## 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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### References:

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