Quantum, Stochastic, and Pseudo Stochastic Languages with Few States

TL;DR

This paper demonstrates that simple automata models like 2-state GFAs, QFAs, and 3-state PFAs can recognize uncountably many languages over a unary alphabet, revealing their rich expressive power.

Contribution

It establishes the minimal number of states needed for unary automata to recognize uncountably many languages and characterizes the class recognized by 1-state GFAs.

Findings

2-state unary GFAs and QFAs recognize uncountably many languages

3-state unary PFAs recognize uncountably many languages

The class of languages recognized by 1-state GFAs is fully characterized

Abstract

Stochastic languages are the languages recognized by probabilistic finite automata (PFAs) with cutpoint over the field of real numbers. More general computational models over the same field such as generalized finite automata (GFAs) and quantum finite automata (QFAs) define the same class. In 1963, Rabin proved the set of stochastic languages to be uncountable presenting a single 2-state PFA over the binary alphabet recognizing uncountably many languages depending on the cutpoint. In this paper, we show the same result for unary stochastic languages. Namely, we exhibit a 2-state unary GFA, a 2-state unary QFA, and a family of 3-state unary PFAs recognizing uncountably many languages; all these numbers of states are optimal. After this, we completely characterize the class of languages recognized by 1-state GFAs, which is the only nontrivial class of languages recognized by 1-state automata. Finally, we consider the variations of PFAs, QFAs, and GFAs based on the notion of inclusive/exclusive cutpoint, and present some results on their expressive power.