### Abstract

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 ℝ, but ⊆ℝ an explicit ordered ring extension of ℤ that is dense in ℝ. 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.

Original language | English (US) |
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Title of host publication | Reliable Implementation of Real Number Algorithms |

Subtitle of host publication | Theory and Practice - International Seminar, Revised Papers |

Pages | 193-237 |

Number of pages | 45 |

DOIs | |

State | Published - Sep 9 2008 |

Event | International Seminar on Reliable Implementation of Real Number Algorithms: Theory and Practice - Dagstuhl Castle, Germany Duration: Jan 8 2006 → Jan 13 2006 |

### Publication series

Name | Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics) |
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Volume | 5045 LNCS |

ISSN (Print) | 0302-9743 |

ISSN (Electronic) | 1611-3349 |

### Other

Other | International Seminar on Reliable Implementation of Real Number Algorithms: Theory and Practice |
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Country | Germany |

City | Dagstuhl Castle |

Period | 1/8/06 → 1/13/06 |

### ASJC Scopus subject areas

- Theoretical Computer Science
- Computer Science(all)

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## Cite this

*Reliable Implementation of Real Number Algorithms: Theory and Practice - International Seminar, Revised Papers*(pp. 193-237). (Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics); Vol. 5045 LNCS). https://doi.org/10.1007/978-3-540-85521-7_12