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 functions) (Soare 1987, Rogers 1967). It was first proved by Stephen Cole Kleene (1943). The name ' 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 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.
The basic form of the theorem applies to functions of two arguments (Nies 2009, p. 6). Given a Gödel numbering 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 and 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:
More generally, for any , there exists a primitive recursive function of arguments that behaves as follows: for every Gödel number e of a partial computable function with arguments, and all values of x<sub>1</sub>, â¦, x<sub>m</sub>:
The function s described above can be taken to be .
Given arities and , for every Turing Machine of arity and for all possible values of inputs , there exists a Turing machine of arity , such that
Furthermore, there is a Turing machine that allows to be calculated from and ; it is denoted .
Informally, finds the Turing Machine that is the result of hardcoding the values of into . The result generalizes to any Turing-complete computing model.
The following Lisp code implements s<sub>11</sub> for Lisp.
For example, evaluates to , where is a "fresh" symbol.