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32
The Lazy Lambda Calculus
 Research Topics in Functional Programming
, 1990
"... Introduction The commonly accepted basis for functional programming is the calculus; and it is folklore that the calculus is the prototypical functional language in puri ed form. But what is the calculus? The syntax is simple and classical; variables, abstraction and application in the pure cal ..."
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Cited by 240 (3 self)
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Introduction The commonly accepted basis for functional programming is the calculus; and it is folklore that the calculus is the prototypical functional language in puri ed form. But what is the calculus? The syntax is simple and classical; variables, abstraction and application in the pure calculus, with applied calculi obtained by adding constants. The further elaboration of the theory, covering conversion, reduction, theories and models, is laid out in Barendregt's already classical treatise [Bar84]. It is instructive to recall the following crux, which occurs rather early in that work (p. 39): Meaning of terms: rst attempt The meaning of a term is its normal form (if it exists). All terms without normal forms are identi ed. This proposal incorporates such a simple and natural interpretation of the calculus as
On reductionbased process semantics
 Theoretical Computer Science
, 1995
"... Abstract. A formulation of semantic theories for processes which is based on reduction relation and equational reasoning is studied. The new construction can induce meaningful theories for processes, both in strong and weak settings. The resulting theories in many cases coincide with, and sometimes ..."
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Cited by 139 (21 self)
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Abstract. A formulation of semantic theories for processes which is based on reduction relation and equational reasoning is studied. The new construction can induce meaningful theories for processes, both in strong and weak settings. The resulting theories in many cases coincide with, and sometimes generalise, observationbased formulation of behavioural equivalence. The basic construction of reductionbased theories is studied, taking a simple name passing calculus called \nucalculus as an example. Results on other calculi are also briefly discussed.
The Lazy Lambda Calculus in a Concurrency Scenario (Extended Abstract)
 Information and Computation
, 1994
"... ) Davide Sangiorgi LFCS  Department of Computer Science Edinburgh University Edinburgh  EH9 3JZ  UK Abstract The use of lambda calculus in richer settings, possibly involving parallelism, is examined in terms of its effect on the equivalence between lambda terms. We concentrate here on Abra ..."
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Cited by 54 (7 self)
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) Davide Sangiorgi LFCS  Department of Computer Science Edinburgh University Edinburgh  EH9 3JZ  UK Abstract The use of lambda calculus in richer settings, possibly involving parallelism, is examined in terms of its effect on the equivalence between lambda terms. We concentrate here on Abramsky's lazy lambda calculus and we follow two directions. First, the lambda calculus is studied within a process calculus by examining the equivalence $ induced by Milner's encoding into the calculus. We give exact operational and denotational characterizations for $. Secondly, we examine Abramsky's applicative bisimulation when the lambda calculus is augmented with (wellformed) operators, i.e. symbols equipped with reduction rules describing their behaviour. Then, maximal discrimination is obtained when all operators are considered; we show that this discrimination coincides with the one given by $ and that the adoption of certain nondeterministic operators is sufficient and necessary...
Strong Stability and the Incompleteness of Stable Models for λCalculus
 ANNALS OF PURE AND APPLIED LOGIC
, 1999
"... We prove that the class of stable models is incomplete with respect to pure λcalculus. More precisely, we show that no stable model has the same theory as the strongly stable version of Park's model. This incompleteness proof can be adapted to the continuous case, giving an incompleteness proof for ..."
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Cited by 20 (0 self)
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We prove that the class of stable models is incomplete with respect to pure λcalculus. More precisely, we show that no stable model has the same theory as the strongly stable version of Park's model. This incompleteness proof can be adapted to the continuous case, giving an incompleteness proof for this case which is much simpler than the original proof by Honsell an Ronchi della Rocca. Moreover, we isolate a very simple finite set, F , of equations and inequations, which has neither a stable nor a continuous model, and which is included in Th(P fs ) and in T
Eager normal form bisimulation
 In Proc. 20th Annual IEEE Symposium on Logic in Computer Science
, 2005
"... Abstract. Normal form bisimulation is a powerful theory of program equivalence, originally developed to characterize LévyLongo tree equivalence and Boehm tree equivalence. It has been adapted to a range of untyped, higherorder calculi, but types have presented a difficulty. In this paper, we prese ..."
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Cited by 14 (4 self)
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Abstract. Normal form bisimulation is a powerful theory of program equivalence, originally developed to characterize LévyLongo tree equivalence and Boehm tree equivalence. It has been adapted to a range of untyped, higherorder calculi, but types have presented a difficulty. In this paper, we present an account of normal form bisimulation for types, including recursive types. We develop our theory for a continuationpassing style calculus, JumpWithArgument (JWA), where normal form bisimilarity takes a very simple form. We give a novel congruence proof, based on insights from game semantics. A notable feature is the seamless treatment of etaexpansion. We demonstrate the normal form bisimulation proof principle by using it to establish a syntactic minimal invariance result and the uniqueness of the fixed point operator at each type.
