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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.
Notions of computability at higher types II
 In preparation
, 2001
"... ntroduce some simple general theory to allow us to talk about notions of highertype computable functional. The following definitions (with minor variations) appear frequently in the literature. Definition 1.1 (Weak partial type structures) A weak partial type structure, or weak PTS A [over a set X ..."
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ntroduce some simple general theory to allow us to talk about notions of highertype computable functional. The following definitions (with minor variations) appear frequently in the literature. Definition 1.1 (Weak partial type structures) A weak partial type structure, or weak PTS A [over a set X], consists of the following data: . for each type #, a set A # of elements of type # [equipped with a canonical bijection A 0 # = X], . for each #, # , a partial application function ## : A ### A # # A # . We usually omit type subscripts from application operations, and often write x y simply as xy. By convention, w
A Physical Example For Teaching Curried Functions
 Chapter 8 of Parallel and Distributed Computing Handbook (A.Y. Zomaya,Editor), McGrawHill
, 1996
"... 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 ..."
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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 153156] studying the concept of a function asked the question...
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 firstorder 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 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...
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, semiinductive definitions, which require nontrivial set theoretic results); 5. (partly) enhancing new axioms in set theory: the case of antifoundation AFA and the mathematics of circular phenomena; 6. suggesting the investigation of nonclassical logical systems, from contractionfree and manyvalued logics to systems with generalized quantifiers; 7. suggesting frameworks with flexible typing for the foundations of Mathematics and Computer Science; 8. applying forms of selfreferential 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.
Before we begin: Thank you, thank you, thank you,... Walter.WassollmanzudiesemMannnochsagen?
"... Doctor rerum naturalium ..."
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AN APPROXIMATION RESULT FOR JAVA VIA STRONGLY NORMALISING INTERSECTION TYPE DERIVATIONS (Extended Abstract)
"... The intersection type discipline (ITD) is wellestablished for the Lambda Calculus (LC) and the functional programming paradigm. It has also been extended to Term Rewriting Systems (TRS) and more recently to object calculi and sequent calculi. We continue this trend by applying the techniques of ITD ..."
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The intersection type discipline (ITD) is wellestablished for the Lambda Calculus (LC) and the functional programming paradigm. It has also been extended to Term Rewriting Systems (TRS) and more recently to object calculi and sequent calculi. We continue this trend by applying the techniques of ITD to the analysis of the (class based) objectoriented (OO) programming paradigm: specifically, we study a small core calculus for Java which is a restriction of Featherweight Java by removing casts. Our main contribution is an approximation result for this programming model, demonstrating a direct correspondence between types and the functional behaviour of programs. This opens the possibility for typebased abstract interpretation and termination analysis for OO. We achieve this result by defining a notion of reduction on type derivations that is strongly normalising, a technique which has also been used for LC and TRS. Finally, we show how the approximation result facilitates a typebased characterisation of (weak) normalisation and termination. We also discuss the relationship between our calculus and TRS, highlighting how our result extends previous work in this area.