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Euclidean geometry in terms of automata theory. (English) Zbl 0678.68055
Summary: The two abstract automata \(GCM_ 0\) and \(A\)-GCM\({}_ 0\) are introduced. A \(GCM_ 0\) is a machine whose primitives are the basic euclidean operations with compass and ruler. It is shown tht some elementary geometric problems can indeed by solved by a \(GCM_ 0\)-algorithm. On the other hand, our definition of \(GCM_ 0\)-constructible functions is so restrictive that not even the perpendicular projection of any arbitrary point \(t\in {\mathbb{E}}^ 2\) onto the x-axis can be \(GCM_ 0\)-constructed. Therefore, the \(A\)-GCM\({}_ 0\) is introduced which has the additional capability to execute jumps under the condition that some x is in A. We shall give a general class of oracle sets A which yield a real extension of the class of constructible functions, and we shall consider another large class of oracles which do not. The last of our theorems deals with a uniform time bound of different constructions effected by nondeterministic \(GCM_ 0\)-operations.

MSC:
68Q45 Formal languages and automata
51M05 Euclidean geometries (general) and generalizations
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