8 research outputs found

    Sequences close to periodic

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    The paper is a survey of notions and results related to classical and new generalizations of the notion of a periodic sequence. The topics related to almost periodicity in combinatorics on words, symbolic dynamics, expressibility in logical theories, algorithmic computability, Kolmogorov complexity, number theory, are discussed.Comment: In Russian. 76 pages, 6 figure

    nn-valued Coset Groups and Dynamics

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    We obtain asymptotic and exact formulae of growth functions for some families of nn-valued coset groups. We also describe connections between the theory of nn-valued groups and Symbolic Dynamics

    On Periodically Iterated Morphisms

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    We investigate the computational power of periodically iterated morphisms, also known as D0L systems with periodic control, PD0L systems for short. These systems give rise to a class of one-sided infinite sequences, called PD0L words. We construct a PD0L word with exponential subword complexity, thereby answering a question raised by Lepisto (1993) on the existence of such words. We solve another open problem concerning the decidability of the first-order theories of PD0L words; we show it is already undecidable whether a certain letter occurs in a PD0L word. This stands in sharp contrast to the situation for D0L words (purely morphic words), which are known to have at most quadratic subword complexity, and for which the monadic theory is decidable. The main result of our paper, leading to these answers, is that every computable word w over an alphabet Sigma can be embedded in a PD0L word u over an extended alphabet Gamma in the following two ways: (i) such that every finite prefix of w is a subword of u, and (ii) such that w is obtained from u by erasing all letters from Gamma not in Sigma. The PD0L system generating such a word u is constructed by encoding a Fractran program that computes the word w; Fractran is a programming language as powerful as Turing Machines. As a consequence of (ii), if we allow the application of finite state transducers to PD0L words, we obtain the set of all computable words. Thus the set of PD0L words is not closed under finite state transduction, whereas the set of D0L words is. It moreover follows that equality of PD0L words (given by their PD0L system) is undecidable. Finally, we show that if erasing morphisms are admitted, then the question of productivity becomes undecidable, that is, the question whether a given PD0L system defines an infinite word

    On cogrowth function of algebras and its logarithmical gap

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    Let AkX/IA \cong k\langle X \rangle / I be an associative algebra. A finite word over alphabet XX is II-reducible if its image in AA is a kk-linear combination of length-lexicographically lesser words. An obstruction is a subword-minimal II-reducible word. If the number of obstructions is finite then II has a finite Gröbner basis, and the word problem for the algebra is decidable. A cogrowth function is the number of obstructions of length n\le n. We show that the cogrowth function of a finitely presented algebra is either bounded or at least logarithmical. We also show that an uniformly recurrent word has at least logarithmical cogrowth

    On cogrowth function of algebras and its logarithmical gap

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    Let AkX/IA \cong k\langle X \rangle / I be an associative algebra. A finite word over alphabet XX is II-reducible if its image in AA is a kk-linear combination of length-lexicographically lesser words. An obstruction is a subword-minimal II-reducible word. If the number of obstructions is finite then II has a finite Gröbner basis, and the word problem for the algebra is decidable. A cogrowth function is the number of obstructions of length n\le n. We show that the cogrowth function of a finitely presented algebra is either bounded or at least logarithmical. We also show that an uniformly recurrent word has at least logarithmical cogrowth

    Sequences close to periodic

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