Computability and λ-Definability

@article{Turing1937ComputabilityA,
  title={Computability and $\lambda$-Definability},
  author={A. Turing},
  journal={J. Symb. Log.},
  year={1937},
  volume={2},
  pages={153-163}
}
  • A. Turing
  • Published 1937
  • Mathematics, Computer Science
  • J. Symb. Log.
Several definitions have been given to express an exact meaning corresponding to the intuitive idea of ‘effective calculability’ as applied for instance to functions of positive integers. The purpose of the present paper is to show that the computable functions introduced by the author are identical with the λ-definable functions of Church and the general recursive functions due to Herbrand and Godel and developed by Kleene. It is shown that every λ-definable function is computable and that… Expand
An Overview of the Computably Enumerable Sets
  • R. Soare
  • Mathematics, Computer Science
  • Handbook of Computability Theory
  • 1999
This chapter summarizes some of the results of the algebraic structure of the computably enumerable (c.e.) sets since 1987, when the subject was covered in Soare. In addition to defining computableExpand
Some Applications of Recursive Functionals to the Foundations of Mathematics and Physics
We consider two applications of recursive functionals. The first application concerns Godel’s theory T , which provides a rudimentary foundation for the formalization of mathematics. T can beExpand
The imperative and functional programming paradigm
TLDR
Major advantages of the functional programming paradigm over the imperative one, that are applicable, provided one is has the mental capacity to explicitly deal with simple abstractions, are presented. Expand
Some Results on Classical Semantics and Polymorphic Types
In the first chapter we consider the simply typed λ-calculus over one ground type with a discriminator δ which distinguishes terms, augmented additionally with an existential quantifier and aExpand
Algebraic structures for the lambda calculus and the propositional logic
Part I Among the unsolvable terms of the lambda calculus, the mute ones are those having the highest degree of undefinedness. For each natural number n ≥ 1, we introduce two infinite and recursiveExpand
Consistency, Turing Computability and Gödel’s First Incompleteness Theorem
TLDR
It is argued that the existence of these algorithms, when conjoined with Gödel’s results and accepted theorems of recursion theory, does provide the basis for an apparent paradox, which illuminates the truth status of axioms in formal models of programs and Turing machines. Expand
A Natural Axiomatization of Computability and Proof of Church's Thesis
TLDR
It is shown that augmenting those postulates about algorithmic computation with an additional requirement regarding basic operations gives a natural axiomatization of computability and a proof of Church's Thesis, as Gödel and others suggested may be possible. Expand
On Undefined and Meaningless in Lambda Definability
TLDR
A natural modification, strict lambda-definability, is obtained, which allows for a Barendregt style of proof in which the representation of composition is truly the composition of representations. Expand
The History and Concept of Computability
  • R. Soare
  • Computer Science, Mathematics
  • Handbook of Computability Theory
  • 1999
TLDR
The chapter considers the Church–Turing thesis that the intuitively computable functions coincide with the formally computable ones and considers using the thesis as a definition. Expand
Global semantic typing for inductive and coinductive computing
TLDR
A Canonicity Theorem is proved, stating that the denotational semantics of an equational program P, understood operationally, has type \tau over the canonical model iff P is understood as a formula, and every intrinsic theory is interpretable in a conservative extension of first-order arithmetic. Expand
...
1
2
3
4
5
...

References

SHOWING 1-6 OF 6 REFERENCES
Some properties of conversion
Our purpose is to establish the properties of conversion which are expressed in Theorems 1 and 2 below. We shall consider first conversion defined by Church's Rules I, II, IIIt and shall then extendExpand
On Computable Numbers, with an Application to the Entscheidungsproblem
1. Computing machines. 2. Definitions. Automatic machines. Computing machines. Circle and circle-free numbers. Computable sequences and numbers. 3. Examples of computing machines. 4. AbbreviatedExpand
An Unsolvable Problem of Elementary Number Theory
Terms and Conditions of Use provides, in part, that unless you have obtained prior permission, you may not download an entire issue of a journal or multiple copies of articles, and you may useExpand
$\lambda$-definability and recursiveness