### Abstract

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 describe specifications. Being non-deterministic, such transducers may not define functions but more generally relations of data $\omega$‑words. In order to increase the expressive power of these machines, we even allow guessing of arbitrary data values when updating their registers.

For functions over data $\omega$‑words, we identify a sufficient condition (the possibility of determining the next letter to be outputted, which we call next letter problem) under which computability (resp. uniform computability) and continuity (resp. uniform continuity) coincide.

We focus on two kinds of data domains: first, the general setting of oligomorphic data, which encompasses any data domain with equality, as well as the setting of rational numbers with linear order; and second, the set of natural numbers equipped with linear order. For both settings, we prove that functionality, i.e. determining whether the relation recognized by the transducer is actually a function, is decidable. We also show that the so called next letter problem is decidable, yielding equivalence between (uniform) continuity and (uniform) computability. Last, we provide characterizations of (uniform) continuity, which allow us to prove that these notions, and thus also (uniform) computability, are decidable. We even show that all these decision problems are PSpace-complete for $(\mathbb{N},\leq)$ and for a large class of oligomorphic data domains, including for instance $(\mathbb{Q},\leq)$.

Publication

*Logical Methods in Computer Science*

This work has been submitted to the LMCS Special Issue dedicated to FoSSaCS 2020, and has been accepted with minor revisions. The preprint is available on
arXiv.
Nathan Lhote joined us to extend the results presented in
On Computability of Data Word Functions Defined by Transducers (FoSSaCS 2020) in the following ways:

- Extension of our study of $(\mathbb{N},=)$ to the setting of
*oligomorphic* data domains. This includes $(\mathbb{Q},\leq)$ in particular.
- Study of the case of $(\mathbb{N},\leq)$, which is not oligomorphic.
- Characterisation of uniform continuity in the case of oligomorphic domains
^{1}.
These contributions are the topic of the
second part of my thesis.

To obtain those generalisations, we shifted to a more abstract presentation of the problem. The reader who is mainly interested in the $(\mathbb{N},=)$ case can consult the
conference version, where the proofs use more concrete objects, although they are arguably less elegant. They might also have a look at
Section 12.1 of my manuscript, where the presentation of the concepts and the proofs have been rewritten and detailed^{2}.