S-M-N Theorem

Overview

The S-M-N theorem, also known as the Iteration theorem or the Parameterization theorem, is a pivotal result in the theory of computable functions. It establishes a fundamental property of computable functions related to their representation and manipulation.

Key Concepts

At its core, the theorem addresses how we can take a computable function that takes multiple arguments and obtain a computable function that takes a single argument (the index) and produces the same result as the original function with specific inputs.

Deep Dive

Formally, for any computable function $f(x, y)$ with two arguments, there exists a computable function $g(x)$ such that $g(x)$ computes $f(e, x)$ for some constant $e$. This constant $e$ effectively acts as an index or a “program” for the function $f$. The theorem guarantees that such an index always exists and can be computed.

More generally, for a computable function $f(x_1, …, x_n, y)$ with $n+1$ arguments, there exists a computable function $g(x_1, …, x_n)$ such that for any $x_1, …, x_n$, $g(x_1, …, x_n)$ computes $f(e, x_1, …, x_n)$ for some constant $e$.

Applications

The S-M-N theorem is crucial for:

• Understanding computability: It provides a formal basis for the Church-Turing thesis.
• Programming language theory: It underpins concepts like partial evaluation and compiler construction.
• Recursion theory: It’s fundamental to proving many other theorems in the field.

Challenges & Misconceptions

A common misconception is that the theorem implies a fixed, universal program for all functions. Instead, it shows that for a given function, we can find a specific program (index) that behaves like it. The index $e$ itself might not be easily interpretable or directly related to a human-readable program.

FAQs

What does ‘S-M-N’ stand for? It refers to the three mathematicians Kleene, Post, and Rogers who independently proved versions of the theorem, often associated with the names Shoenfield, Myhill, and Shepherdson.

How does it relate to Gödel numbering? Gödel numbering assigns unique numbers (indices) to formulas and proofs, and the S-M-N theorem provides a computable way to associate these indices with functions.