Smn theorem

In computability theory the S m
n  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 functions) (Soare 1987, Rogers 1967). It was first proved by Stephen Cole Kleene (1943). The name S m
n  comes from the occurrence of an 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 in essence substitutes the values for the first m 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. 6). Given a Gödel numbering $\phi$ of partial computable functions, there is a primitive recursive function s of two arguments with the following property: for every Gödel number e of a partial computable function f with two arguments, the expressions ${\phi }_{s(e,x)}(y)$ 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:

${\phi }_{s(e,x)}\simeq \lambda y.{\phi }_e(x,y).$

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 e of a partial computable function with m + n arguments, and all values of x₁, …, x_m:

${\phi }_{S_n^m(e,x_1,\dots ,x_m)}\simeq \lambda y_1,\dots ,y_n.{\phi }_e(x_1,\dots ,x_m,y_1,\dots ,y_n).$

The function s described above can be taken to be $S_1^1$.

Formal statement

Given arities m and n, for every Turing Machine ${\text{TM}}_x$ of arity $m+n$ and for all possible values of inputs $y_1,\dots ,y_m$, there exists a Turing machine ${\text{TM}}_k$ of arity n, such that

$\mathrm{\forall }{z}_1,\dots ,z_n:{\text{TM}}_x(y_1,\dots ,y_m,z_1,\dots ,z_n)={\text{TM}}_k(z_1,\dots ,z_n).$

Furthermore, there is a Turing machine S that allows k to be calculated from x and y; it is denoted $k=S_n^m(x,y_1,\dots ,y_m)$.

Informally, S finds the Turing Machine ${\text{TM}}_k$ that is the result of hardcoding the values of y into ${\text{TM}}_x$. The result generalizes to any Turing-complete computing model.

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)), where g42 is a "fresh" symbol.

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. S2CID 120517999.
• Kleene, S. C. (1938). "On Notations for Ordinal Numbers" (PDF). The Journal of Symbolic Logic. 3 (4): 150–155. doi:10.2307/2267778. JSTOR 2267778. S2CID 34314018. (This is the reference that the 1989 edition of Odifreddi's "Classical Recursion Theory" gives on p. 131 for the $S_n^m$ theorem.)
• Nies, A. (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.