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A lambda calculus for real analysis
, 2005
"... Abstract Stone Duality is a revolutionary theory that works directly with computable continuous functions, without using set theory, infinitary lattice theory or a prior theory of discrete computation. Every expression in the calculus denotes both a continuous function and a program, but the reasoni ..."
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Abstract Stone Duality is a revolutionary theory that works directly with computable continuous functions, without using set theory, infinitary lattice theory or a prior theory of discrete computation. Every expression in the calculus denotes both a continuous function and a program, but the reasoning looks remarkably like a sanitised form of that in classical topology. This paper is an introduction to ASD for the general mathematician, and applies it to elementary real analysis. It culminates in the Intermediate Value Theorem, i.e. the solution of equations fx = 0 for continuous f: R → R. As is well known from both numerical and constructive considerations, the equation cannot be solved if f “hovers ” near 0, whilst tangential solutions will never be found. In ASD, both of these failures and the general method of finding solutions of the equation when they exist are explained by the new concept of “overtness”. The zeroes are captured, not as a set, but by highertype operators � and ♦ that remain (Scott) continuous across singularities of a parametric equation. Expressing topology in terms of continuous functions rather than sets of points leads to
Type classes for efficient exact real arithmetic
 IN COQ. CORR ABS/1106.3448
, 2011
"... Floating point operations are fast, but require continuous effort by the user to ensure correctness. This burden can be shifted to the machine by providing a library of exact analysis in which the computer handles the error estimates. Previously, we provided a fast implementation of the exact real ..."
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Floating point operations are fast, but require continuous effort by the user to ensure correctness. This burden can be shifted to the machine by providing a library of exact analysis in which the computer handles the error estimates. Previously, we provided a fast implementation of the exact real numbers in the Coq proof assistant. This implementation incorporates various optimizations to speed up the basic operations of O’Connor’s implementation by a 100 times. We implemented these optimizations in a modular way using type classes to define an abstract specification of the underlying dense set from which the real numbers are built. This abstraction does not hurt the efficiency. This article is a substantially expanded version of (Krebbers/Spitters, Calculemus 2011) in which the implementation is extended in the various ways. First, we implement and verify the sine and cosine function. Secondly, we create an additional implementation of the dense set based on Coq’s fast rational numbers. Thirdly, we extend the hierarchy to capture order on undecidable structures, while it was limited to decidable structures before. This hierarchy, based on type classes, allows us to share theory on the naturals, integers, rationals, dyadics, and reals in a convenient way. Finally, we obtain another dramatic speedup by avoiding evaluation of termination proofs at runtime.
Interval Analysis Without Intervals
, 2006
"... We argue that Dedekind completeness and the Heine–Borel property should be seen as part of the “algebraic ” structure of the real line, along with the usual arithmetic operations and relations. Dedekind cuts provide a uniform and natural way of formulating differentiation, integration and limits. Th ..."
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We argue that Dedekind completeness and the Heine–Borel property should be seen as part of the “algebraic ” structure of the real line, along with the usual arithmetic operations and relations. Dedekind cuts provide a uniform and natural way of formulating differentiation, integration and limits. They and these examples also generalise to intervals. Together with the arithmetic order, cuts enjoy prooftheoretic introduction and elimination rules similar to those for lambda abstraction and application. This system completely axiomatises computably continuous functions on the real line. We show how this calculus (of “single ” points) can be translated formally into Interval Analysis, interpreting the arithmetic operations a la Moore, and compactness as optimisation under constraints. Notice that interval computation is the conclusion and not the starting point. This calculus for the real line is part of a more general recursive axiomatisation of general topology called Abstract Stone Duality. 1
Minima and best approximations in constructive analysis
, 2011
"... Working in Bishop’s constructive mathematics, we first show that minima can be defined as best approximations, in such a way as to preserve the compactness of the underlying metric space when the function is uniformly continuous. Results about finding minima can therefore be carried over to the se ..."
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Working in Bishop’s constructive mathematics, we first show that minima can be defined as best approximations, in such a way as to preserve the compactness of the underlying metric space when the function is uniformly continuous. Results about finding minima can therefore be carried over to the setting of finding best approximations. In particular, the implication from having at most one best approximation to having uniformly at most one best approximation is equivalent to Brouwer’s fan theorem for decidable bars. We then show that for the particular case of finitedimensional subspaces of normed spaces, these two notions do coincide. This gives us a better understanding of Bridges ’ proof that finitedimensional subspaces with at most one best approximation do in fact have one. As a complement we briefly review how the case of best approximations to a convex subset of a uniformly convex normed space fits into the unique existence paradigm.
Tychonov’s Theorem in Abstract Stone Duality
, 2006
"... New constructive definition of compactness in the form of the existence of a continuous ”universal quantifier”. Construction and compactness of Cantor space. Baire space is not definable (locally compact). Examination of the (non) impact of a counterexample due to Kleene that has previously underm ..."
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New constructive definition of compactness in the form of the existence of a continuous ”universal quantifier”. Construction and compactness of Cantor space. Baire space is not definable (locally compact). Examination of the (non) impact of a counterexample due to Kleene that has previously undermined other attempts to define and prove compactness of Cantor space constructively. 1
Unique paths as formal points
, 2011
"... A pointfree formulation of the König Lemma for trees with uniformly at most one infinite path allows for a constructive proof without unique choice. ..."
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A pointfree formulation of the König Lemma for trees with uniformly at most one infinite path allows for a constructive proof without unique choice.
CCA 2008 Efficient Computation with Dedekind Reals
"... Cauchy’s construction of reals as sequences of rational approximations is the theoretical basis for a number of implementations of exact real numbers, while Dedekind’s construction of reals as cuts has inspired fewer useful computational ideas. Nevertheless, we can see the computational content of D ..."
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Cauchy’s construction of reals as sequences of rational approximations is the theoretical basis for a number of implementations of exact real numbers, while Dedekind’s construction of reals as cuts has inspired fewer useful computational ideas. Nevertheless, we can see the computational content of Dedekind reals by constructing them within Abstract Stone Duality (ASD), a computationally meaningful calculus for topology. This provides the theoretical background for a novel way of computing with real numbers in the style of logic programming. Real numbers are defined in terms of (lower and upper) Dedekind cuts, while programs are expressed as statements about real numbers in the language of ASD. By adapting Newton’s method to interval arithmetic we can make the computations as efficient as those based on Cauchy reals. The results reported in this talk are joint work with Paul Taylor. Keywords:
4 Unified Approach to Real Numbers in Various Mathematical Settings
, 2014
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