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 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-c) 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 PTime.
A version with full proofs can be found at https://arxiv.org/abs/2002.08203.
L. Exibard—Funded by a FRIA fellowship from the F.R.S.-FNRS.
E. Filiot—Research associate of F.R.S.-FNRS. Supported by the ARC Project Transform Fédération Wallonie-Bruxelles and the FNRS CDR J013116F; MIS F451019F projects.
P.-A. Reynier—Partly funded by the ANR projects DeLTA (ANR-16-CE40-0007) and Ticktac (ANR-18-CE40-0015).
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Exibard, L., Filiot, E., Reynier, PA. (2020). On Computability of Data Word Functions Defined by Transducers. In: Goubault-Larrecq, J., König, B. (eds) Foundations of Software Science and Computation Structures. FoSSaCS 2020. Lecture Notes in Computer Science(), vol 12077. Springer, Cham. https://doi.org/10.1007/978-3-030-45231-5_12
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