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On Global Dynamics of Optimal Graph Reduction
 1997 ACM International Conference on Functional Programming
, 1997
"... Optimal graph reduction technology for the calculus, as developed by Lamping, with modifications by Asperti, Gonthier, Abadi, and L'evy, has a wellunderstood local dynamics based on a standard menagerie of reduction rules, as well as a global context semantics based on Girard's geometry of interac ..."
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Cited by 7 (1 self)
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Optimal graph reduction technology for the calculus, as developed by Lamping, with modifications by Asperti, Gonthier, Abadi, and L'evy, has a wellunderstood local dynamics based on a standard menagerie of reduction rules, as well as a global context semantics based on Girard's geometry of interaction. However, the global dynamics of graph reduction has not been subject to careful investigation. In particular, graphs lose their structural resemblence to terms after only a few graph reduction steps, and little is known about graph reduction strategies that maintain efficiency or structure. While the context semantics provides global information about the computation, its use as part of a reduction strategy seems computationally infeasible. We propose a tractable graph reduction strategy that preserves computationally relevant global structure, and allows us to efficiently bound the computational resources needed to implement optimal reduction. A simple canonical representation for gr...
Process Semantics of Graph Reduction
 Proc. CONCUR '95, volume 962 of Lecture Notes in Computer Science
, 1995
"... This paper introduces an operational semantics for callbyneed reduction in terms of Milner's ßcalculus. The functional programming interest lies in the use of ßcalculus as an abstract yet realistic target language. The practical value of the encoding is demonstrated with an outline for a paralle ..."
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Cited by 6 (1 self)
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This paper introduces an operational semantics for callbyneed reduction in terms of Milner's ßcalculus. The functional programming interest lies in the use of ßcalculus as an abstract yet realistic target language. The practical value of the encoding is demonstrated with an outline for a parallel code generator. From a theoretical perspective, the ßcalculus representation of computational strategies with shared reductions is novel and solves a problem posed by Milner [13]. The compactness of the process calculus presentation makes it interesting as an alternative definition of callbyneed. Correctness of the encoding is proved with respect to the callbyneed calculus of Ariola et al. [3]. 1 Introduction Graph reduction of extended calculi has become a mature field of applied research. The efficiency of the implementations is due in great measure to a technique known as `sharing', whereby argument values are computed (at most) once and then memoized for future reference. Both...
Innocent Game Models of Untyped λCalculus
 Theoretical Computer Science
, 2000
"... We present a new denotational model for the untyped calculus, using the techniques of game semantics. The strategies used are innocent in the sense of Hyland and Ong [9] and Nickau [17], but the traditional distinction between \question" and \answer" moves is removed. We rst construct models D and ..."
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Cited by 3 (1 self)
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We present a new denotational model for the untyped calculus, using the techniques of game semantics. The strategies used are innocent in the sense of Hyland and Ong [9] and Nickau [17], but the traditional distinction between \question" and \answer" moves is removed. We rst construct models D and DREC as global sections of a reexive object in the categories A and A REC of arenas and innocent and recursive innocent strategies respectively. We show that these are sensible algebras but are neither extensional nor universal. We then introduce a new representation of innocent strategies in an economical form. We show a strong connexion between the economical form of the denotation of a term in the game models and a variablefree form of the Nakajima tree of the term. Using this we show that the denable elements of DREC are precisely what we call eectively almosteverywhere copycat (EAC) strategies. The category A EAC with these strategies as morphisms gives rise to a model D...
Innocent Game Models of Untyped λCalculus
, 2000
"... We present a new denotation model for the untyped λcalculus, using the techniques of game semantics. The strategies used are innocent in the sense of Hyland and Ong [HO94] and Nickau [Nic96], but the traditional distinction between "question" and "answer" moves is removed. We first construct mod ..."
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Cited by 3 (1 self)
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We present a new denotation model for the untyped λcalculus, using the techniques of game semantics. The strategies used are innocent in the sense of Hyland and Ong [HO94] and Nickau [Nic96], but the traditional distinction between "question" and "answer" moves is removed. We first construct models D and DREC as global sections of a reflexive object in the categories A and A REC of arenas and innocent and recursive innocent strategies respectively. We show that these are sensible algebras but are neither extensional nor universal. We then introduce a new representation of innocent strategies in an economical form. We show a stong connexion between the economical form of the denotation of a term in the game models and a variablefree form of the Nakajima tree of the term. Using this we show that the denable elements of DREC are precisely what we call effectively almosteverywhere copycat (EAC) strategies. The category A EAC with these strategies as morphisms gives rise to a ...
Innocent Game Models of Untyped
, 2000
"... We present a new denotation model for the untyped calculus, using the techniques of game semantics. The strategies used are innocent in the sense of Hyland and Ong [HO94] and Nickau [Nic96], but the traditional distinction between \question" and \answer" moves is removed. We rst construct models ..."
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Cited by 1 (0 self)
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We present a new denotation model for the untyped calculus, using the techniques of game semantics. The strategies used are innocent in the sense of Hyland and Ong [HO94] and Nickau [Nic96], but the traditional distinction between \question" and \answer" moves is removed. We rst construct models D and DREC as global sections of a reexive object in the categories A and A REC of arenas and innocent and recursive innocent strategies respectively. We show that these are sensible algebras but are neither extensional nor universal. We then introduce a new representation of innocent strategies in an economical form. We show a stong connexion between the economical form of the denotation of a term in the game models and a variablefree form of the Nakajima tree of the term. Using this we show that the denable elements of DREC are precisely what we call eectively almosteverywhere copycat (EAC) strategies. The category A EAC with these strategies as morphisms gives rise to a ...
Proofs as Graphs
"... This note introduces a method of representing and reasoning about the actions of a class of proof procedures. A graphlike structure, called a proof diagram, is introduced in which conclusions of inferences can be shared. A version of Kruskal's Tree Theorem is developed for these structures and from ..."
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This note introduces a method of representing and reasoning about the actions of a class of proof procedures. A graphlike structure, called a proof diagram, is introduced in which conclusions of inferences can be shared. A version of Kruskal's Tree Theorem is developed for these structures and from there a notion of minimal proof is introduced. The notion of minimal proof allows us to make a link between standard treatments of proof and proofs generated by mechanical theorem provers. We discuss various uses for the proof diagrams including the development of tactics by graph reduction and propose a graph based metalanguage for theorem provers. 1 Introduction Most mechanical proof procedures, such as KnuthBendix completion [10], Grobner bases [3] and resolution [13], have a similar computational behaviour. Starting with a set of assumptions, some subset of those assumptions is used to produce some conclusion which is then added to the set of assumptions. This process then continues ...
CompileTime Pointer Reversal
, 1996
"... This paper introduces an alternative representation for λterms which has the notable property that the search for the leftmost outermost redex is restricted to two steps. This is important in the implementation of a lazy functional programming language, as this search consumes time and space. The r ..."
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This paper introduces an alternative representation for λterms which has the notable property that the search for the leftmost outermost redex is restricted to two steps. This is important in the implementation of a lazy functional programming language, as this search consumes time and space. The representation introduced is similar to that resulting from the implementation technique of reversing pointers in the left spine of a term while traversing it, except that here the pointers in the left spine are reversed before reduction. This paper completely develops this new representation, including rigourous proofs of the correctness as a representation for terms and a number of properties, such as the restriction on the search for the next redex. It is shown that the representation can be used with graphs, hence it can be used as the basis for an implementation of a lazy functional language. An implementation is introduced and its performance is compared with a conventional implementati...