Language Recognition Power and Succinctness of Affine Automata

Abstract

In this work we study a non-linear generalization based on affine transformations of probabilistic and quantum automata proposed recently by Díaz-Caro and Yakaryılmaz [6] 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.

On the Computational Power of Affine Automata

“…Recently, A. Díaz-Caro and A. Yakaryılmaz introduced a new model, called affine automata [4], also investigated in [13] and [3]. It is a purely theoretical model, which means that it cannot be implemented by a physical device like quantum automata.…”

Affine automata verifiers

“…But, bounded-error AfAs can recognize some nonregular languages such as UPAL = {a n b n | n > 0} and PAL = {w ∈ {a, b} * | w = w r } [7]. Moreover, AfAs can be very succinct compared to PFAs and QFAs [28,29], i.e., they can recognize a family of regular languages with bounded-error by using only two states, but the number of states of bounded-error PFAs or QFAs cannot be bounded for this family.…”

“…AfAs was formally defined in [7], and it was shown that they are more powerful than PFAs and quantum finite automata (QFAs) in bounded-error and unbounded-error settings, but their nondeterministic version is equivalent to nondeterministic QFAs. Since then, AfAs and their different generalizations (e.g., OBDDs and using counters) have been investigated in a series of work [28,14,22,17,29,15,16].…”

Computational Limitations of Affine Automata

“…AfAs and their certain generalizations have been investigated in a series of works [5,8,9,21]. In most of the cases, affine models (e.g., bounded-error and unbouded-error AfAs, zero-error affine OBDDs, zero-error affine counter automata, etc.)…”