Compression theorem

In computational complexity theory the compression theorem is an important theorem about the complexity of computable functions.

The theorem states that there exists no largest complexity class, with computable boundary, which contains all computable functions.

Given a Gödel numbering ${\displaystyle \varphi }$ of the computable functions and a Blum complexity measure ${\displaystyle \Phi }$ where a complexity class for a boundary function ${\displaystyle f}$ is defined as

Then there exists a total computable function ${\displaystyle f}$ so that for all ${\displaystyle i}$

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