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18
Analyzing Proofs in Analysis
 LOGIC: FROM FOUNDATIONS TO APPLICATIONS. EUROPEAN LOGIC COLLOQUIUM (KEELE
, 1993
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Guaranteed intervals for Kolmogorov’s theorem (and their possible relation to neural networks
 Interval Computations
, 1993
"... Abstract In 1987, R. HechtNielsen noticed that a theorem that was proved by Kolmogorov in 1957 as a solution to one of Hilbert's problems, actually shows that an arbitrary function f can be implemented by a 3layer neural network with appropriate activation functions and O/. The more accuratel ..."
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Cited by 12 (7 self)
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Abstract In 1987, R. HechtNielsen noticed that a theorem that was proved by Kolmogorov in 1957 as a solution to one of Hilbert's problems, actually shows that an arbitrary function f can be implemented by a 3layer neural network with appropriate activation functions and O/. The more accurately we implement these functions, the better approximation to f we get. Kolmogorov's proof can be transformed into a fast iterative algorithm that converges to the description of a network. However, this algorithm does not provide us with a guaranteed approximation accuracy: namely, if we want to approximate a given function f with a given accuracy ", this algorithm does not tell us after what iteration we can guarantee this accuracy. In 1991, Kurkova proposed a second algorithmic version of Kolmogorov's theorem. Namely, she showed how for every continuous function f, and for every " ? 0, we can construct a neural network that approximates f with a given accuracy ", i.e., whose output belongs to an interval [f (x1;:::; xn) \Gamma "; f (x1;:::; xn) + "]. In the original Kolmogorov's theorem, the design (and, in particular, the number Nhidden of hidden neurons) does not change with ". In Kurkova's algorithm, when " ! 0, the number of hidden neurons increases (Nhidden! 1), and so does the complexity of the approximating network. The natural question is: can we provide a guaranteed approximation property and still keep Nhidden independent on "? Our asnwer is "yes". In this paper, we describe algorithms that generate the functions and O / (from the original
REPRESENTATIONS OF THE REAL NUMBERS AND OF THE OPEN SUBSETS OF THE SET OF REAL NUMBERS
, 1987
"... In previous papers we have presented a unified Type 2 theory of computability and continuity and a theory of representations. In this paper the concepts developed so far are used for the foundation of a new kind of constructive analysis. Different standard representations of the real numbers are com ..."
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Cited by 9 (1 self)
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In previous papers we have presented a unified Type 2 theory of computability and continuity and a theory of representations. In this paper the concepts developed so far are used for the foundation of a new kind of constructive analysis. Different standard representations of the real numbers are compared. It turns out that the crucial differences are of topological nature and that most of the representations (e.g., the decimal representation) are not reasonable for topological reasons. In the second part some effective representations of the open subsets of the real numbers are introduced and compared.
Classical And Constructive Hierarchies In Extended Intuitionistic Analysis
"... This paper introduces an extension of Kleene's axiomatization of Brouwer's intuitionistic analysis, in which the classical arithmetical and analytical hierarchies are faithfully represented as hierarchies of the domains of continuity. A domain of continuity is a relation R(#) on Baire sp ..."
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Cited by 4 (3 self)
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This paper introduces an extension of Kleene's axiomatization of Brouwer's intuitionistic analysis, in which the classical arithmetical and analytical hierarchies are faithfully represented as hierarchies of the domains of continuity. A domain of continuity is a relation R(#) on Baire space with the property that every constructive partial functional defined on {# : R(#)} is continuous there. The domains of continuity for coincide with the stable relations (those equivalent in to their double negations), while every relation R(#) is equivalent in #) for some stable A(#, #) (which belongs to the classical analytical hierarchy). The logic of is intuitionistic. The axioms of include countable comprehension, bar induction, Troelstra's generalized continuous choice, primitive recursive Markov's Principle and a classical axiom of dependent choices proposed by Krauss. Constructive dependent choices, and constructive and classical countable choice, are theorems.
Inaccessible Set Axioms May Have Little Consistency Strength
 Annals of Pure and Applied Logic
"... . The paper investigates inaccessible set axioms and their consistency strength in constructive set theory. In ZFC inaccessible sets are of the form V where is a strongly inaccessible cardinal and V denotes the  th level of the von Neumann hierarchy. Inaccessible sets figure prominently in cat ..."
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Cited by 3 (2 self)
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. The paper investigates inaccessible set axioms and their consistency strength in constructive set theory. In ZFC inaccessible sets are of the form V where is a strongly inaccessible cardinal and V denotes the  th level of the von Neumann hierarchy. Inaccessible sets figure prominently in category theory as Grothendieck universes and are related to universes in type theory. The objective of this paper is to show that the consistency strength of inaccessible set axioms heavily depends on the context in which they are embedded. The context here will be the theory CZF \Gamma of constructive Zermelo Fraenkel set theory but without 2  Induction (foundation). Let INAC be the statement that for every set there is an inaccessible set containing it. CZF \Gamma +INAC is a mathematically rich theory in which one can easily formalize Bishop style constructive mathematics and a great deal of category theory. CZF \Gamma + INAC also has a realizability interpretation in type theory whic...
Compactness in constructive analysis revisited
 Ann. Pure Appl. Logic
, 1987
"... In [5], [3] we have developed a unified Type 2 theory of computability and continuity and a theory of representations. In a third paper [6] representations useful for a new kind of constructive analysis were presented. As an application of these concepts we shall now consider constructive compactnes ..."
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Cited by 3 (0 self)
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In [5], [3] we have developed a unified Type 2 theory of computability and continuity and a theory of representations. In a third paper [6] representations useful for a new kind of constructive analysis were presented. As an application of these concepts we shall now consider constructive compactness. We introduce
A minimalist twolevel foundation for constructive mathematics
, 811
"... We present a twolevel theory to formalize constructive mathematics as advocated in a previous paper with G. Sambin [MS05]. One level is given by an intensional type theory, called Minimal type theory. This theory extends the settheoretic version introduced in [MS05] with collections. The other lev ..."
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Cited by 2 (1 self)
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We present a twolevel theory to formalize constructive mathematics as advocated in a previous paper with G. Sambin [MS05]. One level is given by an intensional type theory, called Minimal type theory. This theory extends the settheoretic version introduced in [MS05] with collections. The other level is given by an extensional set theory that is interpreted in the first one by means of a quotient model. This twolevel theory has two main features: it is minimal among the most relevant foundations for constructive mathematics; it is constructive thanks to the way the extensional level is linked to the intensional one which fulfills the “proofsasprograms ” paradigm and acts as a programming language.