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Quasi-Periodic beta-Expansions and Cut Languages
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SYSNO ASEP 0482160 Druh ASEP J - Článek v odborném periodiku Zařazení RIV J - Článek v odborném periodiku Poddruh J Článek ve WOS Název Quasi-Periodic beta-Expansions and Cut Languages Tvůrce(i) Šíma, Jiří (UIVT-O) RID, SAI, ORCID
Savický, Petr (UIVT-O) SAI, RID, ORCIDZdroj.dok. Theoretical Computer Science. - : Elsevier - ISSN 0304-3975
Roč. 720, 11 April (2018), s. 1-23Poč.str. 23 s. Jazyk dok. eng - angličtina Země vyd. NL - Nizozemsko Klíč. slova beta-expansion ; quasi-periodicity ; Pisot number ; cut language ; Chomsky hierarchy Vědní obor RIV IN - Informatika Obor OECD Computer sciences, information science, bioinformathics (hardware development to be 2.2, social aspect to be 5.8) CEP GBP202/12/G061 GA ČR - Grantová agentura ČR Institucionální podpora UIVT-O - RVO:67985807 UT WOS 000429514000001 EID SCOPUS 85042599451 DOI 10.1016/j.tcs.2018.02.028 Anotace 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. Pracoviště Ústav informatiky Kontakt Tereza Šírová, sirova@cs.cas.cz, Tel.: 266 053 800 Rok sběru 2019
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