We solve the decision problem for simply typed lambda calculus with strong binary sums, equivalently the word problem for free cartesian closed categories with binary coproducts. Our method is based on the semantical technique known as “normalization by evaluation ” and involves inverting the interpretation of the syntax into a suitable sheaf model and from this extracting appropriate unique normal forms. There is no rewriting theory involved, and the proof is completely constructive, allowing program extraction from the proof. 1

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...

Shape theory provides a framework for the study of data types in which shape and data can be manipulated separately. This paper is concerned with shape checking, i.e. the detection of shape errors, such as array bound errors, without handling the data stored within. It can be seen as a form of partial evaluation in which data computations are ignored. We construct a simply-typed lambda-calculus that supports a vector type constructor, whose iteration yields types of arrays. It is expressive enough to construct all of the usual linear algebra operations. All shape errors in a term t can be detected by evaluating its shape #t. Evaluation of #t will terminate if that of t does. Keywords shape analysis, partial evaluation, arrays, higher-order. 1 Introduction Shape theory explores the consequences of manipulating shape and data separately (Jay [14]). Shape refers to the data structure in which the data is stored. For example, the shape of a three-dimensional regular array is a tuple of...

Introduction We present a categorical proof of the normalization theorem for simply typed -calculus, i.e. we derive a computable function nf which assigns to every typed -term a normal form, s.t. M ' N nf(M ) = nf(N ) nf(M ) ' M where ' is fij equality. Both the function nf and its correctness properties can be deduced from the categorical construction. To substantiate this, we present an ML program in the appendix which can be extracted from our argument. We emphasize that this presentation of normalization is reduction free, i.e. we do not mention term rewriting or use properties of term rewriting systems such as the Church-Rosser property. An immediate consequence of normalization is the decidability of ' but there are other useful corollaries; for instance we can show that

. Although the use of expansionary j-rewrite has become increasingly common in recent years, one area where j-contractions have until now remained the only possibility is in the more powerful type theories of the -cube. This paper rectifies this situation by applying j-expansions to the Calculus of Constructions --- we discuss some of the difficulties posed by the presence of dependent types, prove that every term rewrites to a unique long fij-normal form and deduce the decidability of fij-equality, typeability and type inhabitation as corollaries. 1 Introduction Extensional equality for the simply typed -calculus requires j-conversion, whose interpretation as a rewrite rule has traditionally been as a contraction x : T:fx ) f where x 6 2 FV(t). When combined with the usual fi-reduction, the resulting rewrite relation is strongly normalising and confluent, and thus reduction to normal form provides a decision procedure for the associated equational theory. However j-contractions beh...

Term rewriting systems are widely used throughout computer science as they provide an abstract model of computation while retaining a comparatively simple syntax and semantics. In order to reason within large term rewriting systems, structuring operations are used to build large term rewriting systems from smaller ones. Of particular interest is whether key properties are modular, thatis,ifthe components of a structured term rewriting system satisfy a property, then does the term rewriting system as a whole? A body of literature addresses this problem, but most of the results and proofs depend on strong syntactic conditions and do not easily generalize. Although many specific modularity results are known, a coherent framework which explains the underlying principles behind these results is lacking. This thesis posits that part of the problem is the usual, concrete and syntaxoriented semantics of term rewriting systems, and that a semantics is needed which on the one hand elides unnecessary syntactic details but on the other hand still possesses enough expressive power to model the key concepts arising from

The oe-calculus adds explicit substitutions to the -calculus so as to provide a theoretical framework within which the implementation of functional programming languages can be studied. This paper generalises the oe-calculus to provide a linear calculus of explicit substitutions, called xDILL, which analogously describes the implementation of linear functional programming languages. Our main observation is that there are non-trivial interactions between linearity and explicit substitutions and that xDILL is therefore best understood as a synthesis of its underlying logical structure and the technology of explicit substitutions. This is in contrast to the oe-calculus where the explicit substitutions are independent of the underlying logical structure. Keywords: -calculus, explicit substitutions, linear logic 1 Introduction This paper combines the technologies of explicit substitutions and linearity in a mathematically consistent way. We start by describing these technologies and the...

A generalization of positive inductive and coinductive types to monotone inductive and coinductive constructors of rank 1 and rank 2 is described. The motivation is taken from initial algebras and nal coalgebras in a functor category and the Curry-Howard-correspondence. The denition of the system as a -calculus requires an appropriate denition of monotonicity to overcome subtle problems, most notably to ensure that the (co-)inductive constructors introduced via monotonicity of the underlying constructor of rank 2 are also monotone as constructors of rank 1. The problem is solved, strong normalization shown, and the notion proven to be wide enough to cover even highly complex datatypes. 1

. We introduce a notion of Grothendieck logical relation and use it to characterise the definability of morphisms in stable bicartesian closed categories by terms of the simply-typed lambda calculus with finite products and finite sums. Our techniques are based on concepts from topos theory, however our exposition is elementary. Introduction The use of logical relations as a tool for characterising the -definable elements in a model of the simply-typed -calculus originated in the work of Plotkin [10], who obtained such a characterisation of the definable elements in the full type hierarchy using a notion of Kripke logical relation. Subsequently, the more general notion of a Kripke logical relation of varying arity was developed by Jung and Tiuryn, and shown to characterise the definable elements in any Henkin model [4]. Although not emphasised in [4], relations of varying arity are powerful enough to characterise relative definability with respect to any given set of elements con...

The use of expansionary j-rewrite rules in various typed -calculi has become increasingly common in recent years as their advantages over contractive j-rewrite rules have become apparent. Not only does one obtain the decidability of fij-equality, but rewrite relations based on expansions give a natural interpretation of long fij-normal forms, generalise more easily to other type constructors, retain key properties when combined with other rewrite relations, and are supported by a categorical theory of reduction. This paper extends the initial results concerning the simply typed -calculus to System F, that is, we prove strong normalisation and confluence for a rewrite relation consisting of traditional fi-reductions and j-expansions satisfying certain restrictions. Further, we characterise the second order long fij-normal forms as precisely the normal forms of the restricted rewrite relation. These results are an important step towards showing that j-expansions are compatible with the m...