AI 中文总结
本文提出统一模型刻画不同输出域函数正则性，基于豪泽工作推广设定，考虑特定函数类型，证明与已知模型在多域一致，推测其他域也类似，还扩展到无限字母表并研究其表达能力，由爱丽丝和鲍勃通过交换固定消息数计算函数值。
AI 中文摘要
本文旨在为不同输出域函数的正则性提出一个统一的Nerode风格刻画模型。基于豪泽在通信复杂性方面的工作，通过放宽可计算性假设并允许非布尔输出域来推广设定。考虑函数类型为$\Sigma^* \to \domain$，其中$\Sigma$是有限字母表，$\domain$是任意域。对于多个域，证明该模型与已知计算模型一致。进一步推测对于其他缺乏正则性Nerode风格刻画的域也有类似对应，并提供了充分证据。在模型中，输入字符串$w$被拆分为$w = w_1 w_2$分发给合作方爱丽丝和鲍勃，他们通过交换固定数量消息计算函数值。每个消息要么是输出域元素，要么是有限信号集中抽取的信号，各方必须对每个可允许拆分$w = w_1 w_2$产生正确输出。还将框架扩展到名义集设定下的无限字母表，并研究其对含原子单词语言的表达能力。
英文摘要
The goal of this paper is to propose a unifying model for Nerode-style characterizations of regularity across functions with different output domains. Building on Hauser's work in communication complexity, we generalize the setting by relaxing the computability assumptions and allowing non-Boolean output domains. We consider functions of type $Σ^* \to \domain$, where $Σ$ is a finite alphabet and $\domain$ is an arbitrary domain. For several domains, we show that the model coincides with known models of computation. We further conjecture that an analogous correspondence holds for other domains that currently lack a Nerode-style characterization of regularity, and we provide ample supporting evidence. In the model, an input string $w$ is split as $w = w_1 w_2$ and distributed between two cooperating parties, Alice and Bob, who exchange a constant number of messages to compute the value of the function. Each message is either an element of the output domain or a signal drawn from a finite set of signals, and the parties must produce the correct output for every admissible split $w = w_1 w_2$. We further extend the framework to infinite alphabets in the setting of nominal sets, and investigate its expressiveness on languages of words with atoms.
