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Step By Recursive Step: Church's Analysis Of Effective Calculability
 BULLETIN OF SYMBOLIC LOGIC
, 1997
"... Alonzo Church's mathematical work on computability and undecidability is wellknown indeed, and we seem to have an excellent understanding of the context in which it arose. The approach Church took to the underlying conceptual issues, by contrast, is less well understood. Why, for example, was "Ch ..."
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Alonzo Church's mathematical work on computability and undecidability is wellknown indeed, and we seem to have an excellent understanding of the context in which it arose. The approach Church took to the underlying conceptual issues, by contrast, is less well understood. Why, for example, was "Church's Thesis" put forward publicly only in April 1935, when it had been formulated already in February/March 1934? Why did Church choose to formulate it then in terms of G odel's general recursiveness, not his own #definability as he had done in 1934? A number of letters were exchanged between Church and Paul Bernays during the period from December 1934 to August 1937; they throw light on critical developments in Princeton during that period and reveal novel aspects of Church's distinctive contribution to the analysis of the informal notion of e#ective calculability. In particular, they allow me to give informed, though still tentative answers to the questions I raised; the char...
Topology, Domain Theory and Theoretical Computer Science
, 1997
"... In this paper, we survey the use of ordertheoretic topology in theoretical computer science, with an emphasis on applications of domain theory. Our focus is on the uses of ordertheoretic topology in programming language semantics, and on problems of potential interest to topologists that stem from ..."
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In this paper, we survey the use of ordertheoretic topology in theoretical computer science, with an emphasis on applications of domain theory. Our focus is on the uses of ordertheoretic topology in programming language semantics, and on problems of potential interest to topologists that stem from concerns that semantics generates. Keywords: Domain theory, Scott topology, power domains, untyped lambda calculus Subject Classification: 06B35,06F30,18B30,68N15,68Q55 1 Introduction Topology has proved to be an essential tool for certain aspects of theoretical computer science. Conversely, the problems that arise in the computational setting have provided new and interesting stimuli for topology. These problems also have increased the interaction between topology and related areas of mathematics such as order theory and topological algebra. In this paper, we outline some of these interactions between topology and theoretical computer science, focusing on those aspects that have been mo...
Normalisation in Weakly Orthogonal Rewriting
, 1999
"... . A rewrite sequence is said to be outermostfair if every outermost redex occurrence is eventually eliminated. Outermostfair rewriting is known to be (head)normalising for almost orthogonal rewrite systems. In this paper we study (head)normalisation for the larger class of weakly orthogonal rewr ..."
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. A rewrite sequence is said to be outermostfair if every outermost redex occurrence is eventually eliminated. Outermostfair rewriting is known to be (head)normalising for almost orthogonal rewrite systems. In this paper we study (head)normalisation for the larger class of weakly orthogonal rewrite systems. Normalisation is established and a counterexample against headnormalisation is provided. Nevertheless, infinitary normalisation, which is usually obtained as a corollary of headnormalisation, is shown to hold. 1 Introduction The term f(a) in the term rewrite system fa ! a; f(x) ! bg can be rewritten to normal form b, but is also the starting point of the infinite rewrite sequence f(a) ! f(a) ! : : :. It is then of interest to design a normalising strategy, i.e. a restriction on rewriting which guarantees to reach a normal form if one can be reached. How to design a normalising strategy? Observe that in the example the normal form b was reached by contracting the redex closest...
Combinator Shared Reduction and Infinite Objects in Type Theory
, 1996
"... We will present a syntactical proof of correctness and completeness of shared reduction. This work is an application of type theory extended with infinite objects and coinduction. ..."
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We will present a syntactical proof of correctness and completeness of shared reduction. This work is an application of type theory extended with infinite objects and coinduction.
How to Normalize the Jay
 Theoretical Computer Science
"... In this note we give an elementary proof of the strong normalization property of the J combinator by providing an explicit bound for the maximal length of the reduction paths of a term. This result shows nicely that in the theorem of Toyama, Klop and Barendregt on completeness of unions of left line ..."
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In this note we give an elementary proof of the strong normalization property of the J combinator by providing an explicit bound for the maximal length of the reduction paths of a term. This result shows nicely that in the theorem of Toyama, Klop and Barendregt on completeness of unions of left linear term rewriting systems, disjointness is essential. Keywords: Term rewriting systems; Combinatory logic; Strong normalization 1 Introduction The combinators I and J with their reduction rules Ia # a and Jabcd # ab(adc) were introduced by Rosser [2] in 1935. These two combinators are of particular interest since they form a basis for the #Icalculus (cf. e.g. Barendregt [1]). In combinatory logic, it is natural to ask whether a certain system is strongly normalizing, i.e. whether there exists no term with an infinite reduction path. Many standard combinators such as K, B, C and I are strongly normalizing, with the notable exception of S. But surprisingly, it appears to be unknown whe...
Lambda Calculus
, 2002
"... Contents Introduction To The Lecture Notes v 1 The Untyped Lambda Calculus 1 1.1 Inductive Definitions . . . . . . . . . . . . . . . . . . . . . . . 1 1.2 Terms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 1.3 # is a Binding Operator  AlphaConversion . . . . . . . . . 3 1.4 # ..."
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Contents Introduction To The Lecture Notes v 1 The Untyped Lambda Calculus 1 1.1 Inductive Definitions . . . . . . . . . . . . . . . . . . . . . . . 1 1.2 Terms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 1.3 # is a Binding Operator  AlphaConversion . . . . . . . . . 3 1.4 # Performs Substitution  BetaConversion . . . . . . . . . . 4 1.5 Contexts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 1.6 Formal Theory of the Lambda Calculus . . . . . . . . . . . . 6 1.7 A Fixed Point Theorem . . . . . . . . . . . . . . . . . . . . . 7 1.8 Other Lambda Theories . . . . . . . . . . . . . . . . . . . . . 8 2 Reduction, Consistency 11 2.1 Notions of Reduction in the Lambda Calculus . . . . . . . . . 11 2.2 Beta Reduction . . . . . . . . . . . . . . . . . . . . . . . . . . 13 2.3 Properties of Notions of Reduction and Terms . . . . . . . . . 14 2.4 ChurchRosser Property of Beta Reduction . . . . . . . . . . 15 2.5 Consistency . . . . . . . . . . . . . .