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Intersection Types and Lambda Models
, 2005
"... 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 t ..."
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Cited by 11 (1 self)
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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.
Intersection Types and Domain Operators
, 2003
"... We use intersection types as a tool for obtaining λmodels. Relying on the notion of easy intersection type theory we successfully build a λmodel in which the interpretation of an arbitrary simple easy term is any filter which can be described by a continuous predicate. This allows us ..."
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Cited by 6 (3 self)
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We use intersection types as a tool for obtaining λmodels. Relying on the notion of easy intersection type theory we successfully build a λmodel 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 λtheory where the λterm (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 λtheories. The second result is that for any simple easy term there is a λmodel where the interpretation of the term is the minimal fixed point operator.
Easiness in graph models
 Theoretical Computer Science
, 1993
"... We generalize Baeten and Boerboom’s method of forcing, and apply this to show that there is a fixed sequence (uk)k∈ω of closed (untyped) λterms satisfying the following properties: a) For any countable sequence (gk)k∈ω of continuous functions (of arbitrary arity) on the power set of an arbitrary co ..."
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Cited by 5 (3 self)
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We generalize Baeten and Boerboom’s method of forcing, and apply this to show that there is a fixed sequence (uk)k∈ω of closed (untyped) λterms satisfying the following properties: a) For any countable sequence (gk)k∈ω of continuous functions (of arbitrary arity) on the power set of an arbitrary countable set, there is a graph model such that (λx.xx)(λx.xx)uk represents gk in the model. b) For any countable sequence (tk)k∈ω of closed λterms there is a graph model that satisfies (λx.xx)(λx.xx)uk = tk for all k. We apply these two results to show the existence of 1. a finitely axiomatized λtheory L such that the interval lattice constituted by the λtheories extending L is distributive; 2. a continuum of pairwise inconsistent graph theories ( = λtheories that can be realized as theories of graph models); 3. a congruence distributive variety of combinatory algebras (lambda
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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Cited by 3 (1 self)
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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.
Type Preorders and Recursive Terms
, 2004
"... We show how to use intersection types for building models of a #calculus enriched with recursive terms, whose intended meaning is of minimal fixed points. As a byproduct we prove an interesting consistency result. ..."
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Cited by 1 (0 self)
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We show how to use intersection types for building models of a #calculus enriched with recursive terms, whose intended meaning is of minimal fixed points. As a byproduct we prove an interesting consistency result.
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.
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 ..."
Abstract
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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.