In computability theory the S mn 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 computable functions) (Soare 1987, Rogers 1967).
It was first proved by Stephen Cole Kleene (1943).
The name S mn 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 effectively substitutes the values for the first m free variables, leaving the rest of the variables free.
The smn-theorem states that given a function of two arguments
which is computable, there exists a total and computable function such that
basically "fixing" the first argument of
It's like partially applying an argument to a function.
In other words, it addresses the idea of "parametrization" or "indexing" of computable functions.
It's like creating a simplified version of a function that takes an additional parameter (index) to mimic the behavior of a more complex function.
is designed to mimic the behavior of
Essentially, by selecting the right values for
behave like for a specific computation.
Instead of dealing with the complexity of
that captures the essence of the computation.
The basic form of the theorem applies to functions of two arguments (Nies 2009, p. 6).
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
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 m, n > 0, there exists a primitive recursive function
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, …, xm: The function s described above can be taken to be
Given arities m and n, for every Turing Machine
, there exists a Turing machine
of arity n, such that Furthermore, there is a Turing machine S that allows k to be calculated from x and y; it is denoted
Informally, S finds the Turing Machine
that is the result of hardcoding the values of y into
The result generalizes to any Turing-complete computing model.
This can also be extended to total computable functions as follows: Given a total computable function
There is also a simplified version of the same theorem (defined infact as "simplified smn-theorem", which basically uses a total computable function as index as follows: Let
There, there is a total computable function