In this paper, we investigate the problem of synthesizing computable functions of infinite words over an infinite alphabet (data omega-words). The notion of computability is defined through Turing machines with infinite inputs which can produce the corresponding infinite outputs in the limit. We use non-deterministic transducers equipped with registers, an extension of register automata with outputs, to specify functions. Such transducers may not define functions but more generally relations of data omega-words, and we show that it is PSpace-complete to test whether a given transducer defines a function. Then, given a function defined by some register transducer, we show that it is decidable (and again, PSpace-complete) whether such function is computable. As for the known finite alphabet case, we show that computability and continuity coincide for functions defined by register transducers, and show how to decide continuity. We also define a subclass for which those problems are solvable in polynomial time.

中文翻译：

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关于由换能器定义的数据字函数的可计算性

在本文中，我们研究了在无限字母表（数据 omega-words）上合成无限单词的可计算函数的问题。可计算性的概念是通过具有无限输入的图灵机定义的，它可以在极限内产生相应的无限输出。我们使用配备寄存器的非确定性转换器，这是带有输出的寄存器自动机的扩展，来指定功能。这样的转换器可能不定义函数，而是更一般地定义数据 omega-words 的关系，并且我们证明测试给定的转换器是否定义了函数是 PSpace 完全的。然后，给定由某个寄存器转换器定义的函数，我们证明该函数是否可计算是可判定的（再次是 PSpace 完备的）。对于已知的有限字母表，我们证明了由寄存器传感器定义的函数的可计算性和连续性是一致的，并展示了如何确定连续性。我们还定义了一个可以在多项式时间内解决这些问题的子类。