Semicomputable function

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In computability theory, a semicomputable function is a partial function  f : \mathbb{Q} \rightarrow \mathbb{R} that can be approximated either from above or from below by a computable function.

More precisely a partial function  f : \mathbb{Q} \rightarrow \mathbb{R} is upper semicomputable, meaning it can be approximated from above, if there exists a computable function  \phi(x,k) : \mathbb{Q} \times \mathbb{N} \rightarrow \mathbb{Q}, where x is the desired parameter for  f(x) and  k is the level of approximation, such that:

  •  \lim_{k \rightarrow \infty} \phi(x,k) = f(x)
  •  \forall k \in \mathbb{N} : \phi(x,k+1) \leq \phi(x,k)

Completely analogous a partial function  f : \mathbb{Q} \rightarrow \mathbb{R} is lower semicomputable iff  -f(x) is upper semicomputable or equivalently if there exists a computable function  \phi(x,k) such that

  •  \lim_{k \rightarrow \infty} \phi(x,k) = f(x)
  •  \forall k \in \mathbb{N} : \phi(x,k+1) \geq \phi(x,k)

If a partial function is both upper and lower semicomputable it is called computable.

See also[edit]

References[edit]

  • Ming Li and Paul Vitányi, An Introduction to Kolmogorov Complexity and Its Applications, pp 37–38, Springer, 1997.