On the 3-state Mealy Automata over an m-symbol Alphabet of Growth Order [ n ^{{\log n}/{2 \log m}} ]

We consider the sequence ${J_m,m \ge 2}$ of the 3-state Mealy automata over an m-symbol alphabet such that the growth function of $J_m$ has the intermediate growth order $[n ^{{\log n}/{2 \log m}} ]$. For each automaton $J_m$ we describe the automaton transformation monoid $S_{J_m}$, defined by it, provide generating series for the growth functions, and consider primary properties of $S_{J_m}$ and $J_m$.Comment: 38 pages, 5 Postscript figure

Languages, machines, and classical computation

3rd ed, 2021. A circumscription of the classical theory of computation building up from the Chomsky hierarchy. With the usual topics in formal language and automata theory

Quantum Hidden Markov Models based on Transition Operation Matrices

In this work, we extend the idea of Quantum Markov chains [S. Gudder. Quantum Markov chains. J. Math. Phys., 49(7), 2008] in order to propose Quantum Hidden Markov Models (QHMMs). For that, we use the notions of Transition Operation Matrices (TOM) and Vector States, which are an extension of classical stochastic matrices and probability distributions. Our main result is the Mealy QHMM formulation and proofs of algorithms needed for application of this model: Forward for general case and Vitterbi for a restricted class of QHMMs.Comment: 19 pages, 2 figure

The smallest Mealy automaton of intermediate growth

In this paper we study the smallest Mealy automaton of intermediate growth, first considered by the last two authors. We describe the automatic transformation monoid it defines, give a formula for the generating series for its (ball volume) growth function, and give sharp asymptotics for its growth function, namely [ F(n) \sim 2^{5/2} 3^{3/4} \pi^{-2} n^{1/4} \exp{\pi\sqrt{n/6}} ] with the ratios of left- to right-hand side tending to 1 as $n \to \infty$