General recursive functions of natural numbers

@article{Kleene1936GeneralRF,
  title={General recursive functions of natural numbers},
  author={Stephen Cole Kleene},
  journal={Mathematische Annalen},
  year={1936},
  volume={112},
  pages={727-742}
}
  • S. Kleene
  • Published 1 December 1936
  • Mathematics
  • Mathematische Annalen
View on Springer
496 Citations
Highly Influential Citations
26
Background Citations
100
Methods Citations
41
Results Citations
3

496 Citations

A recursion theoretic foundation of computation over real numbers

We define a class of computable functions over real numbers using functional schemes similar to the class of primitive and partial recursive functions defined by Gödel (1931, 1934) and Kleene…

The History and Concept of Computability

  • R. Soare
  • Philosophy
    Handbook of Computability Theory
  • 1999

A SCHEMATIC DEFINITION OF QUANTUM POLYNOMIAL TIME COMPUTABILITY

  • T. Yamakami
  • Computer Science
    The Journal of Symbolic Logic
  • 2020
TLDR
A new, schematic definition of quantum functions mapping finite-dimensional Hilbert spaces to themselves, which avoids the cumbersome introduction of the well-formedness condition imposed on a quantum Turing machine model as well as of the uniformity condition necessary for a quantum circuit model.

Max Dehn, Axel Thue, and the Undecidable

The word problem for finitely presented groups and semigroups is a famous problem in combinatorial group theory. This question originally came up independently in topology and mathematical logic. As…

Where are the data?

  • J. Reich
  • Computer Science
    Nature Structural &Molecular Biology
  • 2016
TLDR
It is argued that the proposed data concept matches the concept of characteristics (Merkmale) of the automation industry and is mathematically conceptualized as typed information based on the two concepts of information and computable functionality.

Ju n 20 14 Probabilistic Recursion Theory and Implicit Complexity ∗

TLDR
It is shown that probabilistic computable functions can be characterized by a natural generalization of Church and Kleene's partial recursive functions, and the obtained algebra can be restricted so as to capture the notion of polytime sampleable distributions, a key concept in average-case complexity and cryptography.

Monoidal computer II: Normal complexity by string diagrams

TLDR
This formalization brings to the foreground the concept of normal complexity measures, which allow decompositions akin to Kleene’s normal form, where evaluating the complexity of a program does not require substantially more resources than evaluating the program itself.

Naming and Diagonalization, from Cantor to Gödel to Kleene

TLDR
A historical reconstruction of the way Godel probably derived his proof from Cantor's diagonalization, through the semantic version of Richard, and how Kleene's recursion theorem is obtained along the same lines is shown.

Diagonalisation and Church's Thesis: Kleene's Homework

In this paper we will discuss the active part played by certain diagonal arguments in the genesis of computability theory. 1 In some cases it is enough to assume the enumerability of Y while in…

Primitive Rewriting

TLDR
Some undecidability results for “primitive” term rewriting systems, which encode primitive-recursive definitions, are presented in the manner suggested by Klop and some results for orthogonal and non-orthogonal rewriting are reprove by applying standard results in recursion theory.
...

References

SHOWING 1-3 OF 3 REFERENCES

Über formal unentscheidbare Sätze der Principia Mathematica und verwandter Systeme I

Über den Zusammenhang der verschiedenen Begriffe der rekursiven Funktion

Zum Hilbertschen Aufbau der reellen Zahlen

Um den Beweis fiir die yon Cantor aufgestellte Vermutung zu e~bringen, dal~ sich die Menge der ree|len Zahlen, d. h. der zaMentheoretischen I~unktionen, mi~ Hilfe der Zahlen de~ zweiten Zahlklasse…