 
Summary: CAT(0) IS AN ALGORITHMIC PROPERTY
MURRAY ELDER AND JON MCCAMMOND1
Abstract. In this article we give an explicit algorithm which will de
termine, in a discrete and computable way, whether a finite piecewise
Euclidean complex is nonpositively curved. In particular, given such
a complex we show how to define a boolean combination of polynomial
equations and inequalities in real variables, i.e. a real semialgebraic
set, which is empty if and only if the complex is nonpositively curved.
Once this equivalence has been shown, the main result follows from a
standard theorem in real algebraic geometry.
Contents
1. Polyhedral geometry 2
2. Comparison geometry 5
3. Galleries 7
4. Detecting piecewise geodesics 13
5. Detecting local geodesics 17
6. Coefficients 20
7. The Main Theorem 21
References 23
In this article we prove that for every finite collection of Euclidean poly
