S^m_n 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 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 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 ${\displaystyle g(x,y)}${\displaystyle g(x,y)} which is computable, there exists a total and computable function such that ${\displaystyle \phi _{s}(x)(y)=g(x,y)}${\displaystyle \phi _{s}(x)(y)=g(x,y)} basically "fixing" the first argument of ${\displaystyle g}${\displaystyle g}. It's like partially applying an argument to a function. This is generalized over ${\displaystyle m,n}${\displaystyle m,n} tuples for ${\displaystyle x,y}${\displaystyle x,y}. 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.

The function ${\displaystyle s_{m}^{n}}${\displaystyle s_{m}^{n}} is designed to mimic the behavior of ${\displaystyle \phi (x,y)}${\displaystyle \phi (x,y)} when given the appropriate parameters. Essentially, by selecting the right values for ${\displaystyle m}${\displaystyle m} and ${\displaystyle n}${\displaystyle n}, you can make ${\displaystyle s}${\displaystyle s} behave like for a specific computation. Instead of dealing with the complexity of ${\displaystyle \phi (x,y)}${\displaystyle \phi (x,y)}, we can work with a simpler ${\displaystyle s_{m}^{n}}${\displaystyle s_{m}^{n}} that captures the essence of the computation.

The basic form of the theorem applies to functions of two arguments (Nies 2009, p. 6). Given a Gödel numbering ${\displaystyle \varphi }${\displaystyle \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 ${\displaystyle \varphi _{s(p,x)}(y)}${\displaystyle \varphi _{s(p,x)}(y)} and ${\displaystyle f(x,y)}${\displaystyle 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:

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

More generally, for any m, n > 0, there exists a primitive recursive function ${\displaystyle S_{n}^{m}}${\displaystyle 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:

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

Given arities m and n, for every Turing Machine ${\displaystyle {\text{TM}}_{x}}${\displaystyle {\text{TM}}_{x}} of arity ${\displaystyle m+n}${\displaystyle m+n} and for all possible values of inputs ${\displaystyle y_{1},\dots ,y_{m}}${\displaystyle y_{1},\dots ,y_{m}}, there exists a Turing machine ${\displaystyle {\text{TM}}_{k}}${\displaystyle {\text{TM}}_{k}} of arity n, such that

${\displaystyle \forall z_{1},\dots ,z_{n}:{\text{TM}}_{x}(y_{1},\dots ,y_{m},z_{1},\dots ,z_{n})={\text{TM}}_{k}(z_{1},\dots ,z_{n}).}${\displaystyle \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 ${\displaystyle k=S_{n}^{m}(x,y_{1},\dots ,y_{m})}${\displaystyle k=S_{n}^{m}(x,y_{1},\dots ,y_{m})}.

Informally, S finds the Turing Machine ${\displaystyle {\text{TM}}_{k}}${\displaystyle {\text{TM}}_{k}} that is the result of hardcoding the values of y into ${\displaystyle {\text{TM}}_{x}}${\displaystyle {\text{TM}}_{x}}. 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 ${\displaystyle s_{m,n}:\mathbb {N} ^{m+1}\rightarrow \mathbb {N} }${\displaystyle s_{m,n}:\mathbb {N} ^{m+1}\rightarrow \mathbb {N} } and ${\displaystyle m,n\geq 1}${\displaystyle m,n\geq 1} such that ${\displaystyle \forall {\vec {x}}\in \mathbb {N} ^{m},\forall {\vec {y}}\in \mathbb {N} ^{n}}${\displaystyle \forall {\vec {x}}\in \mathbb {N} ^{m},\forall {\vec {y}}\in \mathbb {N} ^{n}}, ${\displaystyle \forall e\in \mathbb {N} }${\displaystyle \forall e\in \mathbb {N} }:

${\displaystyle \phi _{e}^{m+n}({\vec {x}},{\vec {y}})=\phi _{s_{m,n}{(e,{\vec {x}})}}^{n}({\vec {y}})}${\displaystyle \phi _{e}^{m+n}({\vec {x}},{\vec {y}})=\phi _{s_{m,n}{(e,{\vec {x}})}}^{n}({\vec {y}})}

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 ${\displaystyle f:\mathbb {N} ^{n+m}\rightarrow \mathbb {N} }${\displaystyle f:\mathbb {N} ^{n+m}\rightarrow \mathbb {N} } be a computable function. There, there is a total computable function ${\displaystyle s:\mathbb {N} ^{m}\rightarrow \mathbb {N} }${\displaystyle s:\mathbb {N} ^{m}\rightarrow \mathbb {N} } such that ${\displaystyle \forall x\in \mathbb {N} ^{m}}${\displaystyle \forall x\in \mathbb {N} ^{m}}, ${\displaystyle {\vec {y}}\in \mathbb {N} ^{n}}${\displaystyle {\vec {y}}\in \mathbb {N} ^{n}}:

${\displaystyle f({\vec {x}},{\vec {y}})=\phi _{S({\vec {x}})}^{(n)}({\vec {y}})}${\displaystyle f({\vec {x}},{\vec {y}})=\phi _{S({\vec {x}})}^{(n)}({\vec {y}})}

The following Lisp code implements s₁₁ for Lisp.

(defuns11(fx)
(let((y(gensym)))
(list'lambda(listy)(listfxy))))

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

• Currying
• Kleene's recursion theorem
• Partial evaluation

• 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 ${\displaystyle S_{n}^{m}}${\displaystyle 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.

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

• Computability theory
• Theorems in theory of computation

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