata in multiple bases.  ...  In this paper, we address the reciprocal property, proving that the sets of vectors that are simultaneously recognizable in all bases, by either weak deterministic or Muller automata, are those definable  ...  In addition, each set S I i is recognizable by a finite-word automaton in base r (operating only on the integer part of r-encodings), and each set S F i is (resp. weakly) r-recognizable if S is (resp.  ... 
doi:10.1007/978-3-642-02959-2_34 fatcat:y3nxkeze2zf5fhb2w3ytkojdgy

On the Sets of Real Numbers Recognized by Finite Automata in Multiple Bases

Bernard Boigelot, Julien Brusten, Véronique Bruyère, Igor Walukiewicz
2010 Logical Methods in Computer Science  
This result extends Cobham's theorem, which characterizes the sets of integer numbers that are recognizable by finite automata in multiple bases.  ...  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  ...  It follows that Theorem 3.1 fully characterizes recognizability in multiple bases for sets of integer numbers.  ... 
doi:10.2168/lmcs-6(1:6)2010 fatcat:u3rssfy4zra3dcbwjue4al5agq

Bertrand numeration systems and recognizability

Véronique Bruyère, Georges Hansel
1997 Theoretical Computer Science  
There exist various well-known characterizations of sets of numbers recognizable by a finite automaton, when they are represented in some integer base p 22.  ...  We show how to modify these characterizations, when integer bases p are replaced by linear numeration systems whose characteristic polynomial is the minimal polynomial of a Pisot number.  ...  U-recognizable sets The aim of this paper is to give three characterizations of sets of positive integers whose normalized U-representations are recognizable by a finite automaton.  ... 
doi:10.1016/s0304-3975(96)00260-5 fatcat:lmf5j3xnlraitemmqrveu2jbvy

Towards a characterization of the star-free sets of integers [article]

Michel Rigo
2001 arXiv   pre-print
Restivo, we obtain a complete logical characterization of the U-star-free sets of integers for suitable numeration systems related to a Pisot number and in particular for integer base systems.  ...  Let U be a numeration system, a set X of integers is U-star-free if the set made up of the U-representations of the elements in X is a star-free regular language.  ...  Logical characterization of recognizable sets of integers In the present section, we consider the binary numeration system.  ... 
arXiv:cs/0111057v1 fatcat:g6e6lnvsrrbdri7tu7rfhigeui

A generalization of Cobham's theorem to automata over real numbers

Bernard Boigelot, Julien Brusten
2009 Theoretical Computer Science  
This result generalizes to real numbers the well-known Cobham's theorem on the finite-state recognizability of sets of integers.  ...  It is known that the sets that are definable in the firstorder additive theory of real and integer variables R, Z, +, < can all be recognized by weak deterministic Büchi automata, regardless of the encoding  ...  Fig. 1 . 1 RNA representing the set of integer powers of a base r > 2.  ... 
doi:10.1016/j.tcs.2008.12.051 fatcat:dcew7fslenh6fhxf3oj67hoq5e

The closure under division and a characterization of the recognizable $\mathcal {Z}$-subsets

Nami Kobayashi
1996 RAIRO - Theoretical Informatics and Applications  
I am also indebted to the référées for their valuable suggestions, some of which led to simplifications of the proofs of Lemmas 11, 13, 19 and Theorem 16, and to Theorems 14 and 15.  ...  The simpler proofs above were suggested by one of the référées.  ...  The second resuit (see Theorem 20) gives a characterization of recognizable Z-subsets through simple Z-subsets.  ... 
doi:10.1051/ita/1996300302091 fatcat:fu3btkz37fbkfldazzokpfppim

$$\langle \mathbb {R}, +,<,1 \rangle $$ Is Decidable in $$\langle \mathbb {R}, +,< ,\mathbb {Z}\rangle $$ [chapter]

Alexis Bès, Christian Choffrut
2020 Lecture Notes in Computer Science  
Consider now recognizability of sets of reals.  ...  A subset X of integers is r−recognizable if there exists a finite automaton accepting precisely the representations in base r of its elements.  ...  A natural issue is to find effective characterizations of subclasses of r−recognizable relations.  ... 
doi:10.1007/978-3-030-40608-0_8 fatcat:po3vzzanrzcyrmkixfp6m7u43i

Decision problems among the main subfamilies of rational relations

Olivier Carton, Christian Choffrut, Serge Grigorieff
2006 RAIRO - Theoretical Informatics and Applications  
We consider the four families of recognizable, synchronous, deterministic rational and rational subsets of a direct product of free monoids.  ...  In particular, adapting a proof of Stearns, we show that it is recursively decidable whether or not a deterministic subset of an arbitrary number of free monoids is recognizable.  ...  The characterization of rational deterministic sets of N k given by Theorem 4.2 is recursive since it involves rational sets of N l for l < k.  ... 
doi:10.1051/ita:2006005 fatcat:wiocjkoaszb2dhgzujscmkis64

Numeration systems on a regular language: Arithmetic operations, Recognizability and Formal power series [article]

Michel Rigo
2000 arXiv   pre-print
For these systems, we obtain a characterization of recognizable sets of integers in terms of rational formal series.  ...  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.  ...  We first characterize the S-recognizable subsets of N in terms of rational series in the noncommuting variables σ ∈ Σ and with coefficients in N.  ... 
arXiv:cs/9911002v2 fatcat:i44jtpls2na5jnnnl3nyszcy6a

On the expressiveness of real and integer arithmetic automata [chapter]

Bernard Boigelot, Stéphane Rassart, Pierre Wolper
1998 Lecture Notes in Computer Science  
We show that the recognizable sets of real vectors are those de nable in the theory of reals and integers with addition and order, extended with a special base-dependent predicate that tests the value  ...  It has been known for some time that, under this encoding, the sets of integer vectors recognizable by nite automata are exactly those de nable in Presburger arithmetic if independence with respect to  ...  As will be shown in Section 6, the converse translation also holds and thus the theory hR; +; ; Z; X r i exactly characterizes the r-recognizable sets of real vectors.  ... 
doi:10.1007/bfb0055049 fatcat:s6yzilh2qbf6dnls2rhrfbgk5u

Formulas, regular languages and boolean circuits

Pierre Péladeau
1992 Theoretical Computer Science  
Logical characterizations of these two classes are given in which the only difference lies in the use of arbitrary numerical predicates in the circuit case.  ...  Several characterizations of this class are given. ' PRC Mathematique et Informatique, France. 0304-3975/92/$05.00 cm  ...  circuit-recognizable languages have very similar logical characterizations.  ... 
doi:10.1016/0304-3975(92)90152-6 fatcat:dwwd4wtexvegjhjjhsehpbbhby
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