The interval in ℕ composed of finite states of the stream version of asymmetric binary systems (ABS) is irreducible if it admits an irreducible finite-state Markov chain. We say that the stream version of ABS is irreducible if its interval is irreducible. Duda gave a necessary condition for the interval to be irreducible. For a probability vector (p,1-p), we assume that p is irrational. Then, we give a necessary and sufficient condition for the interval to be irreducible. The obtained conditions imply that, for a sufficiently small ε, if p∈(1/2,1/2+ε), then the stream version of ABS could not be practically irreducible.

A shift of finite type (SFT) is a set of all bi-infinite sequences over some alphabet which is characterized by a finite set of forbidden words. It is a typical example of sofic shifts and has been used in media storage area, such as CD's or DVD's. The study of sofic shifts is based on graph theory, and the irreducibility of shifts is an important property to be considered for the study. In this paper, we will provide some sufficient conditions for an SFT to be irreducible from the perspective of the antidictionary of a word and the number of forbidden words. We also present a necessary and sufficient condition for an SFT to be irreducible when the number of forbidden words is one less than the alphabet size.