Exact Affine Counter Automata

Abstract

We introduce an affine generalization of counter automata, and analyze their ability as well as affine finite automata. Our contributions are as follows. We show that there is a language that can be recognized by exact realtime affine counter automata but by neither 1-way deterministic pushdown automata nor realtime deterministic [Formula: see text]-counter automata. Then, we show that exact realtime affine counter automata can recognize a family of regular languages with a few states but the states required by realtime deterministic [Formula: see text]-counter automata cannot be bounded. We also show that stateless affine increment-only counter automata can recognize some nonregular languages with one-sided bounded-error. Moreover, we show that a promise problem not solved by two-way quantum finite automata in polynomial time can be solved by Las Vegas affine finite automata. Lastly, we present an affine finite automaton algorithm using a counter cleverly to recognize a language, for which we do not know any bounded-error affine finite automata or two-way quantum finite automata recognizing it.