# 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} }

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

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#### References

SHOWING 1-6 OF 6 REFERENCES

Some properties of conversion

- Mathematics
- 1936

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 extend… Expand

On Computable Numbers, with an Application to the Entscheidungsproblem

- Mathematics
- 1937

1. Computing machines. 2. Definitions. Automatic machines. Computing machines. Circle and circle-free numbers. Computable sequences and numbers. 3. Examples of computing machines. 4. Abbreviated… Expand

An Unsolvable Problem of Elementary Number Theory

- Mathematics
- 1936

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 use… Expand