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On the Role of Implication in Formal Logic
, 1998
"... Evidence is given that implication (and its special case, negation) carry the logical strength of a system of formal logic. This is done by proving normalization and cut elimination for a system based on combinatory logic or #calculus with logical constants for and, or, all, and exists, but with no ..."
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Evidence is given that implication (and its special case, negation) carry the logical strength of a system of formal logic. This is done by proving normalization and cut elimination for a system based on combinatory logic or #calculus with logical constants for and, or, all, and exists, but with none for either implication or negation. The proof is strictly finitary, showing that this system is very weak. The results can be extended to a "classical" version of the system. They can also be extended to a system with a restricted set of rules for implication: the result is a system of intuitionistic higherorder BCK logic with unrestricted comprehension and without restriction on the rules for disjunction elimination and existential elimination. The result does not extend to the classical version of the BCK logic. 1991 AMS (MOS) Classification: 03B40, 03F05, 03B20 Key words: Implication, negation, combinatory logic, lambda calculus, comprehension principle, normalization, cutelimination...
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.
Before we begin: Thank you, thank you, thank you,... Walter.WassollmanzudiesemMannnochsagen?
"... Doctor rerum naturalium ..."
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ON THE FOUNDATIONS OF MATHEMATICAL ECONOMICS
, 2010
"... Kumaraswamy Vela Velupillai [74] presents a constructivist perspective on the foundations of mathematical economics, praising the views of Feynman in developing path integrals and Dirac in developing the delta function. He sees their approach as consistent with the Bishop constructive mathematics an ..."
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Kumaraswamy Vela Velupillai [74] presents a constructivist perspective on the foundations of mathematical economics, praising the views of Feynman in developing path integrals and Dirac in developing the delta function. He sees their approach as consistent with the Bishop constructive mathematics and considers its view on the BolzanoWeierstrass, HahnBanach, and intermediate value theorems, and then the implications of these arguments for such “crown jewels ” of mathematical economics as the existence of general equilibrium and the second welfare theorem. He also relates these ideas to the weakening of certain assumptions to allow for more general results as shown by Rosser [51] in his extension of Gödel’s incompleteness theorem in his opening section. This paper considers these arguments in reverse order, moving from the matters of economics applications to the broader issue of constructivist mathematics, concluding by considering the views of Rosser on these matters, drawing both on his writings and on personal conversations with him. Acknowledgements: I thank K. Vela Velupillai most particularly for his efforts to push me to consider these matters in the most serious manner, as well as my late father, J. Barkley Rosser [Sr.] and also his friend, the late Stephen C. Kleene, for their personal remarks on these matters to me over a long period of time. I also wish to thank Eric Bach, Ken Binmore, Herb Gintis, Jerome Keisler, Roger Koppl, David Levy, and Adrian Mathias for useful comments. The usual caveat holds. I also wish to dedicate this to K. Vela Velupillai who inspired it with his insistence that I finally deal with the work and thought of my father, J. Barkley Rosser [Sr.], as well as ShuHeng Chen, who supported him in this insistence. I thank both of them for this.
Turing’s Approaches to Computability, Mathematical Reasoning and Intelligence
"... Abstract In this paper a distinction is made between Turing’s approach to computability, on the one hand, and his approach to mathematical reasoning and intelligence, on the other hand. Unlike Church’s approach to computability, which is topdown being based on the axiomatic method, Turing’s approac ..."
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Abstract In this paper a distinction is made between Turing’s approach to computability, on the one hand, and his approach to mathematical reasoning and intelligence, on the other hand. Unlike Church’s approach to computability, which is topdown being based on the axiomatic method, Turing’s approach to computability is bottomup, being based on an analysis of the actions of a human computer. It is argued that, for this reason, Turing’s approach to computability is convincing. On the other hand, his approach to mathematical reasoning and intelligence is not equally convincing, because it is based on the assumption that intelligent processes are basically mechanical processes, which however from time to time may require some decision by an external operator, based on intuition. This contrasts with the fact that intelligent processes can be better accounted for in rational terms, specifically, in terms of nondeductive inferences, rather than in term of inscrutable intuition.
On the Role of Implication in Formal Logic
"... Evidence is given that implication (and its special case, negation) carry the logical strength of a system of formal logic. This is done by proving normalization and cut elimination for a system based on combinatory logic or #calculus with logical constants for and, or, all, and exists, but with no ..."
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Evidence is given that implication (and its special case, negation) carry the logical strength of a system of formal logic. This is done by proving normalization and cut elimination for a system based on combinatory logic or #calculus with logical constants for and, or, all, and exists, but with none for either implication or negation. The proof is strictly finitary, showing that this system is very weak. The results can be extended to a "classical" version of the system. They can also be extended to a system with a restricted set of rules for implication: the result is a system of intuitionistic higherorder BCK logic with unrestricted comprehension and without restriction on the rules for disjunction elimination and existential elimination. The result does not extend to the classical version of the BCK logic. 1991 AMS (MOS) Classification: 03B40, 03F05, 03B20 Key words: Implication, negation, combinatory logic, lambda calculus, comprehension principle, normalization, cutelimination...
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