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Synthesizing Computable Functions from Rational Specifications over Infinite Words

Authors Emmanuel Filiot , Sarah Winter

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Emmanuel Filiot
  • Université libre de Bruxelles, Brussels, Belgium
Sarah Winter
  • Université libre de Bruxelles, Brussels, Belgium


The authors want to thank Nathan Lhote for a discussion on two-way transducers and Martin Zimmermann for pointing out a reference.

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Emmanuel Filiot and Sarah Winter. Synthesizing Computable Functions from Rational Specifications over Infinite Words. In 41st IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2021). Leibniz International Proceedings in Informatics (LIPIcs), Volume 213, pp. 43:1-43:16, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2021)


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 by 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 proved 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.

Subject Classification

ACM Subject Classification
  • Theory of computation → Logic and verification
  • Theory of computation → Automata over infinite objects
  • Theory of computation → Transducers
  • uniformization
  • synthesis
  • transducers
  • continuity
  • computability


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