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Improvement in a Lazy Context: An Operational Theory for CallByNeed
 Proc. POPL'99, ACM
, 1999
"... Machine The semantics presented in this section is essentially Sestoft's \mark 1" abstract machine for laziness [Sestoft 1997]. In that paper, he proves his abstract machine 6 A. K. Moran and D. Sands h fx = Mg; x; S i ! h ; M; #x : S i (Lookup) h ; V; #x : S i ! h fx = V g; V; S i (Update) h ; ..."
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Cited by 40 (7 self)
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Machine The semantics presented in this section is essentially Sestoft's \mark 1" abstract machine for laziness [Sestoft 1997]. In that paper, he proves his abstract machine 6 A. K. Moran and D. Sands h fx = Mg; x; S i ! h ; M; #x : S i (Lookup) h ; V; #x : S i ! h fx = V g; V; S i (Update) h ; M x; S i ! h ; M; x : S i (Unwind) h ; x:M; y : S i ! h ; M [ y = x ]; S i (Subst) h ; case M of alts ; S i ! h ; M; alts : S i (Case) h ; c j ~y; fc i ~x i N i g : S i ! h ; N j [ ~y = ~x j ]; S i (Branch) h ; let f~x = ~ Mg in N; S i ! h f~x = ~ Mg; N; S i ~x dom(;S) (Letrec) Fig. 1. The abstract machine semantics for callbyneed. semantics sound and complete with respect to Launchbury's natural semantics, and we will not repeat those proofs here. Transitions are over congurations consisting of a heap, containing bindings, the expression currently being evaluated, and a stack. The heap is a partial function from variables to terms, and denoted in an identical manner to a coll...
Cyclic Lambda Calculi
, 1997
"... . We precisely characterize a class of cyclic lambdagraphs, and then give a sound and complete axiomatization of the terms that represent a given graph. The equational axiom system is an extension of lambda calculus with the letrec construct. In contrast to current theories, which impose restrictio ..."
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Cited by 36 (5 self)
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. We precisely characterize a class of cyclic lambdagraphs, and then give a sound and complete axiomatization of the terms that represent a given graph. The equational axiom system is an extension of lambda calculus with the letrec construct. In contrast to current theories, which impose restrictions on where the rewriting can take place, our theory is very liberal, e.g., it allows rewriting under lambdaabstractions and on cycles. As shown previously, the reduction theory is nonconfluent. We thus introduce an approximate notion of confluence. Using this notion we define the infinite normal form or L'evyLongo tree of a cyclic term. We show that the infinite normal form defines a congruence on the set of terms. We relate our cyclic lambda calculus to the traditional lambda calculus and to the infinitary lambda calculus. Since most implementations of nonstrict functional languages rely on sharing to avoid repeating computations, we develop a variant of our calculus that enforces the ...
Equational Reasoning for Linking with FirstClass Primitive Modules
 In European Symposium on Programming
, 2000
"... . Modules and linking are usually formalized by encodings which use the calculus, records (possibly dependent), and possibly some construct for recursion. In contrast, we introduce the mcalculus, a calculus where the primitive constructs are modules, linking, and the selection and hiding of mo ..."
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Cited by 27 (5 self)
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. Modules and linking are usually formalized by encodings which use the calculus, records (possibly dependent), and possibly some construct for recursion. In contrast, we introduce the mcalculus, a calculus where the primitive constructs are modules, linking, and the selection and hiding of module components. The mcalculus supports smooth encodings of software structuring tools such as functions ( calculus), records, objects (&calculus), and mutually recursive definitions. The mcalculus can also express widely varying kinds of module systems as used in languages like C, Haskell, and ML. We prove the mcalculus is confluent, thereby showing that equational reasoning via the mcalculus is sensible and well behaved. 1 Introduction A long version of this paper [43] which contains full proofs, more details and explanations, and comparisons with more calculi (including the calculus of Ancona and Zucca [4]), is available at http://www.cee.hw.ac.uk/~jbw/papers/. 1.1 Support f...
