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 [Formula: see text] 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.

中文翻译：

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向量和归位向量自动机的新结果

通过对向量自动机和归位向量自动机的研究，我们提出了一些新的结果和有限自动机的各种扩展之间的联系。我们表明，当两者都在 [公式：见文本] 整数矩阵上定义时，归位向量自动机优于扩展有限自动机。我们研究了向量自动机的字符串分离问题，并证明了具有理性条目的广义有限自动机可以仅使用两种状态来分离任何一对字符串。通过研究无状态归位向量自动机，我们证明了一种语言可以被具有乘法的无状态盲确定性实时版本的有限自动机识别，前提是它是可交换的，并且其 Parikh 图像是线性齐次丢番图方程组的非负整数解的集合。