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19
Introduction to lambda calculus
 Group, University of Göteborg and Chalmers University of Technology
, 1988
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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 ..."
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Cited by 44 (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 ...
A New Characterization of Lambda Definability
, 1993
"... . We give a new characterization of lambda definability in Henkin models using logical relations defined over ordered sets with varying arity. The advantage of this over earlier approaches by Plotkin and Statman is its simplicity and universality. Yet, decidability of lambda definability for heredit ..."
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Cited by 26 (1 self)
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. We give a new characterization of lambda definability in Henkin models using logical relations defined over ordered sets with varying arity. The advantage of this over earlier approaches by Plotkin and Statman is its simplicity and universality. Yet, decidability of lambda definability for hereditarily finite Henkin models remains an open problem. But if the variable set allowed in terms is also restricted to be finite then our techniques lead to a decision procedure. 1 Introduction An applicative structure consists of a family (A oe ) oe2T of sets, one for each type oe, together with a family (app oe;ø ) oe;ø 2T of application functions, where app oe;ø maps A oe!ø \Theta A oe into A ø . For an applicative structure to be a model of the simply typed lambda calculus (in which case we call it a Henkin model, following [4]), one requires two more conditions to hold. It must be extensional which means that the elements of A oe!ø are uniquely determined by their behavior under app oe;ø...
Exact Completions and Toposes
 University of Edinburgh
, 2000
"... Toposes and quasitoposes have been shown to be useful in mathematics, logic and computer science. Because of this, it is important to understand the di#erent ways in which they can be constructed. Realizability toposes and presheaf toposes are two important classes of toposes. All of the former and ..."
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Cited by 13 (4 self)
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Toposes and quasitoposes have been shown to be useful in mathematics, logic and computer science. Because of this, it is important to understand the di#erent ways in which they can be constructed. Realizability toposes and presheaf toposes are two important classes of toposes. All of the former and many of the latter arise by adding &quot;good &quot; quotients of equivalence relations to a simple category with finite limits. This construction is called the exact completion of the original category. Exact completions are not always toposes and it was not known, not even in the realizability and presheaf cases, when or why toposes arise in this way. Exact completions can be obtained as the composition of two related constructions. The first one assigns to a category with finite limits, the &quot;best &quot; regular category (called its regular completion) that embeds it. The second assigns to
A Characterization Of Lambda Definability In Categorical Models Of Implicit Polymorphism
 Theoretical Computer Science
, 1995
"... . Lambda definability is characterized in categorical models of simply typed lambda calculus with type variables. A categorytheoretic framework known as glueing or sconing is used to extend the JungTiuryn characterization of lambda definability [JuT93], first to ccc models, and then to categor ..."
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Cited by 11 (0 self)
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. Lambda definability is characterized in categorical models of simply typed lambda calculus with type variables. A categorytheoretic framework known as glueing or sconing is used to extend the JungTiuryn characterization of lambda definability [JuT93], first to ccc models, and then to categorical models of the calculus with type variables. Logical relations are now a wellestablished tool for studying the semantics of various typed lambda calculi. The main lines of research are focused in two areas, the first of which strives for an understanding of Strachey's notion of parametric polymorphism. The main idea is that a parametricly polymorphic function acts independently from the types to which its type variables are instantiated, and that this uniformity may be captured by imposing a relational structure on the types [OHT93, MSd93, MaR91, Wad89, Rey83, Str67]. The other line of research concerns lambda definability and the full abstraction problem for various models of languag...
A Characterization Of The Left Exact Categories Whose Exact Completions Are Toposes
, 1999
"... We characterize the categories with finite limits whose exact completions are toposes. We review the examples in the literature and also find new examples and counterexamples. ..."
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Cited by 5 (2 self)
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We characterize the categories with finite limits whose exact completions are toposes. We review the examples in the literature and also find new examples and counterexamples.
Intuitive Counterexamples for Constructive Fallacies
 Mathematical Foundations of Computer Science 1994  19th International Symposium, MFCS '94, Kosice
, 1994
"... Formal countermodels may be used to justify the unprovability of formulae in the Heyting calculus (the best accepted formal system for constructive reasoning), on the grounds that unprovable formulae are constructively invalid. We argue that the intuitive impact of such countermodels becomes more tr ..."
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Cited by 1 (1 self)
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Formal countermodels may be used to justify the unprovability of formulae in the Heyting calculus (the best accepted formal system for constructive reasoning), on the grounds that unprovable formulae are constructively invalid. We argue that the intuitive impact of such countermodels becomes more transparent and convincing as we move from Kripke/Beth models based on possible worlds, to Lauchli realizability models. We introduce a new semantics for constructive reasoning, called relational realizability, which strengthens further the intuitive impact of Lauchli realizability. But, none of these model theories provides countermodels with the compelling impact of classical truthtable countermodels for classically unprovable formulae. We prove soundness of the Heyting calculus for relational realizability, and conjecture that there is a constructive choicefree proof of completeness. In this respect, relational realizability improves the metamathematical constructivity of Lauchli realizab...