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Domains with Approximating Projections
 Institute of Algebra, Dresden University of Technology
, 1999
"... We investigate approximating posets with projections (approximating pop's). These are triples (D; ; P) consisting of a poset (D; ) and a directed set P of projections with sup P = id D . They carry a canonical uniformity and thus a topology. We relate their properties such as completeness ..."
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We investigate approximating posets with projections (approximating pop's). These are triples (D; ; P) consisting of a poset (D; ) and a directed set P of projections with sup P = id D . They carry a canonical uniformity and thus a topology. We relate their properties 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; ). We call these domains Pdomains and characterize them as inverse limits of posets satisfying the ascending chain condition. Moreover, we describe Pdomains by a certain system of socalled "complete" subsets. We prove that if the set of compact elements of an algebraic domain is mubcomplete, then it is a Pdomain if and only if the mubclosure of every finite set of compact elements fulfils the ascending chain condition. Furthermore, we characte...
Intersection Types and Lambda Theories
 International Workshop on Isomorphisms of Types
, 2002
"... We illustrate the use of intersection types as a semantic tool for showing properties of the lattice of ltheories. 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 desc ..."
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We illustrate the use of intersection types as a semantic tool for showing properties of the lattice of ltheories. 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 wellknow ltheory: this consistency has interesting consequences on the algebraic structure of the lattice of ltheories.
Lambda calculus: models and theories
 Proceedings of the Third AMAST Workshop on Algebraic Methods in Language Processing (AMiLP2003), number 21 in TWLT Proceedings, pages 39–54, University of Twente, 2003. Invited Lecture
"... In this paper we give an outline of recent results concerning theories and models of the untyped lambda calculus. Algebraic and topological methods have been applied to study the structure of the lattice of λtheories, the equational incompleteness of lambda calculus semantics, and the λtheories in ..."
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In this paper we give an outline of recent results concerning theories and models of the untyped lambda calculus. Algebraic and topological methods have been applied to study the structure of the lattice of λtheories, the equational incompleteness of lambda calculus semantics, and the λtheories induced by graph models of lambda calculus.
Injective spaces via adjunction
 J. Pure Appl. Algebra
, 2011
"... Abstract. Our work over the past years shows that not only the collection of (for instance) all topological spaces gives rise to a category, but also each topological space can be seen individually as a category by interpreting the convergence relationx − → x between ultrafilters and points of a top ..."
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Abstract. Our work over the past years shows that not only the collection of (for instance) all topological spaces gives rise to a category, but also each topological space can be seen individually as a category by interpreting the convergence relationx − → x between ultrafilters and points of a topological space X as arrows in X. Naturally, this point of view opens the door to the use of concepts and ideas from (enriched) Category Theory for the investigation of (for instance) topological spaces. In this paper we study cocompleteness, adjoint functors and Kan extensions in the context of topological theories. We show that the cocomplete spaces are precisely the injective spaces, and they are algebras for a suitable monad on Set. This way we obtain enriched versions of known results about injective topological spaces and continuous lattices.
Reflexive domains are not complete for the extensional lambda calculus
 Proc. of LICS’09, IEEE Computer Society Publications
"... Reflexive domains are not complete for the extensional λcalculus ..."
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Reflexive domains are not complete for the extensional λcalculus
Reflexive Scott domains are not complete for the extensional lambda calculus
"... A longstanding open problem is whether there exists a model of the untyped λcalculus in the category Cpo of complete partial orderings and Scott continuous functions, whose theory is exactly the least λtheory λβ or the least extensional λtheory λβη. In this paper we analyze the class of reflexive ..."
