README.md

Synthetic Computability Theory in Coq

Meta

• Author(s):
  • Yannick Forster
  • Felix Jahn
  • Dominik Kirst
  • Fabian Kunze
  • Nils Lauermann
  • Niklas Mück
• Maintainer:
  • Yannick Forster (@yforster)
• License: MIT License
• Compatible Coq versions: 8.17
• Additional dependencies:
  • the stdpp library
  • optionally, the Coq Library of Undecidability Proofs
• Coq namespace: SyntheticComputability
• Related publication(s):
  • Church’s Thesis and Related Axioms in Coq’s Type Theory doi:10.4230/LIPIcs.CSL.2021.21
  • Parametric Church’s Thesis: Synthetic Computability Without Choice doi:10.1007/978-3-030-93100-1_6
  • Computability in Constructive Type Theory doi:10.22028/D291-35758
  • Synthetic Kolmogorov Complexity in Coq doi:10.4230/LIPIcs.ITP.2022.12
  • Constructive and Synthetic Reducibility Degrees: Post's Problem for Many-One and Truth-Table Reducibility in Coq doi:10.4230/LIPIcs.CSL.2023.21
  • A Computational Cantor-Bernstein and Myhill's Isomorphism Theorem in Constructive Type Theory (Proof Pearl) doi:10.1145/3573105.3575690
  • Oracle Computability and Turing Reducibility in the Calculus of Inductive Constructions doi:10.48550/arXiv.2307.15543

Description

This library contains results on synthetic computability theory.

• Equivalence proofs for various axioms of synthetic computability in Axioms/Equivalence.v
• Rice's theorem in Basic/Rice.v
• Myhill's isomorphism theorem in Basic/Myhill.v
• The existence of simple and hypersimple predicates in ReducibilityDegrees.summary_reducibility_degrees.v
• A proof that nonrandom numbers defined via Kolmogorov Complexity form a simple predicate in KolmogorovComplexity/Kolmogorov_gen.v
• A definition of oracle computability and Turing reducibility in TuringReducibility/OracleComputability.v
• A proof of Post's theorem (PT) in TuringReducibility/SemiDec.v
• A proof of Post's theorem about the arithmetical hierarchy in PostsTheorem/PostsTheorem.v
• A proof of the Kleene-Post theorem in PostsTheorem/KleenePostTheorem.v

Installation

opam switch create coq-synthetic-computability --packages=ocaml-variants.4.14.0+options,ocaml-option-flambda
eval $(opam env)
opam repo add coq-released https://coq.inria.fr/opam/released
opam install coq.8.17.0 coq-equations.1.3+8.17 coq-stdpp.1.8.0
cd theories
make
make install

To build the part of the development relying on models of computation, in addition you have to

opam install coq-library-undecidability.1.1+8.17
make models
make install-models