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**11 - 15**of**15**### Abstraction-Based Genetic Programming By

"... This thesis describes a novel method for representing and automatically generating computer programs in an evolutionary computation context. Abstraction-Based Genetic Programming (ABGP) is a typed Genetic Programming representation system that uses System F, an expressive λ-calculus, 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. Abstraction-Based Genetic Programming (ABGP) is a typed Genetic Programming representation system that uses System F, an expressive λ-calculus, 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 Curry-Howard isomorphism. This thesis also presents the results obtained by applying this method to a set of problems. Acknowledgments I would not have been able to complete this work without the encouragements, trust and support of my

### Before we begin: Thank you, thank you, thank you,... Walter.WassollmanzudiesemMannnochsagen?

"... Doctor rerum naturalium ..."

### A semantic approach to illative combinatory logic ∗

"... This work introduces the theory of illative combinatory algebras, which is closely related to systems of illative combinatory logic. We thus provide a semantic interpretation for a formal framework in which both logic and computation may be expressed in a unified manner. Systems of illative combinat ..."

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This work introduces the theory of illative combinatory algebras, which is closely related to systems of illative combinatory logic. We thus provide a semantic interpretation for a formal framework in which both logic and computation may be expressed in a unified manner. Systems of illative combinatory logic consist of combinatory logic extended with constants and rules of inference intended to capture logical notions. Our theory does not correspond strictly to any traditional system, but draws inspiration from many. It differs from them in that it couples the notion of truth with the notion of equality between terms, which enables the use of logical formulas in conditional expressions. We give a consistency proof for first-order illative combinatory algebras. A complete embedding of classical predicate logic into our theory is also provided. The translation is very direct and natural.

### 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 top-down 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 top-down being based on the axiomatic method, Turing’s approach to computability is bottom-up, 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 non-deductive inferences, rather than in term of inscrutable intuition.

### 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 Bolzano-Weierstrass, Hahn-Banach, 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 Shu-Heng Chen, who supported him in this insistence. I thank both of them for this.