# Exact Affine Counter Automata

## TL;DR

This paper introduces affine counter automata, demonstrating their superior recognition power over traditional models and their ability to solve specific problems efficiently, including some conjectured hard for quantum automata.

## Contribution

It presents the first analysis of affine counter automata, showing their enhanced computational capabilities and applications to solving complex recognition and promise problems.

## Key findings

- Recognizes a language not recognizable by 1-way deterministic pushdown automata.
- Solves a promise problem conjectured hard for two-way quantum automata.
- Recognizes MANYTWINS with bounded-error in realtime.

## 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 k-counter automata. We also show that a certain promise problem, which is conjectured not to be solved by two-way quantum finite automata in polynomial time, can be solved by Las Vegas affine finite automata. Lastly, we show that how a counter helps for affine finite automata by showing that the language MANYTWINS, which is conjectured not to be recognized by affine, quantum or classical finite state models in polynomial time, can be recognized by affine counter automata with one-sided bounded-error in realtime.