Smn theorem - Wikipedia - Recent changes [en]

The smn-theorem states that given a function of two arguments <math>g(x,y)</math> which is [[Computable function|computable]], there exists a [[Total function|total]] and computable function such that <math>\phi_s(x)(y)=g(x,y)</math> basically "fixing" the first argument of <math>g</math>. It's like partially applying an argument to a function. This is generalized over <math>m,n</math> tuples for <math>x,y</math>. In other words, it addresses the idea of "parameterization" 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 smn-theorem states that given a function of two arguments <math>g(x,y)</math> which is [[Computable function|computable]], there exists a [[Total function|total]] and computable function such that <math>\phi_s(x)(y)=g(x,y)</math> basically "fixing" the first argument of <math>g</math>. It's like partially applying an argument to a function. This is generalized over <math>m,n</math> tuples for <math>x,y</math>. In other words, it addresses the idea of "parameterization" 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 <math>s_m^n</math> is designed to mimic the behavior of <math>\phi(x,y)</math> when given the appropriate parameters. Essentially, by selecting the right values for <math>m</math> and <math>n</math>, you can make <math>s</math> behave like for a specific computation. Instead of dealing with the complexity of <math>\phi(x,y)</math>, we can work with a simpler <math>s_m^n</math> that captures the essence of the computation.

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

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