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23
Time, Imaginary Value, Paradox, Sign and Space
 in Computing Anticipatory Systems, CASYS  Fifth International Conference, Liege, Belgium (2001) ed. by Daniel Dubois, AIP Conference Proceedings Volume 627
, 2002
"... This paper discusses paradox and imaginary values in relation to the mutuality of sign and space. ..."
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This paper discusses paradox and imaginary values in relation to the mutuality of sign and space.
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.
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.
Intersection Types and Computational Rules
 WoLLIC’03, volume 84 of ENTCS
, 2003
"... The invariance of the meaning of a #term by reduction/expansion w.r.t. the considered computational rules is one of the minimal requirements one expects to hold for a #model. Being the intersection type systems a general framework for the study of semantic domains for the Lambdacalculus, the pres ..."
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The invariance of the meaning of a #term by reduction/expansion w.r.t. the considered computational rules is one of the minimal requirements one expects to hold for a #model. Being the intersection type systems a general framework for the study of semantic domains for the Lambdacalculus, the present paper provides a characterisation of "meaning invariance" in terms of characterisation results for intersection type systems enabling typing invariance of terms w.r.t. various notions of reduction/expansion, like #, # and a number of relevant restrictions of theirs. 1.
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.
Intersection Types for λTrees
, 1999
"... We introduce a type assignment system which is parametric with respect to five families of trees obtained by evaluating λterms (Böhm trees, LevyLongo trees, ...). Then we prove, in an (almost) uniform way, that each type assignment system fully describes the observational equivalences induced by ..."
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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, LevyLongo 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.
Problem 19
"... Abstract. A closed λterm M is easy if, for any other closed term N, the lambda theory generated by M = N is consistent, while it is simple easy if, given an arbitrary intersection type τ, one can find a suitable preorder on types which allows to derive τ for M. Simple easiness implies easiness. Th ..."
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Abstract. A closed λterm M is easy if, for any other closed term N, the lambda theory generated by M = N is consistent, while it is simple easy if, given an arbitrary intersection type τ, one can find a suitable preorder on types which allows to derive τ for M. Simple easiness implies easiness. The question whether easiness implies simple easiness constitutes Problem 19 in the TLCA list of open problems. In this paper we negatively answer the question providing a nonempty cor.e. (complement of a recursively enumerable) set of easy, but non simple easy, λterms. Key words: Lambda calculus, easy terms, simple easy terms, filter models 1
Intersection Types and Lambda Theories ∗
, 2002
"... We illustrate the use of intersection types as a semantic tool for showing properties of the lattice of λtheories. 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 λtheories. 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 predicate. This allows us to prove the consistency of a wellknow λtheory: this consistency has interesting consequences on the algebraic structure of the lattice of λtheories.
TWO CLASSES OF LOCALLY COMPACT SOBER SPACES
, 2005
"... We deal with two classes of locally compact sober spaces, namely, the class of locally spectral coherent spaces and the class of spaces in which every point has a closed spectral neighborhood (CSNspaces, for short). We prove that locally spectral coherent spaces are precisely the coherent sober spa ..."
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We deal with two classes of locally compact sober spaces, namely, the class of locally spectral coherent spaces and the class of spaces in which every point has a closed spectral neighborhood (CSNspaces, for short). We prove that locally spectral coherent spaces are precisely the coherent sober spaces with a basis of compact open sets. We also prove that CSNspaces are exactly the locally spectral coherent spaces in which every compact open set has a compact closure. 1.