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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 ..."
Abstract

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 ..."
Abstract

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