# New Results on Vector and Homing Vector Automata

## TL;DR

This paper explores extensions of finite automata through vector and homing vector automata, demonstrating their capabilities, limitations, and relationships, including superiority in certain cases and characterizations of recognized languages.

## Contribution

It introduces new results connecting vector automata variants, compares their computational power, and characterizes languages recognized by stateless homing vector automata.

## Key findings

- Homing vector automata outperform extended finite automata over 2x2 matrices.
- Two-state rational generalized finite automata can separate any pair of strings.
- Stateless homing vector automata recognize languages with specific algebraic properties.

## Abstract

We present several new results and connections between various extensions of finite automata through the study of vector automata and homing vector automata. We show that homing vector automata outperform extended finite automata when both are defined over $ 2 \times 2 $ integer matrices. We study the string separation problem for vector automata and demonstrate that generalized finite automata with rational entries can separate any pair of strings using only two states. Investigating stateless homing vector automata, we prove that a language is recognized by stateless blind deterministic real-time version of finite automata with multiplication iff it is commutative and its Parikh image is the set of nonnegative integer solutions to a system of linear homogeneous Diophantine equations.