The synthesis problem asks to automatically generates, if it exists, an algorithm from a specification of correct input-output pairs. In this paper, we consider the synthesis of computable functions of infinite words, for a classical Turing computability notion over infinite inputs. We consider specifications which are rational relations of infinite words, i.e., specifications defined non-deterministic parity transducers. We prove that the synthesis problem of computable functions from rational specifications is undecidable. We provide an incomplete but sound reduction to some parity game, such that if Eve wins the game, then the rational specification is realizable by a computable function. We prove that this function is even computable by a deterministic two-way transducer. We provide a sufficient condition under which the latter game reduction is complete. This entails the decidability of the synthesis problem of computable functions, which we prove to be ExpTime-complete, for a large subclass of rational specifications, namely deterministic rational specifications. This subclass contains the class of automatic relations over infinite words, a yardstick in reactive synthesis.

翻译：合成问题要求从正确的输入输出对配的规格中自动生成一个算法, 如果存在的话。 在本文中, 我们考虑将可计算功能的无限字数合成为典型的图灵可计算性概念, 而不是无限输入。 我们考虑的规格是无限字数的合理关系, 即定义非确定性的非确定性对等转换器的规格。 我们证明, 从合理规格中计算可计算函数的合成问题是不可变的。 我们对某种对等游戏提供了不完整但合理的减法, 这样如果Eve胜过游戏, 那么理性的规格就能通过可计算函数实现。 我们证明, 这个函数甚至可以由确定性的双向转换器来计算。 我们为后者的减法提供了充分的条件。 这意味着, 计算函数的综合问题可以衰减, 我们证明, 计算时的合成是完全的, 对于一个大的亚级的合理规格, 即确定性理性规格, 即确定性理性的规格, 我们是一个不完全的。 这个子类包含一个自动关系类别, 超越无限单词, 一个反应合成的准度。