Improved constructions for succinct affine automata

TL;DR

This paper advances the design of affine finite automata by reducing error bounds, minimizing states, and demonstrating their ability to simulate nondeterministic finite automata with high efficiency.

Contribution

It introduces new constructions for AfAs that achieve smaller state complexity and error bounds, and shows AfAs can simulate NFAs with minimal state increase.

Findings

Reduced error bounds to arbitrarily small values.

Achieved lower state complexity in constructions.

Proved AfAs can simulate NFAs with one additional state.

Abstract

Affine finite automata (AfA) can be more succinct than probabilistic and quantum finite automata when recognizing some regular languages with bounded-error. In this paper, we improve previously known constructions given for the succinctness of AfAs in three ways. First, we replace some of fixed error bounds with arbitrarily small error bounds. Second, we present new constructions by using less states than the previous constructions. Third, we show that any language recognized by a nondeterministic finite automaton (NFA) is also recognized by bounded-error AfAs having one more state, and so, AfAs inherit all succinct results by NFAs. As a special case, we also show that any language recognized by a NFA is recognized by AfAs with zero error if the number of accepting path(s) for each member is exactly the same number.