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Data Mining using Genetic Programming Classification and Symbolic Regression J. EggermontData Mining using Genetic Programming Classification and Symbolic Regression
"... proefschrift ter verkrijging van de graad van Doctor aan de Universiteit Leiden, op gezag van de Rector Magnificus Dr. D.D. Breimer, hoogleraar in de faculteit der Wiskunde en Natuurwetenschappen en die der Geneeskunde, volgens besluit van het College voor Promoties te verdedigen op woensdag 14 sept ..."
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proefschrift ter verkrijging van de graad van Doctor aan de Universiteit Leiden, op gezag van de Rector Magnificus Dr. D.D. Breimer, hoogleraar in de faculteit der Wiskunde en Natuurwetenschappen en die der Geneeskunde, volgens besluit van het College voor Promoties te verdedigen op woensdag 14 september 2005 klokke 15.15 uur door
Realizability interpretation of proofs in constructive analysis
, 2006
"... We prove constructively (in the style of Bishop) that every monotone continuous function with a uniform modulus of increase has a continuous inverse. The proof is formalized, and a realizing term extracted. It turns out that even in the logical term language – a version of Gödel’s T – evaluation is ..."
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We prove constructively (in the style of Bishop) that every monotone continuous function with a uniform modulus of increase has a continuous inverse. The proof is formalized, and a realizing term extracted. It turns out that even in the logical term language – a version of Gödel’s T – evaluation is reasonably efficient.
A LargeScale Experiment in Executing Extracted Programs
"... It is a wellknown fact that algorithms are often hidden inside mathematical proofs. If these proofs are formalized inside a proof assistant, then a mechanism called extraction can generate the corresponding programs automatically. Previous work has focused on the difficulties in obtaining a program ..."
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It is a wellknown fact that algorithms are often hidden inside mathematical proofs. If these proofs are formalized inside a proof assistant, then a mechanism called extraction can generate the corresponding programs automatically. Previous work has focused on the difficulties in obtaining a program from a formalization of the Fundamental Theorem of Algebra inside the Coq proof assistant. In theory, this program allows one to compute approximations of roots of polynomials. However, as we show in this work, there is currently a big gap between theory and practice. We study the complexity of the extracted program and analyze the reasons of its inefficiency, showing that this is a direct consequence of the approach used throughout the formalization.
Formalization of the integral calculus in the pvs theorem prover
, 2004
"... The PVS theorem prover is a widely used formal verification tool used for the analysis of safetycritical systems. The PVS prover, though fully equipped to support deduction in a very general logic framework, namely higherorder logic, it must nevertheless, be augmented with the definitions and assoc ..."
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The PVS theorem prover is a widely used formal verification tool used for the analysis of safetycritical systems. The PVS prover, though fully equipped to support deduction in a very general logic framework, namely higherorder logic, it must nevertheless, be augmented with the definitions and associated theorems for every branch of mathematics and computer science that is used in a verification. This is a formidable task, ultimately requiring the contributions of researchers and developers all over the world. This paper reports on the formalization of the integral calculus in the PVS theorem prover. All of the basic definitions and theorems covered in a first course on integral calculus have been completed. The theory and proofs were based on Rosenlicht’s classic text on real analysis and follow the traditional epsilondelta method. The goal of this work was to provide a practical set of PVS theories that could be used for verification of hybrid systems that arise in air traffic management systems and other aerospace applications. All of the basic linearity, integrability, boundedness, and continuity properties of the integral calculus were proved. The work culminated in the proof of the Fundamental Theorem Of Calculus. There is a brief discussion about why mechanically checked proofs are so much longer than standard mathematics textbook proofs.
Program extraction in constructive analysis. Submitted to: Logicism, Intuitionism, and Formalism – What has become of them
, 2006
"... We sketch a development of constructive analysis in Bishop’s style, with special emphasis on low typelevel 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 Int ..."
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We sketch a development of constructive analysis in Bishop’s style, with special emphasis on low typelevel 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
What to Do Next? Analysing and Optimising System Behaviour in Time ACADEMISCH PROEFSCHRIFT
"... any form. No Zebra Finches were harmed during the development of this thesis. For the cover image: © ≫Unfold II≪, 100 x 100 cm, diasec, 2006 by sascha weidner The research in this thesis has been carried out at the Centre for Mathematics ..."
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any form. No Zebra Finches were harmed during the development of this thesis. For the cover image: © ≫Unfold II≪, 100 x 100 cm, diasec, 2006 by sascha weidner The research in this thesis has been carried out at the Centre for Mathematics
Improving Real Analysis in Coq: a UserFriendly Approach to Integrals and Derivatives ⋆
"... Abstract. Verification of numerical analysis programs requires dealing with derivatives and integrals. High confidence in this process can be achieved using a formal proof checker, such as Coq. Its standard library provides an axiomatization of real numbers and various lemmas about real analysis, wh ..."
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Abstract. Verification of numerical analysis programs requires dealing with derivatives and integrals. High confidence in this process can be achieved using a formal proof checker, such as Coq. Its standard library provides an axiomatization of real numbers and various lemmas about real analysis, which may be used for this purpose. Unfortunately, its definitions of derivative and integral are unpractical as they are partial functions that demand a proof term. This proof term makes the handling of mathematical formulas cumbersome and does not conform to traditional analysis. Other proof assistants usually do not suffer from this issue; for instance, they may rely on Hilbert’s epsilon to get total operators. In this paper, we propose a way to define total operators for derivative and integral without having to extend Coq’s standard axiomatization of real numbers. We proved the compatibility of our definitions with the standard library’s in order to leverage existing results. We also greatly improved automation for real analysis proofs that use Coq standard definitions. We exercised our approach on lemmas involving iterated partial derivatives and differentiation under the integral sign, that were missing from the formal proof of a numerical program solving the wave equation. 1
door
, 2007
"... About the cover: picture of the SISYPHUS II machine, constructed by Bruce Shapiro. ..."
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About the cover: picture of the SISYPHUS II machine, constructed by Bruce Shapiro.
prof. dr. Eric R. VerheulFormal and Computational Cryptography:
"... This thesis is typeset using LATEX. Cover: Zimmermann Telegram, written by German Foreign Secretary Arthur Zimmermann, is an encrypted message sent to Mexico, proposing a military alliance against the United States. It was intercepted and decrypted by British cryptographers. The telegram inflamed Am ..."
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This thesis is typeset using LATEX. Cover: Zimmermann Telegram, written by German Foreign Secretary Arthur Zimmermann, is an encrypted message sent to Mexico, proposing a military alliance against the United States. It was intercepted and decrypted by British cryptographers. The telegram inflamed American public opinion and draw the United States to declare war against Germany in 1917. The bookmarker contains the telegram as partially decrypted by the British cryptographers of Room 40.