Let U and V be complete separable metric spaces, Vu compact in V and G : U IR a continuous function. For a large class of (usually non--constructive) proofs of uniqueness theorems Vu G(u, v1 ) = inf G(u, v) = G(u, v2) v1 = v2 one can extract an e#ective modulus of uniqueness # by logical analysis, i.e.

Abstract. For a long time people have been trying to develop probability theory starting from ‘finite ’ events rather than collections of infinite events. In this way one can find natural replacements for measurable sets and integrable functions, but measurable functions seemed to be more difficult. We present a solution. Moreover, our results are constructive (in the sense of Bishop). 1.

We report on the programming system eb that supports computational science and engineering. eb has the constructive philosophy begun by Bishop. This philosophy is explained in enough detail to show how this view is acceptable to, but different from, -calculus and Martin-Lof theories. eb raises a theoretical question of semantics: how to guarantee that the language as implemented works as intended by the constructive reals model? The eb system is currently in "bootstrap" mode. We discuss the implementation of this bootstrap as well as plans for the future. This implementation is a source to source translator to C. Primitive types in eb are multiprecision integers and floating point. As innovations, eb supports both functional and relational models, is nondeterministic, and uses failure as a control mechanism. Keywords Numerical programming (computational science and engineering), functional logic programming. Word Count 4999. 1 Introduction "The life which is unexamined is not wo...

We sketch a development of constructive analysis in Bishop’s style, with special emphasis on low type-level witnesses (using separability of the reals). The goal is to set up things in such a way that realistically executable programs can be extracted from proofs. This is carried out for (1) the Intermediate Value Theorem and (2) the existence of a continuous inverse to a monotonically increasing continuous function. Using the Minlog proof assistant, the proofs leading to the Intermediate Value Theorem are formalized and realizing terms extracted. It turns out that evaluating these terms is a reasonably fast algorithm to compute, say, approximations of √ 2. 1

. The goal of foundational thinking in computer science is to understand the methods and practices of working programmers; we might even be able to improve upon those practices. The investigation outlined here applies the methods of constructive mathematics 'a l`a A. N. Kolmogoroff, L. E. J. Brouwer and Errett Bishop to contemporary computer science. The major approach is to use Kolmogoroff's interpretation of the predicate calculus. This investigation includes an attempt to merge contemporary thoughts on computability and computing semantics with the language of mental constructions proposed by Brouwer. This necessarily forces us to ask about the psychology of language. I present a definition of algorithms that links language, constructive mathematics, and logic. Using the concept of an abstract family of algorithms (Hennie) and principles of constructivity, a definition of problem solving. The constructive requirements for an algorithm are developed and presented. Given this framewor...

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We report on our efforts to explore geometry using constructive mathematics and intuitionistic logic. Such an approach differs greatly from from classical Greek[13] and modern geometry[23]. This change impinges on the theory of computability that uses classical logic and the Church-Turing-Chomsky foundations. We follow the constructive philosophy begun by Bishop[5]. This philosophy is explained in some detail. We hold that these ideas are more natural for computing than classical theories. It is difficult to see how constructive theories are meant to work. We present new ideas based onconstructive proofs of Euclid's Elements. Such work sets a new foundation for geometry and, hence, for graphics and visualization. CR Categories J.2 Computational Science and Engineering, F.4.1 Mathematical Logic, F.1.1 Models of Computation, F.3.1 Reasoning about Programs. 1 Introduction "The life which is unexamined is not worth living." Plato, Apology. "It appears to me that if one wants to make pro...

Abstract. We provide a computer verified exact monadic functional implementation of the Riemann integral in type theory. Together with previous work by O’Connor, this may be seen as the beginning of the realization of Bishop’s vision to use constructive mathematics as a programming language for exact analysis. 1 1.

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