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Parametric Polymorphism and Operational Equivalence
 MATHEMATICAL STRUCTURES IN COMPUTER SCIENCE
, 2000
"... Studies of the mathematical properties of impredicative polymorphic types have for the most part focused on the polymorphic lambda calculus of Girard–Reynolds, which is a calculus of total polymorphic functions. This paper considers polymorphic types from a functional programming perspective, where ..."
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Cited by 75 (2 self)
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Studies of the mathematical properties of impredicative polymorphic types have for the most part focused on the polymorphic lambda calculus of Girard–Reynolds, which is a calculus of total polymorphic functions. This paper considers polymorphic types from a functional programming perspective, where the partialness arising from the presence of fixpoint recursion complicates the nature of potentially infinite (‘lazy’) data types. An approach to Reynolds' notion of relational parametricity is developed that works directly on the syntax of a programming language, using a novel closure operator to relate operational behaviour to parametricity properties of types. Working with an extension of Plotkin's PCF with ∀types, lazy lists and existential types, we show by example how the resulting logical relation can be used to prove properties of polymorphic types up to operational equivalence.
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...
Environmental bisimulations for higherorder languages
 In TwentySecond Annual IEEE Symposium on Logic in Computer Science
, 2007
"... Developing a theory of bisimulation in higherorder languages can be hard. Particularly challenging can be: (1) the proof of congruence, as well as enhancements of the bisimulation proof method with “upto context ” techniques, and (2) obtaining definitions and results that scale to languages with d ..."
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Cited by 35 (11 self)
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Developing a theory of bisimulation in higherorder languages can be hard. Particularly challenging can be: (1) the proof of congruence, as well as enhancements of the bisimulation proof method with “upto context ” techniques, and (2) obtaining definitions and results that scale to languages with different features. To meet these challenges, we present environmental bisimulations, a form of bisimulation for higherorder languages, and its basic theory. We consider four representative calculi: pure λcalculi (callbyname and callbyvalue), callbyvalue λcalculus with higherorder store, and then HigherOrder πcalculus. In each case: we present the basic properties of environmental bisimilarity, including congruence; we show that it coincides with contextual equivalence; we develop some upto techniques, including upto context, as examples of possible enhancements of the associated bisimulation method. Unlike previous approaches (such as applicative bisimulations, logical relations, SumiiPierceKoutavasWand), our method does not require induction/indices on evaluation derivation/steps (which may complicate the proofs of congruence, transitivity, and the combination with upto techniques), or sophisticated methods such as Howe’s for proving congruence. It also scales from the pure λcalculi to the richer calculi with simple congruence proofs. 1
Existential Types: Logical Relations and Operational Equivalence
 In Proceedings of the 25th International Colloquium on Automata, Languages and Programming
, 1998
"... . Existential types have proved useful for classifying various kinds of information hiding in programming languages, such as occurs in abstract datatypes and objects. In this paper we address the question of when two elements of an existential type are semantically equivalent. Of course, it depends ..."
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Cited by 31 (2 self)
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. Existential types have proved useful for classifying various kinds of information hiding in programming languages, such as occurs in abstract datatypes and objects. In this paper we address the question of when two elements of an existential type are semantically equivalent. Of course, it depends what one means by `semantic equivalence'. Here we take a syntactic approachso semantic equivalence will mean some kind of operational equivalence. The paper begins by surveying some of the literature on this topic involving `logical relations'. Matters become quite complicated if the programming language mixes existential types with function types and features involving nontermination (such as recursive definitions). We give an example (suggested by Ian Stark) to show that in this case the existence of suitable relations is sufficient, but not necessary for proving operational equivalences at existential types. Properties of this and other examples are proved using a new form of operatio...
Games and full abstraction for nondeterministic languages
, 1999
"... Abstract Nondeterminism is a pervasive phenomenon in computation. Often it arises as an emergent property of a complex system, typically as the result of contention for access to shared resources. In such circumstances, we cannot always know, in advance, exactly what will happen. In other circumstan ..."
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Cited by 31 (3 self)
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Abstract Nondeterminism is a pervasive phenomenon in computation. Often it arises as an emergent property of a complex system, typically as the result of contention for access to shared resources. In such circumstances, we cannot always know, in advance, exactly what will happen. In other circumstances, nondeterminism is explicitly introduced as a means of abstracting away from implementation details such as precise command scheduling and control flow. However, the kind of behaviours exhibited by nondeterministic computations can be extremely subtle in comparison to those of their deterministic counterparts and reasoning about such programs is notoriously tricky as a result. It is therefore important to develop semantic tools to improve our understanding of, and aid our reasoning about, such nondeterministic programs. In this thesis, we extend the framework of game semantics to encompass nondeterministic computation. Game semantics is a relatively recent development in denotational semantics; its main novelty is that it views a computation not as a static entity, but rather as a dynamic process of interaction. This perspective makes the theory wellsuited to modelling many aspects of computational processes: the original use of game semantics in modelling the simple functional language PCF has subsequently been extended to handle more complex control structures such as references and continuations.
