Synthesis of Computable Regular Functions of Infinite Words

Abstract

Regular functions from infinite words to infinite words can be equivalently specified by MSO-transducers, streaming $\omega$-string transducers as well as deterministic two-way transducers with look-ahead. In their one-way restriction, the latter transducers define the class of rational functions. Even though regular functions are robustly characterised by several finite-state devices, even the subclass of rational functions may contain functions which are not computable (by a Turing machine with infinite input). This paper proposes a decision procedure for the following synthesis problem: given a regular function $f$ (equivalently specified by one of the aforementioned transducer model), is $f$ computable and if it is, synthesize a Turing machine computing it. For regular functions, we show that computability is equivalent to continuity, and therefore the problem boils down to deciding continuity. We establish a generic characterisation of continuity for functions preserving regular languages under inverse image (such as regular functions). We exploit this characterisation to show the decidability of continuity (and hence computability) of rational and regular functions. For rational functions, we show that this can be done in $\mathsf{NLogSpace}$ (it was already known to be in $\mathsf{PTime}$ by Prieur). In a similar fashion, we also effectively characterise uniform continuity of regular functions, and relate it to the notion of uniform computability, which offers stronger efficiency guarantees.

Synthesizing Computable Functions from Rational Specifications Over Infinite Words

“…This is not difficult to see when comparing the definitions of computable and continuous functions. The converse does not hold because the continuity definition does not have any computability requirements (see [12] for a counter-example). However, regarding synthesis, the two notions coincide: E. Filiot and S. Winter 43:15 ▶ Theorem 17.…”

“…First, in [12], the synthesis of computable functions has been shown to be decidable in the particular case of functional relations, i.e., graphs of functions. The main contribution of [12] is to prove that checking whether a function represented by a non-deterministic two-way transducer is computable is decidable, and that computability coincides with continuity (for the Cantor distance) for this large class of functions. The techniques of [12] are different to ours, e.g., games are not needed because output symbols functionally depends on input ones, even in the specification, so, there is not choice to be made by Eve.…”

“…The main contribution of [12] is to prove that checking whether a function represented by a non-deterministic two-way transducer is computable is decidable, and that computability coincides with continuity (for the Cantor distance) for this large class of functions. The techniques of [12] are different to ours, e.g., games are not needed because output symbols functionally depends on input ones, even in the specification, so, there is not choice to be made by Eve.…”