smn theorem

In computability theory the s_mn theorem, (also called the translation lemma, parameter theorem, or parameterization theorem) is a basic result about programming languages (and, more generally, Gödel numberings of the computable functions) (Soare 1987, Rogers 1967). It was first proved by Stephen Cole Kleene (Kleene 1943). The name "s_mn" comes from the occurrence of a s with subscript n and superscript m in the original formulation of the theorem (see below).

In practical terms, the theorem says that for a given programming language and positive integers m and n, there exists a particular algorithm that accepts as input the source code of a program with m + n free variables, together with m values. This algorithm generates source code that effectively substitutes the values for the first m free variables, leaving the rest free.

Details

The basic form of the theorem applies to functions of two arguments (Nies 2009, p. 6). Given a Gödel numbering $\varphi$ of recursive functions, there is a primitive recursive function s of two arguments with the following property: for every Gödel number p of a partial computable function f with two arguments, the expressions $\varphi _{s(p,x)}(y)$ and $f(x,y)$ are defined for the same combinations of natural numbers x and y, and their values are equal for any such combination. In other words, the following extensional equality of functions holds for every x:

$\varphi _{s(p,x)}\simeq \lambda y.\varphi _{p}(x,y).\,$

More generally, for any m, n > 0, there exists a primitive recursive function $s_{n}^{m}$ of m + 1 arguments that behaves as follows: for every Gödel number p of a partial computable function with m + n arguments, and all values of x₁,…,x_m:

$\varphi _{s_{n}^{m}(p,x_{1},\dots ,x_{m})}\simeq \lambda y_{1},\dots ,y_{n}.\varphi _{p}(x_{1},\dots ,x_{m},y_{1},\dots ,y_{n}).\,$

The function s described above can be taken to be $s_{1}^{1}$.

Example

The following Lisp code implements s₁₁ for Lisp.

 (defun s11 (f x)
   (let ((y (gensym)))
     (list 'lambda (list y) (list f x y))))

For example, (s11 '(lambda (x y) (+ x y)) 3) evaluates to (lambda (g42) ((lambda (x y) (+ x y)) 3 g42)).

See also

• Currying
• Kleene's recursion theorem
• Partial evaluation

References

• Kleene, S. C. (1936). "General recursive functions of natural numbers". Mathematische Annalen. 112 (1): 727–742. doi:10.1007/BF01565439.
• Stephen Cole Kleene (1938). "On Notations for Ordinal Numbers" (PDF). The Journal of Symbolic Logic. 3: 150–155. (This is the reference that the 1989 edition of Odifreddi's "Classical Recursion Theory" gives on p.131 for the s_mn theorem.)
• Nies, André (2009). Computability and randomness. Oxford Logic Guides. Vol. 51. Oxford: Oxford University Press. ISBN 978-0-19-923076-1. Zbl 1169.03034.
• Odifreddi, P. (1999). Classical Recursion Theory. North-Holland. ISBN 0-444-87295-7.
• Rogers, H. (1987) [1967]. The Theory of Recursive Functions and Effective Computability. First MIT press paperback edition. ISBN 0-262-68052-1.
• Soare, R. (1987). Recursively enumerable sets and degrees. Perspectives in Mathematical Logic. Springer-Verlag. ISBN 3-540-15299-7.

External links

• Weisstein, Eric W. "Kleene's s-m-n Theorem". MathWorld.