Discrete Normalization and Standardization in Deterministic Residual Structures
 In ALP '96 [ALP96
, 1996
"... . We prove a version of the Standardization Theorem and the Discrete Normalization (DN) Theorem in stable Deterministic Residual Structures, which are Abstract Reduction Systems with axiomatized residual relation, and model orthogonal rewrite systems. The latter theorem gives a strategy for construc ..."
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Cited by 10 (3 self)
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. We prove a version of the Standardization Theorem and the Discrete Normalization (DN) Theorem in stable Deterministic Residual Structures, which are Abstract Reduction Systems with axiomatized residual relation, and model orthogonal rewrite systems. The latter theorem gives a strategy for construction of reductions L'evyequivalent (or permutationequivalent) to a given, finite or infinite, regular (or semilinear) reduction, based on the neededness concept of Huet and L'evy. This and other results of this paper add to the understanding of L'evyequivalence of reductions, and consequently, L'evy's reduction space. We demonstrate how elements of this space can be used to give denotational semantics to known functional languages in an abstract manner. 1 Introduction Long ago, Curry and Feys [CuFe58] proved that repeated contraction of leftmostoutermost redexes in any normalizable term eventually yields its normal form, even if the term possesses infinite reductions as well. The reaso...
Discrimination by Parallel Observers: the Algorithm
 LICS '97 , IEEE Comp. Soc
, 1998
"... The main result of the paper is a constructive proof of the following equivalence: two pure terms are observationally equivalent in the lazy concurrent calculus iff they have the same L'evyLongo trees. An algorithm which allows to build a context discriminating any two pure terms with differe ..."
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Cited by 6 (3 self)
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The main result of the paper is a constructive proof of the following equivalence: two pure terms are observationally equivalent in the lazy concurrent calculus iff they have the same L'evyLongo trees. An algorithm which allows to build a context discriminating any two pure terms with different L'evyLongo trees is described. It follows that contextual equivalence coincides with behavioural equivalence (bisimulation) as considered by Sangiorgi. Another consequence is that the discriminating power of concurrent lambda contexts is the same as that of BoudolLaneve's contexts with multiplicities. 3 1 Introduction The aim of this paper is to improve our understanding of what is the "meaning" of a term in the lazy calculus. To explain our result let us begin with the following few observations borrowed from the paper [2] of Abramsky and Ong. In the ordinary calculus, the most natural understanding of evaluation to a "value" is reduction to a normal form. It is however wellk...
Minimal Relative Normalization in Orthogonal Expression Reduction Systems
 In Proc. of the 16 th International Conference on Foundations of Software Technology and Theoretical Computer Science, FST&TCS'96, Springer LNCS
, 1996
"... . In previous papers, the authors studied normalization relative to desirable sets S of `partial results', where it is shown that such sets must be stable. For example, the sets of normal forms, headnormalforms, and weak headnormalforms in the calculus, are all stable. They showed that, for an ..."
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Cited by 5 (2 self)
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. In previous papers, the authors studied normalization relative to desirable sets S of `partial results', where it is shown that such sets must be stable. For example, the sets of normal forms, headnormalforms, and weak headnormalforms in the calculus, are all stable. They showed that, for any stable S, Sneeded reductions are Snormalizing. This paper continues the investigation into the theory of relative normalization. In particular, we prove existence of minimal normalizing reductions for regular stable sets of results. All the above mentioned sets are regular. We give a sufficient and necessary criterion for a normalizing reduction (w.r.t. a regular stable S) to be minimal. Finally, we establish a relationship between relative minimal and optimal reductions, revealing a conflict between minimality and optimality: for regular stable sets of results, a term need not possess a reduction that is minimal and optimal at the same time. 1 Introduction The Normalization Theorem in ...
Lazy Lambda Calculus: Theories, Models and Local Structure Characterisation
 AUTOMATA, LANGUAGES AND PROGRAMMING, LNCS 623
, 1994
"... Lambda Calculus is commonly thought to be the basis for functional programming. However, there is a fundamental mismatch between the "standard" theory of sensible Lambda Calculus (as in e.g. [Bar84]) and the practice of lazy evaluation which is a distinctive feature of functional programming. This ..."
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Cited by 4 (0 self)
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Lambda Calculus is commonly thought to be the basis for functional programming. However, there is a fundamental mismatch between the "standard" theory of sensible Lambda Calculus (as in e.g. [Bar84]) and the practice of lazy evaluation which is a distinctive feature of functional programming. This paper proposes modification of a number of key notions in the sensible theory along the lines of laziness. Starting from the strongly unsolvables as the meaningless terms, we define and investigate properties of lazy (or weakly sensible) λtheories, lazy λmodels and a number of lazy behavioural preorders on λterms. In the second part, we show that all these notions have a natural place in a class of lazy psemodels. A major result of this paper is a new local structure theorem for lazy psemodels. This characterizes the ordering between denotations of λterms in the model by a new lazy behavioural preorder.