Theory of recursive functions and effective computability
Theory of recursive functions and effective computability
A domain-theoretic approach to computability on the real line
Theoretical Computer Science - Special issue on real numbers and computers
Constructive mathematics: a foundation for computable analysis
Theoretical Computer Science - Special issue on computability and complexity in analysis
Computable analysis: an introduction
Computable analysis: an introduction
The iRRAM: Exact Arithmetic in C++
CCA '00 Selected Papers from the 4th International Workshop on Computability and Complexity in Analysis
The realizability approach to computable analysis and topology
The realizability approach to computable analysis and topology
Unique existence, approximate solutions, and countable choice
Theoretical Computer Science - Topology in computer science
RealLib: An efficient implementation of exact real arithmetic
Mathematical Structures in Computer Science
Implementing Real Numbers With RZ
Electronic Notes in Theoretical Computer Science (ENTCS)
Journal of Logic and Computation
Electronic Notes in Theoretical Computer Science (ENTCS)
Inside Every Model of Abstract Stone Duality Lies an Arithmetic Universe
Electronic Notes in Theoretical Computer Science (ENTCS)
Some reasons for generalising domain theory
Mathematical Structures in Computer Science
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Abstract Stone Duality (ASD) is a direct axiomatisation of general topology, in contrast to the traditional and all other contemporary approaches, which rely on a prior notion of discrete set, type or object of a topos. ASD reconciles mathematical and computational viewpoints, providing an inherently computable calculus that does not sacrifice key properties of real analysis such as compactness of the closed interval. Previous theories of recursive analysis failed to do this because they were based on points; ASD succeeds because, like locale theory and formal topology, it is founded on the algebra of open subspaces. ASD is presented as a lambda calculus, of which we provide a self-contained summary, as the foundational background has been investigated in earlier work. The core of the paper constructs the real line using two-sided Dedekind cuts. We show that the closed interval is compact and overt, where these concepts are defined using quantifiers. Further topics, such as the Intermediate Value Theorem, are presented in a separate paper that builds on this one. The interval domain plays an important foundational role. However, we see intervals as generalised Dedekind cuts, which underly the construction of the real line, not as sets or pairs of real numbers. We make a thorough study of arithmetic, in which our operations are more complicated than Moore's, because we work constructively, and we also consider back-to-front (Kaucher) intervals. Finally, we compare ASD with other systems of constructive and computable topology and analysis.