﻿ Savitch's theorem

# Savitch's theorem

The set of Savitch 's theorem or by Savitch is a set of complexity theory, which was proved in 1970 by Walter Savitch. He says that a solvable by a nondeterministic Turing machine with a certain space complexity problem can be solved on a deterministic Turing machine with a square barrier higher place.

A corollary of the theorem of Savitch is the equality of PSPACE and NPSPACE.

## Formal definition

Be a platzkonstruierbare function. Then:

## Idea of ​​proof

The proof is via a s- space- bounded nondeterministic Turing machine M. are configurations of the Turing machine M. By means of the predicate can be shown that for an input configuration can be converted into a maximum in steps. is the initial configuration of w and the accepting configuration. From the construction of M is an explicit constant c can be extracted, so that applies:

This can be calculated directly for and with the following scheme:

This can be translated into a deterministic program and the maximum space required to be estimated by. This we gain from an induction on i with a certain constant.

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