The fragment of elementary plane Euclidean geometry based on perpendicularity alone with complexity PSPACE-complete

TL;DR

This paper presents a complete axiomatization of elementary plane Euclidean geometry using only perpendicularity, proving it is decidable with PSPACE-complete complexity, contrasting with the exponential complexity of the full theory.

Contribution

It introduces a new axiomatization based solely on perpendicularity and establishes its decidability and PSPACE-completeness, a novel result in geometric logic.

Findings

The theory is not finitely axiomatizable.

The theory is decidable.

Complexity is PSPACE-complete.

Abstract

A. Tarski uses in his system for the elementary geometry only the primitive concept of point, and the two primitive relations betweenness and equidistance. Another approach is the relations to be on lines instead of points. W. Schwabh\"auser and L. Szczerba showed that perpendicularity together with the ternary relation of co-punctuality are sufficient for dimension two, i.e. they may be used as a system of primitive relations for elementary plane Euclidean geometry. In this paper we give a complete axiomatization for the fragment of elementary plane Euclidean geometry based on perpendicularity alone. We show that this theory is not finitely axiomatizable, it is decidable and the complexity is PSPACE-complete. In contrast the complexity of elementary plane Euclidean geometry is exponential.