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Minimal Presentations of Sofic Shifts and Properties of Periodic-Finite-Type ShiftsManada, Akiko 12 August 2009 (has links)
Constrained codes have been used in data storage systems, such as magnetic tapes,
CD’s and DVD’s, in order to reduce the likelihood of errors by predictable noise.
The study of constrained codes is based on the study of sofic shifts, which are sets of
bi-infinite sequences that can be presented using labeled directed graphs called presentations. In this thesis, we will primarily focus on two classes of sofic shifts, namely shifts of finite type (SFT’s) and periodic-finite-type shifts (PFT’s), and examine their properties.
We first consider Shannon covers of sofic shifts. A Shannon cover of a sofic shift
is a deterministic presentation with the smallest number of vertices among all deterministic presentations of the shift. Indeed, a Shannon cover is used as a canonical presentation of a sofic shift, and furthermore, it is used when computing the capacity of the shift or when constructing a finite-state encoder. We follow an algorithm by Crochemore, Mignosi and Restivo which constructs a deterministic presentation of
an SFT and we see how to derive a Shannon cover from the presentation under their
algorithm. Furthermore, as a method to determine whether a given deterministic
presentation is a Shannon cover of a sofic shift, we will provide, based on research by
Jonoska, a sufficient condition for a given presentation to have the smallest number
of vertices among all presentations of the shift.
We then move our focus towards PFT’s, and investigate new properties of PFT’s from various perspectives. We define three types of periods that can be associated with a PFT and do pairwise comparisons between them. Also, we consider the zeta function of a PFT, which is a generating function for the number of periodic sequences in the PFT, and present a simple formula to compute the zeta function of a PFT. / Thesis (Ph.D, Mathematics & Statistics) -- Queen's University, 2009-08-08 14:08:36.876
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Class degree and measures of relative maximal entropyAllahbakhshi, Mahsa 16 March 2011 (has links)
Given a factor code [pi] from a shift of finite type X onto an irreducible sofic shift Y, and a fully supported ergodic measure v on Y we give an explicit upper bound on the number of ergodic measures on X which project to v and have maximal entropy among all measures in the fiber [pi]-1{v}. This bound is invariant under conjugacy. We relate this to an important construction for finite-to-one symbolic factor maps.
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