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Star-free sets of integers

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. ...

##
###
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]

2010
*
Lecture Notes in Computer Science
*

We discuss the notion

doi:10.1007/978-3-642-14455-4_6
fatcat:nl3tcpwgcfdfxffsyyvzkeher4
*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. ...##
###
Numeration Systems: a Link between Number Theory and Formal Language Theory
[article]

2012
*
arXiv
*
pre-print

We discuss the notion

arXiv:1204.5887v1
fatcat:xoju4renerhqzcr245luz7klye
*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. ...##
###
A Generalization of Semenov's Theorem to Automata over Real Numbers
[chapter]

2009
*
Lecture Notes in Computer Science
*

This result can be seen as a generalization to the mixed

doi:10.1007/978-3-642-02959-2_34
fatcat:y3nxkeze2zf5fhb2w3ytkojdgy
*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. ...##
###
On the Sets of Real Numbers Recognized by Finite Automata in Multiple Bases

2010
*
Logical Methods in Computer Science
*

This result extends Cobham's theorem, which

doi:10.2168/lmcs-6(1:6)2010
fatcat:u3rssfy4zra3dcbwjue4al5agq
*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. ...##
###
Bertrand numeration systems and recognizability

1997
*
Theoretical Computer Science
*

There exist various well-known

doi:10.1016/s0304-3975(96)00260-5
fatcat:lmf5j3xnlraitemmqrveu2jbvy
*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. ...##
###
Towards a characterization of the star-free sets of integers
[article]

2001
*
arXiv
*
pre-print

Restivo, we obtain a complete logical

arXiv:cs/0111057v1
fatcat:g6e6lnvsrrbdri7tu7rfhigeui
*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. ...##
###
A generalization of Cobham's theorem to automata over real numbers

2009
*
Theoretical Computer Science
*

This result generalizes to real numbers the well-known Cobham's theorem on the finite-state

doi:10.1016/j.tcs.2008.12.051
fatcat:dcew7fslenh6fhxf3oj67hoq5e
*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. ...##
###
The closure under division and a characterization of the recognizable $\mathcal {Z}$-subsets

1996
*
RAIRO - Theoretical Informatics and Applications
*

I am also indebted to the référées for their valuable suggestions, some

doi:10.1051/ita/1996300302091
fatcat:fu3btkz37fbkfldazzokpfppim
*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. ...##
###
$$\langle \mathbb {R}, +,<,1 \rangle $$ Is Decidable in $$\langle \mathbb {R}, +,< ,\mathbb {Z}\rangle $$
[chapter]

2020
*
Lecture Notes in Computer Science
*

Consider now

doi:10.1007/978-3-030-40608-0_8
fatcat:po3vzzanrzcyrmkixfp6m7u43i
*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. ...##
###
Decision problems among the main subfamilies of rational relations

2006
*
RAIRO - Theoretical Informatics and Applications
*

We consider the four families

doi:10.1051/ita:2006005
fatcat:wiocjkoaszb2dhgzujscmkis64
*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. ...##
###
Numeration systems on a regular language: Arithmetic operations, Recognizability and Formal power series
[article]

2000
*
arXiv
*
pre-print

For these systems, we obtain a

arXiv:cs/9911002v2
fatcat:i44jtpls2na5jnnnl3nyszcy6a
*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. ...##
###
On the expressiveness of real and integer arithmetic automata
[chapter]

1998
*
Lecture Notes in Computer Science
*

We show that the

doi:10.1007/bfb0055049
fatcat:s6yzilh2qbf6dnls2rhrfbgk5u
*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. ...##
###
Formulas, regular languages and boolean circuits

1992
*
Theoretical Computer Science
*

Logical

doi:10.1016/0304-3975(92)90152-6
fatcat:dwwd4wtexvegjhjjhsehpbbhby
*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*. ...
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