We investigate the interactions of subtyping and recursive types, in a simply typed λ-calculus. The two fundamental questions here are whether two (recursive) types are in the subtype relation, and whether a term has a type. To address the first question, we relate various definitions of type equivalence and subtyping that are induced by a model, an ordering on infinite trees, an algorithm, and a set of type rules. We show soundness and completeness between the rules, the algorithm, and the tree semantics. We also prove soundness and a restricted form of completeness for the model. To address the second question, we show that to every pair of types in the subtype relation we can associate a term whose denotation is the uniquely determined coercion map between the two types. Moreover, we derive an algorithm that, when given a term with implicit coercions, can infer its least

Abstract. We develop a powerdomain construction, [.], which is analogous to the powerset construction and also fits in with the usual sum, product and exponentiation constructions on domains. The desire for such a construction arises when considering programming languages with nondeterministic features or parallel features treated in a nondeterministic way. We hope to achieve a natural, fully abstract semantics in which such equivalences as (pparq)=(qparp) hold. The domain (D Truthvalues) is not the right one, and instead we take the (finitely) generable subsets of D. When D is discrete they are ordered in an elementwise fashion. In the general case they are given the coarsest ordering consistent, in an appropriate sense, with the ordering given in the discrete case. We then find a restricted class of algebraic inductive partial orders which is closed under [. as well as the sum, product and exponentiation constructions. This class permits the solution of recursive domain equations, and we give some illustrative semantics using 5[.]. It remains to be seen if our powerdomain construction does give rise to fully abstract semantics, although such natural equivalences as the above do hold. The major deficiency is the lack of a convincing treatment of the fair parallel construct. 1. Introduction. When one follows the Scott-Strachey approach to the

. We introduce a bicartesian closed category of what we call profinite domains. Study of these domains is carried out through the use of an equivalent category of pre-orders in a manner similar to the information systems approach advocated by Dana Scott and others. A class of universal profinite domains is defined and used to derive sufficient conditions for the profinite solution of domain equations involving continuous operators. As a special instance of this construction, a universal domain for the category SFP is demonstrated. Necessary conditions for the existence of solutions for domain equations over the profinites are also given and used to derive results about solutions of some equations. A new universal bounded complete domain is also demonstrated using an operator which has bounded complete domains as its fixed points. 1 Introduction. For our purposes a domain equation has the form X ¸ = F (X) where F is an operator on a class of semantic domains (typically, F is an endof...

Introduction Much recent work in artificial intelligence has required formal techniques for working with incomplete knowledge. The new approach clearly necessitated that a distinction be made between logic and the action (i.e. two components of intelligence: the epistemological and the heuristic; cf. [MCC 69]). However, certain new inference procedures associated with the new approach (for example the familiar example of default logic) have, like classical logic, been based on an underlying consistent ontology. (Cf. [GIN 87a]: "It is precisely this `absence of information to the contrary' that makes the inference non-monotonic...") 1 Philosophers were the first to call attention to the new situation. (Cf. [RES 79]: "...we can live with the prospect of inconsistency --- not only in epistemology, but even in ontology"; and further: "The invocation of ontology here is significant... Thus if t(P ) were to be construed not as `P

M. DEZANI-CIANCAGLINI Universita di Torino, Italy F. HONSELL Universita di Udine, Italy F. ALESSI Universita di Udine, Italy Abstract We characterize those intersection-type theories which yield complete intersection-type assignment systems for l-calculi, with respect to the three canonical set-theoretical semantics for intersection-types: the inference semantics, the simple semantics and the F-semantics. Keywords Lambda Calculus, Intersection Types, Semantic Completeness, Filter Structures. 1 Introduction Intersection-types disciplines originated in [6] to overcome the limitations of Curry 's type assignment system and to provide a characterization of strongly normalizing terms of the l-calculus. But very early on, the issue of completeness became crucial. Intersection-type theories and filter l-models have been introduced, in [5], precisely to achieve the completeness for the type assignment system l" BCD W , with respect to Scott's simple semantics. And this result, ...

Ralph Kummetz 1 Institut fur Algebra Technische Universitat Dresden D-01062 Dresden, Germany Abstract We study approximating partial orders with projections (approximating pop's). These are triples (D; ; P) consisting of a poset (D; ) and a directed set P of projections such that the supremum of P exists and sup P = id D . We derive a canonical uniformity U on D and relate properties of U such as completeness and compactness to properties of the poset and the projection set. We show that each monotone net in D is convergent if and only if (D; ) is an algebraic domain such that the images of the projections are precisely the compact elements of (D; ). Furthermore, the bifinite domains arise exactly as approximating pop's where U is compact. 1 Introduction In the theory of denotational semantics of programming languages, various classes of domains and their approximations have been intensively investigated. Scott [11] constructed a domain-theoretic model of the type-free -calculus ...

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We build a lambda model which characterizes completely (persistently) normalizing, (persistently) head normalizing, and (persistently) weak head normalizing terms.

Abstract. We investigate the logical aspects of the partial λ-calculus with equality, exploiting an equivalence between partial λ-theories and partial cartesian closed categories (pcccs) established here. The partial λ-calculus with equality provides a full-blown intuitionistic higher order logic, which in a precise sense turns out to be almost the logic of toposes, the distinctive feature of the latter being unique choice. We give a linguistic proof of the generalization of the fundamental theorem of toposes to pcccs with equality; type theoretically, one thus obtains that the partial λ-calculus with equality encompasses a Martin-Löf-style dependent type theory. This work forms part of the semantical foundations for the higher order algebraic specification language HasCasl.

We illustrate the use of intersection types as a semantic tool for showing properties of the lattice of l-theories. Relying on the notion of easy intersection type theory we successfully build a filter model in which the interpretation of an arbitrary simple easy term is any filter which can be described in an uniform way by a recursive predicate. This allows us to prove the consistency of a well-know l-theory: this consistency has interesting consequences on the algebraic structure of the lattice of l-theories.