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A longstanding open problem is whether there exists a model of the untyped λcalculus in the category Cpo of complete partial orderings and Scott continuous functions, whose theory is exactly the least λtheory λβ or the least extensional λtheory λβη. In this paper we analyze the class of reflexive Scott domains, the models of λcalculus living in the category of Scott domains (a full subcategory of Cpo). The following are the main results of the paper: (i) Extensional reflexive Scott domains are not complete for the λβηcalculus, i.e., there are equations not in λβη which hold in all extensional reflexive Scott domains. (ii) The order theory of an extensional reflexive Scott domain is never recursively enumerable. These results have been obtained by isolating among the reflexive Scott domains a class of webbed models arising from Scott’s information systems, called iwebmodels. The class of iwebmodels includes all extensional reflexive Scott domains, all preordered coherent models and all filter models living in Cpo. Based on a finegrained study of an “effective” version of Scott’s information systems, we have shown that there are equations not in λβ (resp. λβη) which hold in all (extensional) iwebmodels.
Recursive Domain Equations of Filter Models
 In SOFSEM 2008, LNCS 4910
, 2008
"... Abstract. Filter models and (solutions of) recursive domain equations are two different ways of constructing lambda models. Many partial results have been shown about the equivalence between these two constructions (in some specific cases). This paper deepens the connection by showing that the equ ..."
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Abstract. Filter models and (solutions of) recursive domain equations are two different ways of constructing lambda models. Many partial results have been shown about the equivalence between these two constructions (in some specific cases). This paper deepens the connection by showing that the equivalence can be shown in a general framework. We will introduce the class of disciplined intersection type theories and its four subclasses: natural split, lazy split, natural equated and lazy equated. We will prove that each class corresponds to a different recursive domain equation. For this result, we are extracting the essence of the specific proofs for the particular cases of intersection type theories and making one general construction that encompasses all of them. This general approach puts together all these results which may appear scattered and sometimes with incomplete proofs in the literature. 1
Intersection Types for λTrees
"... We introduce a type assignment system which is parametric with respect to five families of trees obtained by evaluating λterms (Böhm trees, LévyLongo trees,...). Then we prove, in an (almost) uniform way, that each type assignment system fully describes the observational equivalences induced by th ..."
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We introduce a type assignment system which is parametric with respect to five families of trees obtained by evaluating λterms (Böhm trees, LévyLongo trees,...). Then we prove, in an (almost) uniform way, that each type assignment system fully describes the observational equivalences induced by the corresponding tree representation of terms. More precisely, for each family of trees, two terms have the same tree if and only if they get assigned the same types in the corresponding type assignment system.
TU as a Universal Domain
, 1977
"... In mathematical semantics, in the sense of Scott, the question arises of what domains of interpretation should be chosen. It has been felt by the author, and others, that lattices are the wrong choice and instead one should use complete partial orders (cpo’s), which do not necessarily have the embar ..."
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In mathematical semantics, in the sense of Scott, the question arises of what domains of interpretation should be chosen. It has been felt by the author, and others, that lattices are the wrong choice and instead one should use complete partial orders (cpo’s), which do not necessarily have the embarrassing top element. So far, however, no mathematical theory as pleasant as that developed for Pw in the paper “Data Types as Lattices ” has been available. The present paper is intended to fill this gap and is a close analog of the Pw paper, replacing Pw by 8”‘, the wpower of the threeelement truthvalue cpo, T. 1.
Effective λmodels versus recursively enumerable λtheories
"... A longstanding open problem is whether there exists a nonsyntactical model of the untyped λcalculus whose theory is exactly the least λtheory λβ. In this paper we investigate the more general question of whether the equational/order theory of a model of the untyped λcalculus can be recursively e ..."
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A longstanding open problem is whether there exists a nonsyntactical model of the untyped λcalculus whose theory is exactly the least λtheory λβ. In this paper we investigate the more general question of whether the equational/order theory of a model of the untyped λcalculus can be recursively enumerable (r.e. for brevity). We introduce a notion of effective model of λcalculus, which covers in particular all the models individually introduced in the literature. We prove that the order theory of an effective model is never r.e.; from this it follows that its equational theory cannot be λβ, λβη. We then show that no effective model living in the stable or strongly stable semantics has an r.e. equational theory. Concerning Scott’s semantics, we investigate the class of graph models and prove that no order theory of a graph model can be r.e., and that there exists an effective graph model whose equational/order theory is the minimum among the theories of graph models. Finally, we show that the class of graph models enjoys a kind of downwards LöwenheimSkolem theorem.