Regular functions from infinite words to infinite words can be equivalently specified by MSO-transducers, streaming $\omega$-string transducers as well as deterministic two-way transducers with look-ahead. In their one-way restriction, the latter transducers define the class of rational functions. Even though regular functions are robustly characterised by several finite-state devices, even the subclass of rational functions may contain functions which are not computable (by a Turing machine with infinite input). This paper proposes a decision procedure for the following synthesis problem: given a regular function $f$ (equivalently specified by one of the aforementioned transducer model), is $f$ computable and if it is, synthesize a Turing machine computing it. For regular functions, we show that computability is equivalent to continuity, and therefore the problem boils down to deciding continuity. We establish a generic characterisation of continuity for functions preserving regular languages under inverse image (such as regular functions). We exploit this characterisation to show the decidability of continuity (and hence computability) of rational and regular functions. For rational functions, we show that this can be done in $\mathsf{NLogSpace}$ (it was already known to be in $\mathsf{PTime}$ by Prieur). In a similar fashion, we also effectively characterise uniform continuity of regular functions, and relate it to the notion of uniform computability, which offers stronger efficiency guarantees.

翻译：从无限单词到无限单词的常规功能可以由 MSO- Transporters 等量指定, 流出 $\ omega$-string Transporters 和 确定性的双向双向传输器, 外观显示。 在单向限制中, 后一传输器定义了理性功能的类别 。 尽管常规功能由若干限定状态装置严格定性, 即使是理性功能的亚类, 也可能包含无法比较的功能( 由带有无限输入的图灵机 ) 。 本文为以下合成问题提出了一个决定程序: 给一个常规函数$f( 由上述一个 Transporter{ 模式 等量), 折成 折成美元( 等值) 。 对于常规字符来说, 共性功能的兼容性与连续性是相当的, 直径直径不一等值 。 在 Rialf\ 格式中, 直径可显示 直径( 直径) 直径/ 直径/ 直径直径 函数的稳定性。