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New tools are presented for reasoning about properties of recursively defined domains. We work within a general, category-theoretic framework for various notions of `relation' on domains and for actions of domain constructors on relations. Freyd's analysis of recursive types in terms of a property of mixed initiality/finality is transferred to a corresponding property of invariant relations. The existence of invariant relations is proved under completeness assumptions about the notion of relation. We show how this leads to simpler proofs of the computational adequacy of denotational semantics for functional programming languages with user-declared datatypes. We show how the initiality/finality property of invariant relations can be specialized to yield an induction principle for admissible subsets of recursively defined domains, generalizing the principle of structural induction for inductively defined sets. We also show how the initiality /finality property gives rise to the co-induct...

Machines Th. STREICHER Fachbereich 4 Mathematik, TU Darmstadt, Schlossgartenstr. 7, 64289 Darmstadt, streiche@mathematik.th-darmstadt.de B. REUS Institut fur Informatik, Ludwig-Maximilians-Universitat, Oettingenstr. 67, D-80538 Munchen, reus@informatik.uni-muenchen.de Abstract One of the goals of this paper is to demonstrate that denotational semantics is useful for operational issues like implementation of functional languages by abstract machines. This is exemplified in a tutorial way by studying the case of extensional untyped call-byname -calculus with Felleisen's control operator C. We derive the transition rules for an abstract machine from a continuation semantics which appears as a generalization of the ::-translation known from logic. The resulting abstract machine appears as an extension of Krivine's Machine implementing head reduction. Though the result, namely Krivine's Machine, is well known our method of deriving it from continuation semantics is new and applicable to other languages (as e.g. call-by-value variants).

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While Game Semantics has been remarkably successful at modelling, often in a fully abstract manner, a wide range of features of programming languages, there has to date been no attempt at applying it to subtyping. We show how the simple device of explicitly introducing error values in the syntax of the calculus leads to a notion of subtyping for game semantics. We construct an interpretation of a simple -calculus with subtyping and show how the range of the interpretation of types is a complete lattice thus yielding an interpretation of bounded quantification.

. We study the longstanding problem of semantics for input /output (I/O) expressed using side-effects. Our vehicle is a small higher-order imperative language, with operations for interactive character I/O and based on ML syntax. Unlike previous theories, we present both operational and denotational semantics for I/O effects. We use a novel labelled transition system that uniformly expresses both applicative and imperative computation. We make a standard definition of bisimilarity and prove it is a congruence using Howe's method. Next, we define a metalogical type theory M in which we may give a denotational semantics to O. M generalises Crole and Pitts' FIX-logic by adding in a parameterised recursive datatype, which is used to model I/O. M comes equipped both with judgements of equality of expressions, and an operational semantics; M itself is given a domain-theoretic semantics in the category CPPO of cppos (bottom-pointed posets with joins of !-chains) and Scott continuous functions...

This note gives a new proof of the `operational extensionality' property of Abramsky's lazy lambda calculus---namely the coincidence of contextual equivalence with a co-inductively defined notion of `applicative bisimilarity'. This purely syntactic result is here proved using a logical relation (due to Plotkin) between the syntax and its denotational semantics. The proof exploits a mixed inductive/coinductive characterisation of the logical relation recently discovered by the author.

Abstract. We present a Coq formalization of constructive ω-cpos (extending earlier work by Paulin-Mohring) up to and including the inverselimit construction of solutions to mixed-variance recursive domain equations, and the existence of invariant relations on those solutions. We then define operational and denotational semantics for both a simplytyped CBV language with recursion and an untyped CBV language, and establish soundness and adequacy results in each case. 1

We show that the standard normalization-by-evaluation construction for the simply-typed ### -calculus has a natural counterpart for the untyped ## -calculus, with the central type-indexed logical relation replaced by a "recursively defined" invariant relation, in the style of Pitts. In fact, the construction can be seen as generalizing a computational-adequacy argument for an untyped, call-by-name language to normalization instead of evaluation. In the

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