Erratic Fudgets: A Semantic Theory for an Embedded Coordination Language
 SCIENCE OF COMPUTER PROGRAMMING
, 2003
"... The powerful abstraction mechanisms of functional programming languages provide the means to develop domainspecific programming languages within the language itself. Typically, this is realised by designing a set of combinators (higherorder reusable programs) for an application area, and by constr ..."
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Cited by 20 (3 self)
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The powerful abstraction mechanisms of functional programming languages provide the means to develop domainspecific programming languages within the language itself. Typically, this is realised by designing a set of combinators (higherorder reusable programs) for an application area, and by constructing individual applications by combining and coordinating individual combinators. This paper is concerned with a successful example of such an embedded programming language, namely Fudgets, a library of combinators for building graphical user interfaces in the lazy functional language Haskell. The Fudget library has been used to build a number of substantial applications, including a web browser and a proof editor interface to a proof checker for constructive type theory. This paper develops a semantic theory for the nondeterministic stream processors that are at the heart of the Fudget concept. The interaction of two features of stream processors makes the development of such a semantic theory problematic: (i) the sharing of computation provided by the lazy evaluation mechanism of the underlying host language, and (ii) the addition of nondeterministic choice needed to handle the natural concurrency that reactive applications entail We demonstrate that this combination of features in a higherorder functional language can be tamed to provide a tractable semantic theory and induction principles suitable for reasoning about contextual equivalence of Fudgets.
CallByPushValue: A Subsuming Paradigm
 in Proc. TLCA ’99
, 1999
"... . Callbypushvalue is a new paradigm that subsumes the callbyname and callbyvalue paradigms, in the following sense: both operational and denotational semantics for those paradigms can be seen as arising, via translations that we will provide, from similar semantics for callbypushvalue. To ..."
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Cited by 15 (0 self)
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. Callbypushvalue is a new paradigm that subsumes the callbyname and callbyvalue paradigms, in the following sense: both operational and denotational semantics for those paradigms can be seen as arising, via translations that we will provide, from similar semantics for callbypushvalue. To explain callbypushvalue, we first discuss general operational ideas, especially the distinction between values and computations, using the principle that "a value is, a computation does". Using an example program, we see that the lambdacalculus primitives can be understood as push/pop commands for an operandstack. We provide operational and denotational semantics for a range of computational effects and show their agreement. We hence obtain semantics for callbyname and callbyvalue, of which some are familiar, some are new and some were known but previously appeared mysterious. 1 Introduction 1.1 Contribution In his invited lecture at POPL '98 [32], Reynolds, surveying over 30 year...
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.
Normal form bisimulation for parametric polymorphism
 In LICS
, 2008
"... This paper presents a new bisimulation theory for parametric polymorphism which enables straightforward coinductive proofs of program equivalences involving existential types. The theory is an instance of typed normal form bisimulation and demonstrates the power of this recent framework for modeling ..."
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Cited by 10 (2 self)
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This paper presents a new bisimulation theory for parametric polymorphism which enables straightforward coinductive proofs of program equivalences involving existential types. The theory is an instance of typed normal form bisimulation and demonstrates the power of this recent framework for modeling typed lambda calculi as labelled transition systems. We develop our theory for a continuationpassing style calculus, JumpWithArgument, where normal form bisimulation takes a simple form. We equip the calculus with both existential and recursive types. An “ultimate pattern matching theorem ” enables us to define bisimilarity and we show it to be a congruence. We apply our theory to proving program equivalences, type isomorphisms and genericity. 1
Imprecise Exceptions, CoInductively
"... In a recent paper, Peyton Jones et al. proposed a design for imprecise exceptions in the lazy functional programming language Haskell [PJRH + 99]. The main contribution of the design was that it allowed the language to continue to enjoy its current rich algebra of transformations. However, the den ..."
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Cited by 8 (2 self)
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In a recent paper, Peyton Jones et al. proposed a design for imprecise exceptions in the lazy functional programming language Haskell [PJRH + 99]. The main contribution of the design was that it allowed the language to continue to enjoy its current rich algebra of transformations. However, the denotational semantics used to formalise the design does not combine easily with other extensions, most notably that of concurrency. We present an alternative semantics for a lazy functional language with imprecise exceptions which is entirely operational in nature, and combines well with other extensions, such as I/O and concurrency. The semantics is based upon a convergence relation, which describes evaluation, and an exceptional convergence relation, which describes the raising of exceptions. Convergence and exceptional convergence lead naturally to a simple notion of renement, where a term M is re ned by N whenever they have identical convergent behaviour, and any exception raised by N c...