Computability of Data-Word Transductions over Different Data Domains

In this paper, we investigate the problem of synthesizing computable functions infinite words over an alphabet (data $\omega$-words). The notion computability is defined through Turing machines with inputs which can produce corresponding outputs in limit. We use non-deterministic transducers equipped registers, extension register automata outputs, to describe specifications. Being non-deterministic, such may not define but more generally relations data $\omega$-words. order increase expressive power these machines, even allow guessing arbitrary values when updating their registers. For $\omega$-words, identify a sufficient condition (the possibility determining next letter be outputted, call problem) under (resp. uniform computability) and continuity continuity) coincide. focus on two kinds domains: first, general setting oligomorphic data, encompasses any domain equality, as well rational numbers linear order; second, set natural order. both settings, prove that functionality, i.e. whether relation recognized by transducer actually function, decidable. also show so-called decidable, yielding equivalence between (uniform) computability. Last, provide characterizations continuity, us notions, thus computability, are all decision problems PSpace-complete for $(\mathbb{N},<)$ large class domains, including instance $(\mathbb{Q},<)$.