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KripkeStyle Models for Typed Lambda Calculus
 Annals of Pure and Applied Logic
, 1996
"... The semantics of typed lambda calculus is usually described using Henkin models, consisting of functions over some collection of sets, or concrete cartesian closed categories, which are essentially equivalent. We describe a more general class of Kripkestyle models. In categorical terms, our Kripke ..."
Abstract

Cited by 45 (3 self)
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The semantics of typed lambda calculus is usually described using Henkin models, consisting of functions over some collection of sets, or concrete cartesian closed categories, which are essentially equivalent. We describe a more general class of Kripkestyle models. In categorical terms, our Kripke lambda models are cartesian closed subcategories of the presheaves over a poset. To those familiar with Kripke models of modal or intuitionistic logics, Kripke lambda models are likely to seem adequately \semantic." However, when viewed as cartesian closed categories, they do not have the property variously referred to as concreteness, wellpointedness, or having enough points. While the traditional lambda calculus proof system is not complete for Henkin models that may have empty types, we prove strong completeness for Kripke models. In fact, every set of equations that is closed under implication is the theory of a single Kripke model. We also develop some properties of logical relations ...
Retractions of types with many atoms
, 2001
"... We dene a sound and complete proof system for aÆne retractions in simple types (built over many atoms), and we state a necessary condition for arbitrary retractions in simple types. We also show a simple necessary condition for polymorphic retractability and we disprove an earlier conjecture abou ..."
Abstract

Cited by 3 (0 self)
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We dene a sound and complete proof system for aÆne retractions in simple types (built over many atoms), and we state a necessary condition for arbitrary retractions in simple types. We also show a simple necessary condition for polymorphic retractability and we disprove an earlier conjecture about a stronger necessary condition. 1