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Parametric and TypeDependent Polymorphism
, 1995
"... Data Types, though, as Reynolds stresses, is not perfectly suited for higher type or higher order systems and, thus, he proposes a "relational" treatment of invariance: computations do not depend on types in the sense that they are "invariant" w.r.t. arbitrary relations on types ..."
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Cited by 10 (5 self)
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Data Types, though, as Reynolds stresses, is not perfectly suited for higher type or higher order systems and, thus, he proposes a "relational" treatment of invariance: computations do not depend on types in the sense that they are "invariant" w.r.t. arbitrary relations on types and between types. Reynolds's approach set the basis for most of the current work on parametricity, as we will review below (.3). Some twelve years earlier, Girard had given just a simple hint towards another understanding of the properties of "computing with types". In [Gir71], it is shown, as a side remark, that, given a type A, if one defines a term J A such that, for any type B, J A B reduces to 1, if A = B, and reduces to 0, if A ¹ B, then F + J A does not normalize. In particular, then, J A is not definable in F. This remark on how terms may depend on types is inspired by a view of types which is quite different from Reynolds's. System F was born as the theory of proofs of second order intuitionis...
Type Theory via Exact Categories (Extended Abstract)
 In Proceedings of the 13th Annual IEEE Symposium on Logic in Computer Science LICS '98
, 1998
"... Partial equivalence relations (and categories of these) are a standard tool in semantics of type theories and programming languages, since they often provide a cartesian closed category with extended definability. Using the theory of exact categories, we give a categorytheoretic explanation of why ..."
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Partial equivalence relations (and categories of these) are a standard tool in semantics of type theories and programming languages, since they often provide a cartesian closed category with extended definability. Using the theory of exact categories, we give a categorytheoretic explanation of why the construction of a category of partial equivalence relations often produces a cartesian closed category. We show how several familiar examples of categories of partial equivalence relations fit into the general framework. 1 Introduction Partial equivalence relations (and categories of these) are a standard tool in semantics of programming languages, see e.g. [2, 5, 7, 9, 15, 17, 20, 22, 35] and [6, 29] for extensive surveys. They are usefully applied to give proofs of correctness and adequacy since they often provide a cartesian closed category with additional properties. Take for instance a partial equivalence relation on the set of natural numbers: a binary relation R ` N\ThetaN on th...
REFLECTIONS ON FORMALISM AND REDUCTIONISM IN LOGIC AND COMPUTER SCIENCE
"... This report contains a preprint (paper 1) and a reprint (paper 2). The first develops some epistemological views which were hinted in the second, in particular by stressing the need of a greater role of geometric insight and images in foundational studies and in approaches to cognition. The second p ..."
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This report contains a preprint (paper 1) and a reprint (paper 2). The first develops some epistemological views which were hinted in the second, in particular by stressing the need of a greater role of geometric insight and images in foundational studies and in approaches to cognition. The second paper is the "philosophical " part of a lecture in Type Theory, whose technical sections, omitted here, have been largely subsumed by
Notes on Restriction Categories
, 2014
"... This is a summary of some notes taken by me during the course given ..."
1 An expression of closure to efficient causation in terms of lambdacalculus1
"... In this paper, we propose a mathematical expression of closure to efficient causation in terms of λcalculus; we argue that this opens up the perspective of developing principled computer simulations of systems closed to efficient causation in an appropriate programming language. An important implic ..."
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In this paper, we propose a mathematical expression of closure to efficient causation in terms of λcalculus; we argue that this opens up the perspective of developing principled computer simulations of systems closed to efficient causation in an appropriate programming language. An important implication of our formulation is that, by exhibiting an expression in λcalculus, which is a paradigmatic formalism for computability and programming, we show that there are no conceptual or principled problems in realizing a computer simulation or model of closure to efficient causation. We conclude with a brief discussion of the question whether closure to efficient causation captures all relevant properties of living systems. We suggest that it might not be the case, and that more complex definitions could indeed create some obstacles to computability.
Contents
, 2007
"... Abstract. Message passing is a key ingredient of concurrent programming. The purpose of this paper is to describe the equivalence between the proof theory, the categorical semantics, and term calculus of message passing. In order to achieve this we introduce the categorical notion of a linear actego ..."
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Abstract. Message passing is a key ingredient of concurrent programming. The purpose of this paper is to describe the equivalence between the proof theory, the categorical semantics, and term calculus of message passing. In order to achieve this we introduce the categorical notion of a linear actegory and the related polycategorical notion of a polyactegory. Not surprisingly the notation used for the term calculus borrows heavily from the (synchronous) πcalculus. The cut elimination procedure for the system provides an operational
Writing Worth?
"... Abstract. In one of the many and fundamental sideremarks made by Turing in his 1950 paper (The Imitation Game paper), an analogy is made between Mechanism and Writing. Turing is aware that his Machine is a writing/rewriting mechanism, but he doesn’t go deeper into the comparison. Striding along th ..."
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Abstract. In one of the many and fundamental sideremarks made by Turing in his 1950 paper (The Imitation Game paper), an analogy is made between Mechanism and Writing. Turing is aware that his Machine is a writing/rewriting mechanism, but he doesn’t go deeper into the comparison. Striding along the history of writing, we will hint here at the nature and the role of alphabetic writing in the invention of Turing’s (and today’s) notion of computability. We will stress that computing is a matter of alphabetic sequence checking and replacement, far away from the physical world, yet related to it once the role of physical measurement is taken into account. Turing Morphogenesis paper, 1952, provides the guidelines for the modern analysis of “continuous dynamics ” at the core Turing’s late and innovative approach to biophysical processes 3.
Some Topologies for Computations∗
, 2003
"... 1.1 Computations The applications of topological and order structures in Theory of Computation, a key aspect of Foundations of Mathematics and of Theoretical Computer Science, has various origins and it is largely due to the relevance of these struc ..."
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1.1 Computations The applications of topological and order structures in Theory of Computation, a key aspect of Foundations of Mathematics and of Theoretical Computer Science, has various origins and it is largely due to the relevance of these struc