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Adjoint Rewriting
, 1995
"... This thesis concerns rewriting in the typed calculus. Traditional categorical models of typed calculus use concepts such as functor, adjunction and algebra to model type constructors and their associated introduction and elimination rules, with the natural categorical equations inherent in these s ..."
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Cited by 25 (11 self)
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This thesis concerns rewriting in the typed calculus. Traditional categorical models of typed calculus use concepts such as functor, adjunction and algebra to model type constructors and their associated introduction and elimination rules, with the natural categorical equations inherent in these structures providing an equational theory for terms. One then seeks a rewrite relation which, by transforming terms into canonical forms, provides a decision procedure for this equational theory. Unfortunately the rewrite relations which have been proposed, apart from for the most simple of calculi, either generate the full equational theory but contain no decision procedure, or contain a decision procedure but only for a subtheory of that required. Our proposal is to unify the semantics and reduction theory of the typed calculus by generalising the notion of model from categorical structures based on term equality to categorical structures based on term reduction. This is accomplished via...
Normalization and the Yoneda Embedding
"... this paper we describe a new, categorical approach to normalization in typed  ..."
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Cited by 22 (3 self)
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this paper we describe a new, categorical approach to normalization in typed 
Topological Representation of the &ambda;Calculus
, 1998
"... The calculus can be represented topologically by assigning certain spaces to the types and certain continuous maps to the terms. Using a recent result from category theory, the usual calculus of conversion is shown to be deductively complete with respect to such topological semantics. It is al ..."
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The calculus can be represented topologically by assigning certain spaces to the types and certain continuous maps to the terms. Using a recent result from category theory, the usual calculus of conversion is shown to be deductively complete with respect to such topological semantics. It is also shown to be functionally complete, in the sense that there is always a "minimal" topological model, in which every continuous function is definable. These results subsume earlier ones using cartesian closed categories, as well as those employing socalled Henkin and Kripke models. Introduction The calculus originates with Church [6]; it is intended as a formal calculus of functional application and specification. In this paper, we are mainly interested in the version known as simply typed calculus ; as is now wellknown, the untyped version can be treated as a special case of this ([17]). We present here a topological representation of the calculus: types are represented by cert...
New Semantics for the Simply Typed lambdacalculus
, 2003
"... The simply typed calculus is known to be complete with respect to models of the form Sets . More formally, that means that given any simply typed theory T, T ` t 1 = t 2 i for all Tmodels [[ ]] in Sets , for all categories C , we have [[t 1 ]] = [[t 2 ]]. It follows from a result by Awodey ..."
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The simply typed calculus is known to be complete with respect to models of the form Sets . More formally, that means that given any simply typed theory T, T ` t 1 = t 2 i for all Tmodels [[ ]] in Sets , for all categories C , we have [[t 1 ]] = [[t 2 ]]. It follows from a result by Awodey that it is enough to look at models in Sets for P a poset. In this thesis, I will describe explicitly how this more powerful completeness result follows from a result in [2]. As models of the form Sets for P a poset resemble the Kripke models familiar from intuitionistic logic, they are relatively easy for noncategory theorists to understand. We hope that the simpler semantics result in new applications of the simply typed calculus. We also describe how this gives a complete semantics of the simply typed calculus in a certain category of posets.
New Semantics for the Simply Typed
"... The simply typed calculus is known to be complete with respect to models of the form Sets . More formally, that means that given any simply typed  theory T, T ` t 1 = t 2 i for all Tmodels [[ ]] in Sets we have [[t 1 ]] = [[t 2 ]]. It follows from a result by Awodey that it is enough to loo ..."
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The simply typed calculus is known to be complete with respect to models of the form Sets . More formally, that means that given any simply typed  theory T, T ` t 1 = t 2 i for all Tmodels [[ ]] in Sets we have [[t 1 ]] = [[t 2 ]]. It follows from a result by Awodey that it is enough to look at models in Sets for P a poset. For my thesis, I will describe explicitly how this more powerful completeness result follows from his recent paper. As models of the form for P a poset resemble the Kripke models familiar from intuitionistic logic, they are relatively easy for noncategory theorists to understand. We hope that the simpler semantics result in new applications of the simply typed calculus. We also describe how this gives a complete semantics of the simply typed calculus in Pos=P.
Completeness of Conversion between Reactive Programs for Ultrametric Models
"... Abstract. In 1970 Friedman proved completeness of beta eta conversion in the simplytyped lambda calculus for the settheoretical model. Recently Krishnaswami and Benton have captured the essence of Hudak’s reactive programs in an extension of simply typed lambda calculus with causal streams and a t ..."
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Abstract. In 1970 Friedman proved completeness of beta eta conversion in the simplytyped lambda calculus for the settheoretical model. Recently Krishnaswami and Benton have captured the essence of Hudak’s reactive programs in an extension of simply typed lambda calculus with causal streams and a temporal modality and provided this typed lambda calculus for reactive programs with a sound ultrametric semantics. We show that beta eta conversion in the typed lambda calculus of reactive programs is complete for the ultrametric model. 1