On the Zeta Function of a Periodic-Finite-Type Shift


IEICE TRANSACTIONS on Fundamentals of Electronics, Communications and Computer Sciences   Vol.E96-A   No.6   pp.1024-1031
Publication Date: 2013/06/01
Online ISSN: 1745-1337
DOI: 10.1587/transfun.E96.A.1024
Print ISSN: 0916-8508
Type of Manuscript: Special Section PAPER (Special Section on Discrete Mathematics and Its Applications)
periodic-finite-type shift,  zeta function,  word-based graph,  Mobius inversion formula,  

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Periodic-finite-type shifts (PFT's) are sofic shifts which forbid the appearance of finitely many pre-specified words in a periodic manner. The class of PFT's strictly includes the class of shifts of finite type (SFT's). The zeta function of a PFT is a generating function for the number of periodic sequences in the shift. For a general sofic shift, there exists a formula, attributed to Manning and Bowen, which computes the zeta function of the shift from certain auxiliary graphs constructed from a presentation of the shift. In this paper, we derive an interesting alternative formula computable from certain “word-based graphs” constructed from the periodically-forbidden word description of the PFT. The advantages of our formula over the Manning-Bowen formula are discussed.