### Lambda-Calculus and Functional Programming tions.

"... The lambda-calculus is a formalism for representing func-By the second half of the nineteenth century, the concept of function as used in mathematics had reached the point at which the standard notation had become ambiguous. For example, consider the operator P defined on real functions as follows: ..."

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The lambda-calculus is a formalism for representing func-By the second half of the nineteenth century, the concept of function as used in mathematics had reached the point at which the standard notation had become ambiguous. For example, consider the operator P defined on real functions as follows: ⎧f(x) – f(0) for x 0 P[f(x)] = ⎨ x ⎩f ′(0) for x = 0 What is P[f(x + 1)]? To see that this is ambiguous, let f(x) = x 2. Then if g(x) = f(x + 1), P[g(x)] = P[x 2 + 2x + 1] = x + 2. But if h(x) = P[f(x)] = x, then h(x + 1) = x + 1 P[g(x)]. This ambiguity has actually led to an error in the published literature; see the discussion in (Curry and Feys

### Functional Organization in Molecular Systems

"... and the *-calculus Dissertation zur Erlangung des akademischen Grades Doctor rerum naturalium ..."

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and the *-calculus Dissertation zur Erlangung des akademischen Grades Doctor rerum naturalium

### PIM Service Description Deliverable 16.3 Report on Adaptive Service Generator Contract N o 507953 Project funded by the European Community under the “Information Society Technology”

"... This report presents the state of the art of research in adaptive service generation as an enabling step toward automatic code generation. The relevant scientific results, computational problems and infrastructural implications are presented with the aim of providing the basis for the foundation of ..."

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This report presents the state of the art of research in adaptive service generation as an enabling step toward automatic code generation. The relevant scientific results, computational problems and infrastructural implications are presented with the aim of providing the basis for the foundation of a computational theory for Digital Ecosystems.

### De Bruijn's Automath and Pure Type Systems

"... We study the position of the Automath systems within the framework of Pure Type Systems (PTSs). In [2, 22], a rough relationship has been given between Automath and PTSs. That relationship ignores three of the most important features of Automath: definitions, parameters and -reduction, because at th ..."

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We study the position of the Automath systems within the framework of Pure Type Systems (PTSs). In [2, 22], a rough relationship has been given between Automath and PTSs. That relationship ignores three of the most important features of Automath: definitions, parameters and -reduction, because at the time, formulations of PTSs did not have these features. Since, PTSs have been extended with these features and in view of this, we revisit the correspondence between Automath and PTSs. This paper gives the most accurate description of Automath as a PTS so far.

### 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.

### 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.

### 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.

### 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 higher-order 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, cut-elimination...