First Steps in Synthetic Computability Theory

  title={First Steps in Synthetic Computability Theory},
  author={A. Bauer},
  • A. Bauer
  • Published in MFPS 1 May 2006
  • Computer Science
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.
An Analysis of Tennenbaum's Theorem in Constructive Type Theory
Tennenbaum’s theorem states that the only countable model of Peano arithmetic (PA) with computable arithmetical operations is the standard model of natural numbers. In this paper, we use constructive
Partial Elements and Recursion via Dominances in Univalent Type Theory
We begin by revisiting partiality in univalent type theory via the notion of dominance. We then perform first steps in constructive computability theory, discussing the consequences of working with
Parametric Church's Thesis: Synthetic Computability Without Choice
This work introduces various parametric strengthenings of CTφ, which are equivalent to assuming CT φ and an S n operator for φ like in the S n theorem, and explains the novel axioms and proofs of Rice’s theorem.
On fixed-point theorems in synthetic computability
Abstract Lawvere’s fixed point theorem captures the essence of diagonalization arguments. Cantor’s theorem, Gödel’s incompleteness theorem, and Tarski’s undefinability of truth are all instances of
Partial functions and recursion in univalent type theory
This work investigates partial functions and computability theory from within a constructive, univalent type theory, using the notion of dominance, which is used in synthetic domain theory to discuss classes of partial maps, and suggests an alternative notion of partial function the authors call disciplined maps.
An intrinsic treatment of ubiquity phenomena in higher-order computability models ( Part I , draft version )
We introduce some simple conditions on typed partial combinatory algebras (viewed as models of higher-order computability) which suffice for an axiomatic development of some non-trivial computability
Computational Back-And-Forth Arguments in Constructive Type Theory
The back-and-forth method is a well-known technique to establish isomorphisms of countable structures. In this proof pearl, we formalise this method abstractly in the framework of constructive type
A Brief Critique of Pure Hypercomputation
Hypercomputation—the hypothesis that Turing-incomputable objects can be computed through infinitary means—is ineffective, as the unsolvability of the halting problem for Turing machines depends just
On synthetic undecidability in Coq, with an application to the Entscheidungsproblem
Developing a basic framework for synthetic computability theory in Coq, this work proves the equivalence of Post's theorem with Markov's principle and provides a convenient technique for establishing the enumerability of inductive predicates such as the considered proof systems and PCP.


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.
Axioms for Computation Theories-First Draft
The theory of functions and sets of natural numbers
Recursiveness and Computability. Induction. Systems of Equations. Arithmetical Formal Systems. Turing Machines. Flowcharts. Functions as Rules. Arithmetization. Church's Thesis. Basic Recursion
Total Sets and Objects in Domain Theory
Practical Foundations of Mathematics
  • P. Taylor
  • Philosophy
    Cambridge studies in advanced mathematics
  • 1999
The aim of the book is to exhibit and study the mathematical principles behind logic and induction as needed and used for the formalisation of (the main parts of) Mathematics and Computer Science.
On effective topological spaces
  • D. Spreen
  • Mathematics
    Journal of Symbolic Logic
  • 1998
Abstract Starting with D. Scott's work on the mathematical foundations of programming language semantics, interest in topology has grown up in theoretical computer science, under the slogan ‘open
Axiomatic Recursive Function Theory
The Effective Topos