Parametric Church's Thesis: Synthetic Computability Without Choice

  title={Parametric Church's Thesis: Synthetic Computability Without Choice},
  author={Yannick Forster},
  • Y. Forster
  • Published 16 December 2021
  • Philosophy
  • ArXiv
In synthetic computability, pioneered by Richman, Bridges, and Bauer, one develops computability theory without an explicit model of computation. This is enabled by assuming an axiom equivalent to postulating a function φ to be universal for the space N→N (CTφ, a consequence of the constructivist axiom CT), Markov’s principle, and at least the axiom of countable choice. Assuming CT and countable choice invalidates the law of excluded middle, thereby also invalidating classical intuitions… 
Towards a Mechanized Theory of Computation for Education
This project includes full proofs of results from a textbook, such as the undecidability of the halting problem and Rice’s theorem, and presents a simple and expressive calculus that allows for the essence of informal proofs of classic theorems in a mechanized setting.
Synthetic Turing Reducibility in CIC
A definition of Turing reducibility in synthetic computability carried out in CIC, the type theory underlying Coq, is discussed, allowing for a defined via continuous Turing functionals based on an idea of Bauer.


Church's thesis without tears
The purpose of this paper is to show how the main results of the Church-Markov-Turing theory of computable functions may quickly be derived and understood without recourse to the largely irrelevant theories of recursive functions, Markov algorithms, or Turing machines.
Consistency of the intensional level of the Minimalist Foundation with Church’s thesis and axiom of choice
It is shown that consistency with the formal Church’s thesis and the axiom of choice are satisfied by the intensional level of the two-level Minimalist Foundation, for short MF, completed in 2009 by the second author.
A Formal and Constructive Theory of Computation
This thesis presents a formal development of basic computability theory in constructive type theory on a minimal functional programming language obtained as a variant of the weak call-by-value lambda calculus and proves that termination for all arguments is neither acceptable nor co-acceptable.
Church's thesis and related axioms in Coq's type theory
The paper provides a partial answer to the question which axioms may preserve computational intuitions inherent to type theory, and which certainly do not, and is read as a broad survey ofAxioms in type theory.
An introduction to mathematical logic and type theory - to truth through proof
This volume will be of interest to mathematicians, computer scientists, and philosophers in universities, as well as to computer scientists in industry who wish to use higher-order logic for hardware and software specification and verification.
This chapter explores the limits of mechanical computation as defined by Turing machines and Chomsky’s hierarchy of formal grammars, and describes the schemes of primitive recursion and μ-recursion, which enable a concise, mathematical description of computable functions that is independent of any machine model.
Game semantics of Martin-Löf type theory, part III: its consistency with Church's thesis
This work proves consistency of intensional Martin-Lof type theory with formal Church's thesis (CT) by novel realizability a la game semantics, which is based on the author's previous work.
Formalizing computability theory via partial recursive functions
An extension to the $\mathtt{mathlib}$ library of the Lean theorem prover formalizing the foundations of computability theory is presented, which includes the construction of a universal partial recursive function and a proof of the undecidability of the halting problem.
Game Semantics for Martin-Löf Type Theory
A category with families of a novel variant of games is proposed, which induces a surjective and injective interpretation of the intensional variant of MLTT equipped with unit-, empty-, N-, dependent product, dependent sum and Id-types as well as the cumulative hierarchy of universes for the first time in the literature.