Computability

About: Computability is a research topic. Over the lifetime, 2829 publications have been published within this topic receiving 85162 citations.

Theory of Real Computation According to EGC

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 ?, but $\mathbb{F}\subseteq \mathbb{R}$, an explicit ordered ring extension of ? that is dense in ? 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 Schonhage'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

Computability Theory: An Introduction to Recursion Theory

Abstract: Computability Theory: An Introduction to Recursion Theory, provides a concise, comprehensive, and authoritative introduction to contemporary computability theory, techniques, and results. The basic concepts and techniques of computability theory are placed in their historical, philosophical and logical context. This presentation is characterized by an unusual breadth of coverage and the inclusion of advanced topics not to be found elsewhere in the literature at this level. The text includes both the standard material for a first course in computability and more advanced looks at degree structures, forcing, priority methods, and determinacy. The final chapter explores a variety of computability applications to mathematics and science. Computability Theory is an invaluable text, reference, and guide to the direction of current research in the field. Nowhere else will you find the techniques and results of this beautiful and basic subject brought alive in such an approachable way. Frequent historical information presented throughout More extensive motivation for each of the topics than other texts currently available Connects with topics not included in other textbooks, such as complexity theory

A simplified proof of the characterization theorem for Gröbner-bases

Abstract: In /2/ a certain type of bases ("Grobner-Bases") for polynomial ideals has been introduced whose usefulness stems from the fact that a number of important computability problems in the theory of polynomial ideals are reducible to the construction of bases of this type. The key to an algorithmic construction of Grobner-bases is a characterization theorem for Grobner-bases whose proof in /2/is rather complex.In this paper a simplified proof is given. The simplification is based on two new lemmas that are of some interest in themselves. The first lemma characterizes the congruence relation modulo a polynomial ideal as the reflexive-transitive closure of a particular reduction relation ("M-reduction") used in the definition of Grobner-bases and its inverse. The second lemma is a lemma on general reduction relations, which allows to guarantee the Church-Rosser property under very weak assumptions.