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ion for the Lazy -calculus Samson Abramsky Guy McCusker Department of Computing Imperial College of Science, Technology and Medicine 180 Queen's Gate London SW7 2BZ United Kingdom Abstract We define a category of games G, and its extensional quotient E . A model of the lazy -calculus, a type-free functional language based on evaluation to weak head normal form, is given in G, yielding an extensional model in E . This model is shown to be fully abstract with respect to applicative simulation. This is, so far as we know, the first purely semantic construction of a fully abstract model for a reflexively-typed sequential language. 1 Introduction Full Abstraction is a key concept in programming language semantics [9, 12, 23, 26]. The ingredients are as follows. We are given a language L, with an `observational preorder' - on terms in L such that P - Q means that every observable property of P is also satisfied by Q; and a denotational model MJ\DeltaK. The model M is then said to be f...

The literature on programming languages contains an abundance of informal claims on the relative expressive power of programming languages, but there is no framework for formalizing such statements nor for deriving interesting consequences. As a first step in this direction, we develop a formal notion of expressiveness and investigate its properties. To validate the theory, we analyze some widely held beliefs about the expressive power of several extensions of functional languages. Based on these results, we believe that our system correctly captures many of the informal ideas on expressiveness, and that it constitutes a foundation for further research in this direction. 1 Comparing Programming Languages The literature on programming languages contains an abundance of informal claims on the expressive power of programming languages. Arguments in these contexts typically assert the expressibility or non-expressibility of programming constructs relative to a language. Unfortunately, pro...

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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.

this paper, we present a new formal embodiment of Girard's programme, with the following salient features. 1. Our formalisation is based on elementary Domain Theory rather than C --algebras. It exposes precisely what structure is required of the ambient category in order to carry out the interpretation. Furthermore, we show how the interpretation arises from the construction of a categorical model of Linear Logic; this provides the basis for a rational reconstruction which makes the structure of the interpretation much easier to understand. 2. The key definitions in our interpretation differ from Girard's. Most notably, we replace the "execution formula" by a least fixpoint, essentially a generalisation of Kahn's semantics for feedback in dataflow networks [Kah77, KM77]. This, coupled with the use of the other distinctive construct of Domain theory, the lifting monad, enables us to interpret the whole of Linear Logic, and to prove soundness in full generality. 3. Our general notion of interpretation has simple examples, providing a suitable basis for concrete implementations. In fact, we sketch a computational interpretation of the Geometry of Interaction in terms of dataflow networks. Recall that computation in dataflow networks is asynchronous, i.e. "no global time", and proceeds by purely local "firing rules" that manipulate tokens. The further structure of this paper is as follows. In Section 2, we review the syntax of Linear Logic, and present the basic, and quite simple intuitions underlying the interpretation. In Section 3, we use these ideas to construct models of Linear Logic. In Section 4 we define the Geometry of Interaction interpretations, and how that they arise from the model constructed previously in a natural fashion. In Section 5, we give a computati...

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Plotkin has advocated the combination of linear lambda calculus, polymorphism and fixed point recursion as an expressive semantic metalanguage. We study its expressive power from an operational point of view. We show that the naturally call-by-value operators of linear lambda calculus can be given a call-by-name semantics without affecting termination at exponential types and hence without affecting ground contextual equivalence. This result is used to prove properties of a logical relation that provides a new extensional characterisation of ground contextual equivalence and relational parametricity properties of polymorphic types.

Reversibility is a key issue in the interface between computation and physics, and of growing importance as miniaturization progresses towards its physical limits. Most foundational work on reversible computing to date has focussed on simulations of low-level machine models. By contrast, we develop a more structural approach. We show how high-level functional programs can be mapped compositionally (i.e. in a syntax-directed fashion) into a simple kind of automata which are immediately seen to be reversible. The size of the automaton is linear in the size of the functional term. In mathematical terms, we are building a concrete model of functional computation. This construction stems directly from ideas arising in Geometry of Interaction and Linear Logic—but can be understood without any knowledge of these topics. In fact, it serves as an excellent introduction to them. At the same time, an interesting logical delineation between reversible and irreversible forms of computation emerges from our analysis. 1