### This document in subdirectoryRS/96/56/ Modeling Sharing and Recursion for Weak Reduction Strategies using Explicit Substitution

, 1996

"... Reproduction of all or part of this work is permitted for educational or research use on condition that this copyright notice is included in any copy. See back inner page for a list of recent publications in the BRICS Report Series. Copies may be obtained by contacting: BRICS ..."

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Reproduction of all or part of this work is permitted for educational or research use on condition that this copyright notice is included in any copy. See back inner page for a list of recent publications in the BRICS Report Series. Copies may be obtained by contacting: BRICS

### Handbook of the History of Logic. Volume 6

"... ABSTRACT: Here is a crude list, possibly summarizing the role of paradoxes within the framework of mathematical logic: 1. directly motivating important theories (e.g. type theory, axiomatic set theory, combinatory logic); 2. suggesting methods of proving fundamental metamathematical results (fixed p ..."

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ABSTRACT: Here is a crude list, possibly summarizing the role of paradoxes within the framework of mathematical logic: 1. directly motivating important theories (e.g. type theory, axiomatic set theory, combinatory logic); 2. suggesting methods of proving fundamental metamathematical results (fixed point theorems, incompleteness, undecidability, undefinability); 3. applying inductive definability and generalized recursion; 4. introducing new semantical methods (e. g. revision theory, semi-inductive definitions, which require non-trivial set theoretic results); 5. (partly) enhancing new axioms in set theory: the case of anti-foundation AFA and the mathematics of circular phenomena; 6. suggesting the investigation of non-classical logical systems, from contraction-free and many-valued logics to systems with generalized quantifiers; 7. suggesting frameworks with flexible typing for the foundations of Mathematics and Computer Science; 8. applying forms of self-referential truth and in Artificial Intelligence, Theoretical Linguistics, etc. Below we attempt to shed some light on the genesis of the issues 1–8 through the history of the paradoxes in the twentieth century, with a special emphasis on semantical aspects.

### Neural Algebra and Consciousness: A Theory of Structural Functionality in Neural Nets

"... Abstract. Thoughts are spatio-temporal patterns of coalitions of firing neurons and their interconnections. Neural algebras represent these patterns as formal algebraic objects, and a suitable composition operation reflects their interaction. Thus, a neural algebra is associated with any neural net. ..."

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Abstract. Thoughts are spatio-temporal patterns of coalitions of firing neurons and their interconnections. Neural algebras represent these patterns as formal algebraic objects, and a suitable composition operation reflects their interaction. Thus, a neural algebra is associated with any neural net. The present paper presents this formalization and develops the basic algebraic tools for formulating and solving the problem of finding the neural correlates of concepts such as reflection, association, coordination, etc. The main application is to the notion of consciousness, whose structural and functional basis is made explicit as the emergence of a set of solutions to a fixpoint equation. Key words: neural nets, combinatory algebra, functional structures, emergent properties, models of consciousness

### 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

### 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

### Modeling Sharing and Recursion for Weak Reduction Strategies using Explicit Substitution

- In Proc. PLILP'96, the 8 th International Symposium on Programming Languages, Implementations, Logics, and Programs, volume 1140 of LNCS
, 1996

"... We present the oe w -calculus, a formal synthesis of the concepts of sharing and explicit substitution for weak reduction. We show how w can be used as a foundation of implementations of functional programming languages by modeling the essential ingredients of such implementations, namely wea ..."

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We present the oe w -calculus, a formal synthesis of the concepts of sharing and explicit substitution for weak reduction. We show how w can be used as a foundation of implementations of functional programming languages by modeling the essential ingredients of such implementations, namely weak reduction strategies, recursion, space leaks, recursive data structures, and parallel evaluation, in a uniform way.

### Approximation and Normalization Results for Typeable Combinator Systems

, 1995

"... This paper studies the relation between types and normalization in the context of Combinator Systems. It presents notions of approximants for terms, intersection type assignment, and reduction on derivations; the last will be proved to be strongly normalizable. With this result, it is proved that, f ..."

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This paper studies the relation between types and normalization in the context of Combinator Systems. It presents notions of approximants for terms, intersection type assignment, and reduction on derivations; the last will be proved to be strongly normalizable. With this result, it is proved that, for every typeable term, there exists an approximant of that term with the same type, and a characterization of the normalization behaviour of terms using their assignable types is given. Introduction In this paper we will focus on the relation between assignable intersection types for terms and, respectively, types for their approximants and normalization in the framework of Combinator Systems (CS). This topic has been studied extensively in the setting of Lambda Calculus (LC) [6] (see [9, 7, 1, 3]), but, perhaps surprisingly, to our knowledge it has never been studied for CS. Moreover, in systems without explicit abstraction, these results are harder to obtain; in fact, in order to prove t...

### TEACHING CURRIED FUNCTIONS

, 1995

"... Curried functions are an important topic in Computing courses that teach functional programming, including courses that study programming languages. Good motivating examples for teaching curried functions and their utility can be taken from Physics. BACKGROUND Curried functions and currying are an i ..."

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Curried functions are an important topic in Computing courses that teach functional programming, including courses that study programming languages. Good motivating examples for teaching curried functions and their utility can be taken from Physics. BACKGROUND Curried functions and currying are an important topic in computer science courses that teach functional programming [14, section 7.3]. Such courses include undergraduate courses in programming paradigms (unit PL11 in the ACM’s Computing Curricula 1991 [15]), with titles such as “Principles of Programming Languages ” [11, p. 388] [10, p. 100]. Also included are undergraduate and graduate courses in programming language semantics (unit PL10 in [15]), with titles such as “Essentials of Programming Languages ” [7, p.27]. Curried functions are also directly supported by some modern functional programming languages, such as Haskell [4]. Curry [2,3] and others [13] [5, pages 153-156] studying the concept of a function asked the question: “does a programming language need to provide functions with