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57
Inductive and Coinductive types with Iteration and Recursion
 Proceedings of the 1992 Workshop on Types for Proofs and Programs, Bastad
, 1992
"... We study (extensions of) simply and polymorphically typed lambda calculus from a point of view of how iterative and recursive functions on inductive types are represented. The inductive types can usually be understood as initial algebras in a certain category and then recursion can be defined in ter ..."
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Cited by 56 (1 self)
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We study (extensions of) simply and polymorphically typed lambda calculus from a point of view of how iterative and recursive functions on inductive types are represented. The inductive types can usually be understood as initial algebras in a certain category and then recursion can be defined in terms of iteration. However, in the syntax we often have only weak initiality, which makes the definition of recursion in terms of iteration inefficient or just impossible. We propose a categorical notion of (primitive) recursion which can easily be added as computation rule to a typed lambda calculus and gives us a clear view on what the dual of recursion, corecursion, on coinductive types is. (The same notion has, independently, been proposed by [Mendler 1991].) We look at how these syntactic notions work out in the simply typed lambda calculus and the polymorphic lambda calculus. It will turn out that in the syntax, recursion can be defined in terms of corecursion and vice versa using polymo...
Computability and recursion
 BULL. SYMBOLIC LOGIC
, 1996
"... We consider the informal concept of “computability” or “effective calculability” and two of the formalisms commonly used to define it, “(Turing) computability” and “(general) recursiveness.” We consider their origin, exact technical definition, concepts, history, general English meanings, how they b ..."
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Cited by 38 (1 self)
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We consider the informal concept of “computability” or “effective calculability” and two of the formalisms commonly used to define it, “(Turing) computability” and “(general) recursiveness.” We consider their origin, exact technical definition, concepts, history, general English meanings, how they became fixed in their present roles, how they were first and are now used, their impact on nonspecialists, how their use will affect the future content of the subject of computability theory, and its connection to other related areas. After a careful historical and conceptual analysis of computability and recursion we make several recommendations in section §7 about preserving the intensional differences between the concepts of “computability” and “recursion.” Specifically we recommend that: the term “recursive ” should no longer carry the additional meaning of “computable” or “decidable;” functions defined using Turing machines, register machines, or their variants should be called “computable” rather than “recursive;” we should distinguish the intensional difference between Church’s Thesis and Turing’s Thesis, and use the latter particularly in dealing with mechanistic questions; the name of the subject should be “Computability Theory” or simply Computability rather than
'Computing' as information compression by multiple alignment, unification and search
 Journal of Universal Computer Science
, 1999
"... This paper argues that the operations of a `Universal Turing Machine' (UTM) and equivalent mechanisms such as the `Post Canonical System' (PCS)  which are widely accepted as definitions of the concept of `computing'  may be interpreted as information compression by multiple alig ..."
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Cited by 29 (14 self)
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This paper argues that the operations of a `Universal Turing Machine' (UTM) and equivalent mechanisms such as the `Post Canonical System' (PCS)  which are widely accepted as definitions of the concept of `computing'  may be interpreted as information compression by multiple alignment, unification and search (ICMAUS). The motivation for this interpretation is that it suggests ways in which the UTM/PCS model may be augmented in a proposed new computing system designed to exploit the ICMAUS principles as fully as possible. The provision of a relatively sophisticated search mechanism in the proposed `SP' system appears to open the door to the integration and simplification of a range of functions including unsupervised inductive learning, bestmatch pattern recognition and information retrieval, probabilistic reasoning, planning and problem solving, and others. Detailed consideration of how the ICMAUS principles may be applied to these functions is outside the scope of this article but relevant sources are cited in this article.
The Impact of the Lambda Calculus in Logic and Computer Science
 BULLETIN OF SYMBOLIC LOGIC
, 1997
"... One of the most important contributions of A. Church to logic is his invention of the lambda calculus. We present the genesis of this theory and its two major areas of application: the representation of computations and the resulting functional programming languages on the one hand and the represent ..."
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Cited by 28 (1 self)
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One of the most important contributions of A. Church to logic is his invention of the lambda calculus. We present the genesis of this theory and its two major areas of application: the representation of computations and the resulting functional programming languages on the one hand and the representation of reasoning and the resulting systems of computer mathematics on the other hand.
Notions of computability at higher types I
 In Logic Colloquium 2000
, 2005
"... We discuss the conceptual problem of identifying the natural notions of computability at higher types (over the natural numbers). We argue for an eclectic approach, in which one considers a wide range of possible approaches to defining higher type computability and then looks for regularities. As a ..."
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Cited by 19 (5 self)
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We discuss the conceptual problem of identifying the natural notions of computability at higher types (over the natural numbers). We argue for an eclectic approach, in which one considers a wide range of possible approaches to defining higher type computability and then looks for regularities. As a first step in this programme, we give an extended survey of the di#erent strands of research on higher type computability to date, bringing together material from recursion theory, constructive logic and computer science. The paper thus serves as a reasonably complete overview of the literature on higher type computability. Two sequel papers will be devoted to developing a more systematic account of the material reviewed here.
Binary Lambda Calculus and Combinatory Logic.” Sep 14, 2004. http://homepages. cwi.nl/ ∼ tromp/cl/LC.pdf [64] Tadaki, K. “Upper bound by Kolmogorov complexity for the probability
 in computable POVM measurement.” Proceedings of the 5th Conference on Real Numbers and Computers, RNC5
, 2003
"... In the first part, we introduce binary representations of both lambda calculus and combinatory logic terms, and demonstrate their simplicity by providing very compact parserinterpreters for these binary languages. Along the way we also present new results on list representations, bracket abstractio ..."
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Cited by 14 (0 self)
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In the first part, we introduce binary representations of both lambda calculus and combinatory logic terms, and demonstrate their simplicity by providing very compact parserinterpreters for these binary languages. Along the way we also present new results on list representations, bracket abstraction, and fixpoint combinators. In the second part we review Algorithmic Information Theory, for which these interpreters provide a convenient vehicle. We demonstrate this with several concrete upper bounds on programsize complexity, including an elegant selfdelimiting code for binary strings. 1
Algoristic foundations to cognitive psychology
 In
, 1974
"... What is known and how it is known are relative questions that make no sense independent of the question of who knows. Indeed, our opinion is that the central question of cognitive psychology concerns the essential nature of a knowingagent, rather than just what is known or even how what is known is ..."
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Cited by 12 (3 self)
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What is known and how it is known are relative questions that make no sense independent of the question of who knows. Indeed, our opinion is that the central question of cognitive psychology concerns the essential nature of a knowingagent, rather than just what is known or even how what is known is known. Only a certain kind of sceptic can hold that all things are relative without falling into the absurdity that if everything were relative, there would of course be nothing for it to be In the past decade or so psychology has relinquished its obsessive concern for the question of how organisms behave, or can be made to behave, in favor of a broader set of questions. It is now popular to assume that what people process is
Physics, Topology, Logic and Computation: A Rosetta Stone
, 2009
"... Category theory is a very general formalism, but there is a certain special way that physicists use categories which turns out to have close analogues in topology, logic and computation. A category has objects and morphisms, which represent things and ways to go between things. In physics, the objec ..."
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Cited by 11 (1 self)
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Category theory is a very general formalism, but there is a certain special way that physicists use categories which turns out to have close analogues in topology, logic and computation. A category has objects and morphisms, which represent things and ways to go between things. In physics, the objects are often physical systems, and the morphisms are processes turning a state of one physical system into a state of another system — perhaps