Using the prover ANDP to simplify orthogonality

In the 1920s, Heyting attempted at axiomatizing constructive geometry. Recently, von Plato used di4erent concepts to axiomatize the geometry: he used 14 axioms to describe the axiomatization for apartness geometry. Then he added axioms A1 and A2 to his apartness geometry to get his a8ne geometry, then he added axioms O1, O2, O3 and O4 to the a8ne geometry to get orthogonality. In total, this gives 22 axioms. von Plato used four relations to describe the concept of orthogonality in O1, O2 and O4. That is, all the three relations of two lines, which are convergence, unorthogonality and di4erence, and the relation of a point and a line. ANDP is an automated natural deduction prover developed over the years at our institute. After doing a lot of experiments using ANDP, much shorter and more intuitive axioms were found for axioms O1, O2 and O4, respectively. For example, O2 can be replaced by one of its four conjuncts. This paper shows that it is enough to use two relations on lines, which are convergence and unorthogonality, to describe the concept of orthogonality. c © 2003 Elsevier B.V. All rights reserved.