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The Impact of the Lambda Calculus in Logic and Computer Science
 BULLETIN OF SYMBOLIC LOGIC
, 1997
"... One of the most important contributions of A. Church to logic is his invention of the lambda calculus. We present the genesis of this theory and its two major areas of application: the representation of computations and the resulting functional programming languages on the one hand and the represent ..."
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One of the most important contributions of A. Church to logic is his invention of the lambda calculus. We present the genesis of this theory and its two major areas of application: the representation of computations and the resulting functional programming languages on the one hand and the representation of reasoning and the resulting systems of computer mathematics on the other hand.
Step By Recursive Step: Church's Analysis Of Effective Calculability
 BULLETIN OF SYMBOLIC LOGIC
, 1997
"... Alonzo Church's mathematical work on computability and undecidability is wellknown indeed, and we seem to have an excellent understanding of the context in which it arose. The approach Church took to the underlying conceptual issues, by contrast, is less well understood. Why, for example, wa ..."
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Alonzo Church's mathematical work on computability and undecidability is wellknown indeed, and we seem to have an excellent understanding of the context in which it arose. The approach Church took to the underlying conceptual issues, by contrast, is less well understood. Why, for example, was "Church's Thesis" put forward publicly only in April 1935, when it had been formulated already in February/March 1934? Why did Church choose to formulate it then in terms of G odel's general recursiveness, not his own #definability as he had done in 1934? A number of letters were exchanged between Church and Paul Bernays during the period from December 1934 to August 1937; they throw light on critical developments in Princeton during that period and reveal novel aspects of Church's distinctive contribution to the analysis of the informal notion of e#ective calculability. In particular, they allow me to give informed, though still tentative answers to the questions I raised; the char...
Why sets?
 PILLARS OF COMPUTER SCIENCE: ESSAYS DEDICATED TO BORIS (BOAZ) TRAKHTENBROT ON THE OCCASION OF HIS 85TH BIRTHDAY, VOLUME 4800 OF LECTURE NOTES IN COMPUTER SCIENCE
, 2008
"... Sets play a key role in foundations of mathematics. Why? To what extent is it an accident of history? Imagine that you have a chance to talk to mathematicians from a faraway planet. Would their mathematics be setbased? What are the alternatives to the settheoretic foundation of mathematics? Besi ..."
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Sets play a key role in foundations of mathematics. Why? To what extent is it an accident of history? Imagine that you have a chance to talk to mathematicians from a faraway planet. Would their mathematics be setbased? What are the alternatives to the settheoretic foundation of mathematics? Besides, set theory seems to play a significant role in computer science; is there a good justification for that? We discuss these and some related issues.
Curry's Anticipation of the Types Used in Programming Languages
, 2003
"... This paper shows that H. B. Curry anticipated both the kind of data types used in computer programming languages and also the dependent function type. ..."
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This paper shows that H. B. Curry anticipated both the kind of data types used in computer programming languages and also the dependent function type.
Course Notes in Typed Lambda Calculus
, 1998
"... this paper is clearly stated, after recalling how the logical connectives can be explained in term of the Sheffer connective: "We are led to the idea, which at first glance certainly appears extremely bold of attempting to eliminate by suitable reduction the remaining fundamental notions, those ..."
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this paper is clearly stated, after recalling how the logical connectives can be explained in term of the Sheffer connective: "We are led to the idea, which at first glance certainly appears extremely bold of attempting to eliminate by suitable reduction the remaining fundamental notions, those of proposition, propositional function, and variable, from those contexts in which we are dealing with completely arbitrary, logical general propositions . . . To examine this possibility more closely and to pursue it would be valuable not only from the methodological point of view that enjoins us to strive for the greatest possible conceptual uniformity but also from a certain philosophic, or if you wish, aesthetic point of view."
An Interactive Proof System for Map Theory
, 2002
"... This dissertation is submitted in partial ful llment of the requirements for the Danish Ph.D. degree. It documents work done between August 1999 and September 2002 at the Department of Computer Science at the University of Copenhagen (DIKU) ..."
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This dissertation is submitted in partial ful llment of the requirements for the Danish Ph.D. degree. It documents work done between August 1999 and September 2002 at the Department of Computer Science at the University of Copenhagen (DIKU)
Before we begin: Thank you, thank you, thank you,... Walter.WassollmanzudiesemMannnochsagen?
"... Doctor rerum naturalium ..."
AbstractionBased Genetic Programming
, 2009
"... This thesis describes a novel method for representing and automatically generating computer programs in an evolutionary computation context. AbstractionBased Genetic Programming (ABGP) is a typed Genetic Programming representation system that uses System F, an expressive λcalculus, to represent th ..."
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This thesis describes a novel method for representing and automatically generating computer programs in an evolutionary computation context. AbstractionBased Genetic Programming (ABGP) is a typed Genetic Programming representation system that uses System F, an expressive λcalculus, to represent the computational components from which the evolved programs are assembled. ABGP is based on the manipulation of closed, independent modules expressing computations with effects that have the ability to affect the whole genotype. These modules are plugged into other modules according to precisely defined rules to form complete computer programs. The use of System F allows the straightforward representation and use of many typical computational structures and behaviors (such as iteration, recursion, lists and trees) in modular form. This is done without introducing additional external symbols in the set of predefined functions and terminals of the system. In fact, programming structures typically included in GP terminal sets, such as if then else, may be removed and represented as abstractions in ABGP for the same problems. ABGP also provides a search space partitioning system based on the structure of the genotypes, similar to the species partitioning system of living organisms and derived from the CurryHoward isomorphism. This thesis also presents the results obtained by applying this method to a set of problems.
and Niels Hellraiser Christensen for being good friends and colleagues.
, 2002
"... As any other Ph.D. student, I have met plenty of people during the course of my studies, that should be thanked in some way or the other. First and foremost, my heartfelt thanks to Neil D. Jones for being my strict supervisor and kind boss, during both my master’s thesis and my Ph.D. studies. Also t ..."
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As any other Ph.D. student, I have met plenty of people during the course of my studies, that should be thanked in some way or the other. First and foremost, my heartfelt thanks to Neil D. Jones for being my strict supervisor and kind boss, during both my master’s thesis and my Ph.D. studies. Also to Klaus Grue, who suggested the topic to me in early 1999, when I was complaining about total functions in HOL, and for helping me with numerous technical questions regarding map theory. Tobias Nipkow at Technische Universität München having me as a guest in the first half of 2000. Thanks also to the rest of the Isabelle group:
Kurt Gödel and Computability Theory
"... Abstract. Although Kurt Gödel does not figure prominently in the history of computabilty theory, he exerted a significant influence on some of the founders of the field, both through his published work and through personal interaction. In particular, Gödel’s 1931 paper on incompleteness and the meth ..."
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Abstract. Although Kurt Gödel does not figure prominently in the history of computabilty theory, he exerted a significant influence on some of the founders of the field, both through his published work and through personal interaction. In particular, Gödel’s 1931 paper on incompleteness and the methods developed therein were important for the early development of recursive function theory and the lambda calculus at the hands of Church, Kleene, and Rosser. Church and his students studied Gödel 1931, and Gödel taught a seminar at Princeton in 1934. Seen in the historical context, Gödel was an important catalyst for the emergence of computability theory in the mid 1930s. 1