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The Longest Perpetual Reductions in Orthogonal Expression Reduction Systems
 In: Proc. of the 3 rd International Conference on Logical Foundations of Computer Science, LFCS'94, A. Nerode and Yu.V. Matiyasevich, eds., Springer LNCS
, 1994
"... We consider reductions in Orthogonal Expression Reduction Systems (OERS), that is, Orthogonal Term Rewriting Systems with bound variables and substitutions, as in the calculus. We design a strategy that for any given term t constructs a longest reduction starting from t if t is strongly normaliza ..."
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Cited by 18 (8 self)
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We consider reductions in Orthogonal Expression Reduction Systems (OERS), that is, Orthogonal Term Rewriting Systems with bound variables and substitutions, as in the calculus. We design a strategy that for any given term t constructs a longest reduction starting from t if t is strongly normalizable, and constructs an infinite reduction otherwise. The Conservation Theorem for OERSs follows easily from the properties of the strategy. We develop a method for computing the length of a longest reduction starting from a strongly normalizable term. We study properties of pure substitutions and several kinds of similarity of redexes. We apply these results to construct an algorithm for computing lengths of longest reductions in strongly persistent OERSs that does not require actual transformation of the input term. As a corollary, we have an algorithm for computing lengths of longest developments in OERSs. 1 Introduction A strategy is perpetual if, given a term t, it constructs an infinit...
Finite Family Developments
"... Associate to a rewrite system R having rules l → r, its labelled version R ω having rules l ◦ m+1 → r • , for any natural number m m ∈ ω. These rules roughly express that a lefthand side l carrying labels all larger than m can be replaced by its righthand side r carrying labels all smaller than o ..."
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Cited by 13 (6 self)
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Associate to a rewrite system R having rules l → r, its labelled version R ω having rules l ◦ m+1 → r • , for any natural number m m ∈ ω. These rules roughly express that a lefthand side l carrying labels all larger than m can be replaced by its righthand side r carrying labels all smaller than or equal to m. A rewrite system R enjoys finite family developments (FFD) if R ω is terminating. We show that the class of higher order pattern rewrite systems enjoys FFD, extending earlier results for the lambda calculus and first order term rewrite systems.
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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Cited by 11 (0 self)
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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...
A Deterministic Notation for Cooperating Processes
 Preprint MCSP3460193, Mathematics and Computer Science Division, Argonne National Laboratory, Argonne, Ill
, 1993
"... This paper proposes extensions of sequential programming languages for parallel programming that have the following features: 1. Dynamic Structures The process structure is dynamic: Processes and variables can be created and deleted. 2. Paradigm Integration The programming notation allows shared mem ..."
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Cited by 9 (3 self)
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This paper proposes extensions of sequential programming languages for parallel programming that have the following features: 1. Dynamic Structures The process structure is dynamic: Processes and variables can be created and deleted. 2. Paradigm Integration The programming notation allows shared memory and message passing. 3. Determinism Demonstrating that a program is deterministic  all executions with the same input produce the same output  is straightforward. A program can be written so that the compiler can verify that the program is deterministic. Nondeterministic constructs can be introduced in a sequence of refinement steps to obtain greater efficiency if required. The ideas have been incorporated in an extension of Fortran, but the underlying sequential imperative language is not central to the ideas described here. Keywords: parallel programming languages, determinism, functional programming, multicomputers, debugging Supported by NSF Center for Research in Parallel ...
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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Cited by 8 (4 self)
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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...
Lazy XSL Transformations
 In ACM DocEng
, 2003
"... We introduce a lazy XSLT interpreter that provides random access to the transformation result. This allows e# cient pipelining of transformation sequences. Nodes of the result tree are computed only upon initial access. As these computations have limited fanin, sparse output coverage propagates ba ..."
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Cited by 8 (0 self)
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We introduce a lazy XSLT interpreter that provides random access to the transformation result. This allows e# cient pipelining of transformation sequences. Nodes of the result tree are computed only upon initial access. As these computations have limited fanin, sparse output coverage propagates backwards through the pipeline.
