In computability theory the theorem, written also as "smn-theorem" or "s-m-n theorem" (also called the translation lemma, parameter theorem, and the parameterization theorem) is a basic result about programming languages (and, more generally, Gödel numberings of the partial computable functionss) (Soare 1987, Rogers 1967). It was first proved by Stephen Cole Kleene (1943). The name in the original formulation of the theorem (see below).

, there exists a particular algorithm that accepts as input the source code free variables free variables, leaving the rest of the variables free.

Details

The basic form of the theorem applies to functions of two arguments (Nies 2009, p. of partial computable functions, there is a primitive recursive function, and their values are equal for any such combination. In other words, the following extensional equality:

:

.

Formal statement

, such that

.

. The result generalizes to any Turing-complete computing model.

Example

The following Lisp code implements s11 for Lisp. For example, evaluates to , where is a "fresh" symbol.

See also

• Currying

• Kleene's recursion theorem

• Partial evaluation

References

• (This is the reference that the 1989 edition of Odifreddi's "Classical Recursion Theory" gives on p. theorem.)

External links