Quasi-Periodic beta-Expansions and Cut Languages

Result code in IS VaVaI

<a href="https://www.isvavai.cz/riv?ss=detail&h=RIV%2F67985807%3A_____%2F18%3A00482160" target="_blank" >RIV/67985807:_____/18:00482160 - isvavai.cz</a>

Result on the web

<a href="http://dx.doi.org/10.1016/j.tcs.2018.02.028" target="_blank" >http://dx.doi.org/10.1016/j.tcs.2018.02.028</a>

DOI - Digital Object Identifier

<a href="http://dx.doi.org/10.1016/j.tcs.2018.02.028" target="_blank" >10.1016/j.tcs.2018.02.028</a>

Result language

angličtina

Original language name

Quasi-Periodic beta-Expansions and Cut Languages

Original language description

Motivated by the analysis of neural net models between integer and rational weights, we introduce a so-called cut language over a real digit alphabet, which contains finite beta-expansions (i.e. base-beta representations) of the numbers less than a given threshold. We say that an infinite beta-expansion is eventually quasi-periodic if its tail sequence formed by the numbers whose representations are obtained by removing leading digits, contains an infinite constant subsequence. We prove that a cut language is regular iff its threshold is a quasi-periodic number whose all beta-expansions are eventually quasi-periodic, by showing that altogether they have a finite number of tail values. For algebraic bases beta, we prove that there is an eventually quasi-periodic beta-expansion with an infinite number of tail values iff there is a conjugate of beta on the unit circle. For transcendental beta combined with algebraic digits, a beta-expansion is eventually quasi-periodic iff it has a finite number of tail values. For a Pisot base beta and digits from the smallest field extension Q(beta) over rational numbers including beta, we show that any number from Q(beta) is quasi-periodic. In addition, we achieve a dichotomy that a cut language is either regular or non-context-free and we show that any cut language with rational parameters is context-sensitive.

Czech name

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Czech description

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Type

J_imp - Article in a specialist periodical, which is included in the Web of Science database

CEP classification

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OECD FORD branch

10201 - Computer sciences, information science, bioinformathics (hardware development to be 2.2, social aspect to be 5.8)

Project

<a href="/en/project/GBP202%2F12%2FG061" target="_blank" >GBP202/12/G061: Center of excellence - Institute for theoretical computer science (CE-ITI)</a><br>

Continuities

P - Projekt vyzkumu a vyvoje financovany z verejnych zdroju (s odkazem do CEP)

Publication year

2018

Confidentiality

S - Úplné a pravdivé údaje o projektu nepodléhají ochraně podle zvláštních právních předpisů

Name of the periodical

Theoretical Computer Science

ISSN

0304-3975

e-ISSN

—

Volume of the periodical

720

Issue of the periodical within the volume

11 April

Country of publishing house

NL - THE KINGDOM OF THE NETHERLANDS

Number of pages

23

Pages from-to

1-23

UT code for WoS article

000429514000001

EID of the result in the Scopus database

2-s2.0-85042599451