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Can one design a geometry engine? Can one design a geometry engine? On the (un)decidability of certain affine Euclidean geometries. (English) Zbl 1439.03033
Written for “practitioners of symbolic computation or automated theorem proving,” this is a survey of undecidability results for elementary (first-order) geometries, with particular emphasis on “affine incidence geometry, Hilbert-style Euclidean geometry, Wu’s orthogonal geometry and Origami geometry.” The author shows
$$\bullet$$ how the segment calculus allows for a transfer of the undecidability results known to hold for theories of fields to that of elementary geometries, $$\bullet$$ how the result by M. Ziegler [Enseign. Math., II. Sér. 28, 269–280 (1982; Zbl 0519.12018)], which states that every finitely axiomatized field theory that has $$\mathbb{R}$$ or $$\mathbb{C}$$ as a model is hereditarily undecidable, can be used to prove the undecidability of a plethora of finitely axiomatized elementary geometries, and
$$\bullet$$ that the consequences in the universal fragment of geometries over fields are decidable.
He also indicates that these results can be extended to projective, elliptic, hyperbolic, and higher-dimensional geometries.
##### MSC:
 03B25 Decidability of theories and sets of sentences 03D35 Undecidability and degrees of sets of sentences 51M05 Euclidean geometries (general) and generalizations
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