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 \textsc{NLogSpace} (it was already known to be in \textsc{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函数可以由 MSO- Transporters 指定等效的 MSO- Transporters 指定, 流出 $\ omega$ string Transporters 和 确定性双向转导器 。 在单向限制中, 后者的转导器定义了理性功能的类别 。 尽管经常函数由若干限定状态装置强有力地定性为从无限单向, 即使是理性函数的亚级, 理性函数的分类也可能包含无法比较的功能( 由具有无限投入的图解机器 ) 。 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( 由上述导导导导模式之一指定 ), 是可计算 $f$f( 等值) 和 确定性双向 双向 双向 双向 。 对于常规函数来说, 我们展示的是, 直向 直线性 和 直线性 函数 。