Correctness of Monadic State: An Imperative CallbyNeed Calculus
 In Proc. 25th ACM Symposium on Principles of Programming Languages
, 1998
"... The extension of Haskell with a builtin state monad combines mathematical elegance with operational efficiency: ffl Semantically, at the source language level, constructs that act on the state are viewed as functions that pass an explicit store data structure around. ffl Operationally, at the imp ..."
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Cited by 20 (2 self)
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The extension of Haskell with a builtin state monad combines mathematical elegance with operational efficiency: ffl Semantically, at the source language level, constructs that act on the state are viewed as functions that pass an explicit store data structure around. ffl Operationally, at the implementation level, constructs that act on the state are viewed as statements whose evaluation has the sideeffect of updating the implicit global store in place. There are several unproven conjectures that the two views are consistent. Recently, we have noted that the consistency of the two views is far from obvious: all it takes for the implementation to become unsound is one judiciouslyplaced betastep in the optimization phase of the compiler. This discovery motivates the current paper in which we formalize and show the correctness of the implementation of monadic state. For the proof, we first design a typed callbyneed language that models the intermediate language of the compiler, to...
Confluent Equational Reasoning for Linking with FirstClass Primitive Modules
 In ESOP 2000  European Symposium on Programming 2000, number 1782 in Lecture Notes in Computer Science
, 1999
"... Modules and linking are usually formalized by encodings which use the lambda calculus, records (possibly dependent), and possibly some construct for recursion. In contrast, we present the mcalculus, a calculus where the primitive constructs are modules, linking, and the selection and hiding of modu ..."
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Cited by 17 (3 self)
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Modules and linking are usually formalized by encodings which use the lambda calculus, records (possibly dependent), and possibly some construct for recursion. In contrast, we present the mcalculus, a calculus where the primitive constructs are modules, linking, and the selection and hiding of module components. In addition to supporting equational reasoning about modules and linking, the mcalculus allows smooth encodings of software structuring tools such as the lambda calculus, mutually recursive definitions, records (including operations like extension and concatenation), and objects. The mcalculus is extremely well behaved  we show not only that the mcalculus is confluent but also that it satisfies the strong finite developments property.
Skew and ωSkew Confluence and Abstract Böhm Semantics
"... Abstract. Skew confluence was introduced as a characterization of nonconfluent term rewriting systems that had unique infinite normal forms or Böhm like trees. This notion however is not expressive enough to deal with all possible sources of nonconfluence in the context of infinite terms or terms e ..."
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Cited by 1 (0 self)
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Abstract. Skew confluence was introduced as a characterization of nonconfluent term rewriting systems that had unique infinite normal forms or Böhm like trees. This notion however is not expressive enough to deal with all possible sources of nonconfluence in the context of infinite terms or terms extended with letrec. We present a new notion called ωskew confluence which constitutes a sufficient and necessary condition for uniqueness. We also present a theory that can lift uniqueness results from term rewriting systems to rewriting systems on terms with letrec. We present our results in the setting of Abstract Böhm Semantics, which is a generalization of Böhm like trees to abstract reduction systems. 1
Skew and ωSkew Confluence and Infinite Normal Forms
"... Abstract. The notion of skew confluence was introduced to characterize nonconfluent term rewriting systems that had unique infinite normal forms or Böhm like trees. This notion however is not expressive enough to deal with all possible sources of nonconfluence in the context of infinite terms or t ..."
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Abstract. The notion of skew confluence was introduced to characterize nonconfluent term rewriting systems that had unique infinite normal forms or Böhm like trees. This notion however is not expressive enough to deal with all possible sources of nonconfluence in the context of infinite terms or terms extended with letrec. To that end, we present a new notion called ωskew confluence. We show that ωskew confluence constitutes a sufficient and necessary condition for uniqueness of infinite normal forms over infinite terms or terms with letrec. 1