Oracle Computability and Turing Reducibility in the Calculus of Inductive Constructions

Yannick Forster, Dominik Kirst, Niklas Mück

Research output: Chapter in Book/Report/Conference proceedingConference contributionpeer-review


We develop synthetic notions of oracle computability and Turing reducibility in the Calculus of Inductive Constructions (CIC), the constructive type theory underlying the Coq proof assistant. As usual in synthetic approaches, we employ a definition of oracle computations based on meta-level functions rather than object-level models of computation, relying on the fact that in constructive systems such as CIC all definable functions are computable by construction. Such an approach lends itself well to machine-checked proofs, which we carry out in Coq. There is a tension in finding a good synthetic rendering of the higher-order notion of oracle computability. On the one hand, it has to be informative enough to prove central results, ensuring that all notions are faithfully captured. On the other hand, it has to be restricted enough to benefit from axioms for synthetic computability, which usually concern first-order objects. Drawing inspiration from a definition by Andrej Bauer based on continuous functions in the effective topos, we use a notion of sequential continuity to characterise valid oracle computations. As main technical results, we show that Turing reducibility forms an upper semilattice, transports decidability, and is strictly more expressive than truth-table reducibility, and prove that whenever both a predicate p and its complement are semi-decidable relative to an oracle q, then p Turing-reduces to q.

Original languageEnglish
Title of host publicationProgramming Languages and Systems - 21st Asian Symposium, APLAS 2023, Proceedings
EditorsChung-Kil Hur
PublisherSpringer Science and Business Media Deutschland GmbH
Number of pages27
ISBN (Print)9789819983100
StatePublished - 1 Jan 2023
Event21st Asian Symposium on Programming Languages and Systems, APLAS 2023 - Taipei, Taiwan, Province of China
Duration: 26 Nov 202329 Nov 2023

Publication series

NameLecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics)
Volume14405 LNCS
ISSN (Print)0302-9743
ISSN (Electronic)1611-3349


Conference21st Asian Symposium on Programming Languages and Systems, APLAS 2023
Country/TerritoryTaiwan, Province of China


  • Coq proof assistant
  • Logical foundations
  • Synthetic computability theory
  • Type theory

ASJC Scopus subject areas

  • Theoretical Computer Science
  • General Computer Science


Dive into the research topics of 'Oracle Computability and Turing Reducibility in the Calculus of Inductive Constructions'. Together they form a unique fingerprint.

Cite this