Language recognition power and succintness of affine automata

TL;DR

This paper explores affine automata, a non-linear extension of probabilistic and quantum automata, demonstrating their efficiency in recognizing certain languages and analyzing their succinctness compared to quantum and probabilistic models.

Contribution

It introduces affine automata, shows their ability to simulate probabilistic and quantum automata, and characterizes unary languages recognized by two-state affine automata.

Findings

Affine automata can efficiently simulate probabilistic and quantum automata.

An infinite family of unary languages can be recognized by 2-state affine automata.

Unary languages recognized by two-state affine automata are fully characterized.

Abstract

In this work we study a non-linear generalization based on affine transformations of probabilistic and quantum automata proposed recently by D\'iaz-Caro and Yakary{\i}lmaz \cite{DCY16A} referred as affine automata. First, we present efficient simulations of probabilistic and quantum automata by means of affine automata which allows us to characterize the class of exclusive stochastic languages. Then, we initiate a study on the succintness of affine automata. In particular, we show that an infinite family of unary regular languages can be recognized by 2-state affine automata but the state numbers of quantum and probabilistic automata cannot be bounded. Finally, we present the characterization of all (regular) unary languages recognized by two-state affine automata.