. In this paper we address the problem of solving recursive domain equations using uncountable limits of domains. These arise for instance, when dealing with the ! 1 -continuous function-space constructor and are used in the denotational semantics of programming languages which feature unbounded choice constructs. Surprisingly, the category of cpo's and ! 1 -continuous embeddings is not ! 0 -cocomplete. Hence the standard technique for solving reflexive domain equations fails. We give two alternative methods. We discuss also the issue of completeness of the fij-calculus w.r.t reflexive domain models. We show that among the reflexive domain models in the category of cpo's and ! 0 -continuous functions there is one which has a minimal theory. We give a reflexive domain model in the category of cpo's and ! 1 -continuous functions whose theory is precisely the fij theory. So ! 1 -continuous -models are complete for the fij-calculus. CR Classification: F.3.2, F.4.1, D.3.3 Key words: count...

We show how to characterize compositionally a number of evaluation properties of λ-terms using Intersection Type assignment systems. In particular, we focus on termination properties, such as strong normalization, normalization, head normalization, and weak head normalization. We consider also the persistent versions of such notions. By way of example, we consider also another evaluation property, unrelated to termination, namely reducibility to a closed term. Many of these characterization results are new, to our knowledge, or else they streamline, strengthen, or generalize earlier results in the literature. The completeness parts of the characterizations are proved uniformly for all the properties, using a set-theoretical semantics of intersection types over suitable kinds of stable sets. This technique generalizes Krivine 's and Mitchell's methods for strong normalization to other evaluation properties.

. Proof principles for reasoning about various semantics of untyped -calculus are discussed. The semantics are determined operationally by fixing a particular reduction strategy on -terms and a suitable set of values, and by taking the corresponding observational equivalence on terms. These principles arise naturally as co-induction principles, when the observational equivalences are shown to be induced by the unique mapping into a final F -coalgebra, for a suitable functor F . This is achieved either by induction on computation steps or exploiting the properties of some, computationally adequate, inverse limit denotational model. The final F -coalgebras cannot be given, in general, the structure of a "denotational" -model. Nevertheless the "final semantics" can count as compositional in that it induces a congruence. We utilize the intuitive categorical setting of hypersets and functions. The importance of the principles introduced in this paper lies in the fact that they often allow...

Dipartimento di Informatica Universit`a di Venezia

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, ...

Peter Selinger Department of Mathematics University of Pennsylvania 209 S. 33rd Street Philadelphia, PA 19104-6395 selinger@math.upenn.edu Abstract Many familiar models of the type-free lambda calculus are constructed by order theoretic methods. This paper provides some basic new facts about ordered models of the lambda calculus. We show that in any partially ordered model that is complete for the theory of fi- or fij-conversion, the partial order is trivial on term denotations. Equivalently, the open and closed term algebras of the type-free lambda calculus cannot be non-trivially partially ordered. Our second result is a syntactical characterization, in terms of so-called generalized Mal'cev operators, of those lambda theories which cannot be induced by any non-trivially partially ordered model. We also consider a notion of finite models for the type-free lambda calculus. We introduce partial syntactical lambda models, which are derived from Plotkin's syntactical models of redu...

In this paper we show that the Stone representation theorem for Boolean algebras can be generalized to combinatory algebras. In every combinatory algebra there is a Boolean algebra of central elements (playing the role of idempotent elements in rings), whose operations are defined by suitable combinators. Central elements are used to represent any combinatory algebra as a Boolean product of directly indecomposable combinatory algebras (i.e., algebras which cannot be decomposed as the Cartesian product of two other nontrivial algebras). Central elements are also used to provide applications of the representation theorem to lambda calculus. We show that the indecomposable semantics (i.e., the semantics of lambda calculus given in terms of models of lambda calculus, which are directly indecomposable as combinatory algebras) includes the continuous, stable and strongly stable semantics, and the term models of all semisensible lambda theories. In one of the main results of the paper we show that the indecomposable semantics is equationally incomplete, and this incompleteness is as wide as possible: for every recursively enumerable lambda theory T, there is a continuum of lambda theories including T which are omitted by the indecomposable semantics. 1

Invariance of interpretation by #-conversion is one of the minimal requirements for any standard model for the #-calculus. With the intersection type systems being a general framework for the study of semantic domains for the #-calculus, the present paper provides a (syntactic) characterisation of the above mentioned requirement in terms of characterisation results for intersection type assignment systems.

We use intersection types as a tool for obtaining λ-models. Relying on the notion of easy intersection type theory we successfully build a λ-model in which the interpretation of an arbitrary simple easy term is any filter which can be described by a continuous predicate. This allows us to prove two results. The first gives a proof of consistency of the λ-theory where the λ-term (λx.xx)(λx.xx) is forced to behave as the join operator. This result has interesting consequences on the algebraic structure of the lattice of λ-theories. The second result is that for any simple easy term there is a λ-model where the interpretation of the term is the minimal fixed point operator.

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