🤖 AI Summary
This work addresses the absence of a unified Nerode-style characterization of regularity applicable to arbitrary output domains. Building on a communication complexity framework, it partitions input strings between two parties, Alice and Bob, who collaboratively compute function values by exchanging a constant number of messages drawn either from the output domain or a finite signal set. This approach relaxes traditional computability assumptions, accommodating non-Boolean output domains and infinite alphabets, and extends—within the nominal sets framework—to languages over atomic words. The study demonstrates that the proposed model aligns with established computational models across diverse output domains, thereby offering a unified conjecture and theoretical foundation for settings previously lacking a Nerode-type characterization.
📝 Abstract
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.