Theory of Real Computation According to EGC

• Authors: Chee Yap
• Venue: Reliable Implementation of Real Number Algorithms: Theory and Practice
• Year: 2008

The Exact Geometric Computation (EGC) mode of computation has been developed over the last decade in response to the widespread problem of numerical non-robustness in geometric algorithms. Its technology has been encoded in libraries such as LEDA, CGAL and Core Library. The key feature of EGC is the necessity to decide zero in its computation. This paper addresses the problem of providing a foundation for the EGC mode of computation. This requires a theory of real computation that properly addresses the Zero Problem. The two current approaches to real computation are represented by the analytic school and algebraic school. We propose a variant of the analytic approach based on real approximation.To capture the issues of representation, we begin with a reworking of van der Waerden's idea of explicit rings and fields. We introduce explicit sets and explicit algebraic structures.Explicit rings serve as the foundation for real approximation: our starting point here is not 驴, but $\mathbb{F}\subseteq \mathbb{R}$, an explicit ordered ring extension of 驴 that is dense in 驴. We develop the approximability of real functions within standard Turing machine computability, and show its connection to the analytic approach.Current discussions of real computation fail to address issues at the intersection of continuous and discrete computation. An appropriate computational model for this purpose is obtained by extending Schönhage's pointer machines to support both algebraic and numerical computation.Finally, we propose a synthesis wherein both the algebraic and the analytic models coexist to play complementary roles. Many fundamental questions can now be posed in this setting, including transfer theorems connecting algebraic computability with approximability.