Multilateral Researches on Computability Problems on the Continuum

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The purpose of this project is the computability structure on the continuum in Pour-E1 style; its extension, application, and formalization. Most of the objectives have been steadily achieved. We here report our results.The major research target of this project is the computability problems of real discontinuous functions, especially piecewise continuous functions, that is, the foundations of computation of function values at discontinuous points.1. Limit computation: This is a computation method by taking the limits of recursive functions. We have shown that many of piecewise continuous functions are computable with this method.2. Effective uniform space: The theory of the computability structure on the uniform space obtained by isolating discontinuous points has been developed. Many of piecewise continuous functions have been shown to be computable in this theory. The equivalence of effective convergences of a function with regards to respectively uniformity and its metrization.3. Limit computation and uniform space: Under a certain condition, sequential computabilities of a piecewise continuous function with regards to respectively limit computation and uniformity.4. Method of Walsh analysis: The theory of representing computability notions in terms of Fine metric has been developed, and various notions of computability have been defined.5. A formal system of limit computable mathematics: A formal system in which limiting computable mathematics can be executed has been defined, and its functional interpretation has been carried out.6. Computability in functional analysis: Effectivity of solving the invisid partial differential equation and effectivity of some linear operators on the interpolation space have been solved affirmatively.7. Theories of representing real numbers: Theories of representing a complete uniform space by a uniform domain, and representations of real numbers by respectively Gray codes and a certain category have been developed.