Towards Concurrent Type Theory
 INVITED TALK AT TLDI’12
, 2012
"... We review progress in a recent line of research that provides a concurrent computational interpretation of (intuitionistic) linear logic. Propositions are interpreted as session types, sequent proofs as processes in the πcalculus, cut reductions as process reductions, and vice versa. The strong pro ..."
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Cited by 8 (7 self)
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We review progress in a recent line of research that provides a concurrent computational interpretation of (intuitionistic) linear logic. Propositions are interpreted as session types, sequent proofs as processes in the πcalculus, cut reductions as process reductions, and vice versa. The strong prooftheoretic foundation of this type system provides immediate opportunities for uniform generalization, specifically, to embed terms from a functional type theory. The resulting system satisfies the properties of type preservation, progress, and termination, as expected from a language derived via a CurryHoward isomorphism. While very expressive, the language is strictly stratified so that dependent types for functional terms can be enforced during communication, but neither processes nor channels can appear in functional terms. We briefly speculate on how this limitation might be overcome to arrive at a fully dependent concurrent type theory.
E.: The Intensional Lambda Calculus
 LFCS
, 2007
"... Abstract. We introduce a natural deduction formulation for the Logic of Proofs, a refinement of modal logic S4 in which the assertion ✷A is replaced by [s]A whose intended reading is “s is a proof of A”. A term calculus for this formulation yields a typed lambda calculus λ I that internalises intens ..."
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Cited by 8 (2 self)
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Abstract. We introduce a natural deduction formulation for the Logic of Proofs, a refinement of modal logic S4 in which the assertion ✷A is replaced by [s]A whose intended reading is “s is a proof of A”. A term calculus for this formulation yields a typed lambda calculus λ I that internalises intensional information on how a term is computed. In the same way that the Logic of Proofs internalises its own derivations, λ I internalises its own computations. Confluence and strong normalisation of λ I is proved. This system serves as the basis for the study of type theories that internalise intensional aspects of computation. 1
Perpetual Reductions in λCalculus
, 1999
"... This paper surveys a part of the theory of fireduction in calculus which might aptly be called perpetual reductions. The theory is concerned with perpetual reduction strategies, i.e., reduction strategies that compute infinite reduction paths from terms (when possible), and with perpetual red ..."
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Cited by 7 (0 self)
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This paper surveys a part of the theory of fireduction in calculus which might aptly be called perpetual reductions. The theory is concerned with perpetual reduction strategies, i.e., reduction strategies that compute infinite reduction paths from terms (when possible), and with perpetual redexes, i.e., redexes whose contraction in terms preserves the possibility (when present) of infinite reduction paths. The survey not only recasts classical theorems in a unified setting, but also offers new results, proofs, and techniques, as well as a number of applications to problems in calculus and type theory. 1. Introduction Considerable attention has been devoted to classification of reduction strategies in typefree calculus [4, 6, 7, 15, 38, 44, 81]see also [2, Ch. 13]. We are concerned with strategies differing in the length of reduction paths. This paper draws on several sources. In late 1994, van Raamsdonk and Severi [59] and Srensen [66, 67] independently developed ...
The Mechanisation of BarendregtStyle Equational Proofs (the Residual Perspective)
, 2001
"... We show how to mechanise equational proofs about higherorder languages by using the primitive proof principles of firstorder abstract syntax over onesorted variable names. We illustrate the method here by proving (in Isabelle/HOL) a technical property which makes the method widely applicable for ..."
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Cited by 7 (4 self)
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We show how to mechanise equational proofs about higherorder languages by using the primitive proof principles of firstorder abstract syntax over onesorted variable names. We illustrate the method here by proving (in Isabelle/HOL) a technical property which makes the method widely applicable for the λcalculus: the residual theory of β is renamingfree upto an initiality condition akin to the socalled Barendregt Variable Convention. We use our results to give a new diagrambased proof of the development part of the strong finite development property for the λcalculus. The proof has the same equational implications (e.g., confluence) as the proof of the full property but without the need to prove SN. We account for two other uses of the proof method, as presented elsewhere. One has been mechanised in full in Isabelle/HOL.