Synthesizing Computable Functions from Rational Specifications over Infinite Words

TL;DR

This paper investigates the synthesis of computable functions from rational specifications over infinite words, establishing undecidability in general but identifying decidable subclasses with practical implications in reactive synthesis.

Contribution

It proves undecidability of the general synthesis problem, introduces a game-based reduction, and identifies a large decidable subclass for rational specifications.

Findings

Synthesis of computable functions from rational specifications is undecidable in general.

Decidability is achieved for deterministic rational specifications, including automatic relations.

The synthesis problem in this subclass is ExpTime-complete.

Abstract

The synthesis problem asks to automatically generate, if it exists, an algorithm from a specification of correct input-output pairs. In this paper, we consider the synthesis of computable functions of infinite words, for a classical Turing computability notion over infinite inputs. We consider specifications which are rational relations of infinite words, i.e., specifications defined non-deterministic parity transducers. We prove that the synthesis problem of computable functions from rational specifications is undecidable. We provide an incomplete but sound reduction to some parity game, such that if Eve wins the game, then the rational specification is realizable by a computable function. We prove that this function is even computable by a deterministic two-way transducer. We provide a sufficient condition under which the latter game reduction is complete. This entails the decidability of the synthesis problem of computable functions, which we prove to be ExpTime-complete, for a large subclass of rational specifications, namely deterministic rational specifications. This subclass contains the class of automatic relations over infinite words, a yardstick in reactive synthesis.