Quasi-Periodic beta-Expansions and Cut Languages

0482160 - ÚI 2019 RIV NL eng J - Journal Article
Šíma, Jiří - Savický, Petr
Quasi-Periodic beta-Expansions and Cut Languages.
Theoretical Computer Science. Roč. 720, 11 April (2018), s. 1-23. ISSN 0304-3975. E-ISSN 1879-2294
R&D Projects: GA ČR GBP202/12/G061
Institutional support: RVO:67985807
Keywords : beta-expansion * quasi-periodicity * Pisot number * cut language * Chomsky hierarchy
OECD category: Computer sciences, information science, bioinformathics (hardware development to be 2.2, social aspect to be 5.8)
Impact factor: 0.718, year: 2018

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.
Permanent Link: http://hdl.handle.net/11104/0277554