We present a higher-order calculus ECC which can be seen as an extension of the calculus of constructions [CH88] by adding strong sum types and a fully cumulative type hierarchy. ECC turns out to be rather expressive so that mathematical theories can be abstractly described and abstract mathematics may be adequately formalized. It is shown that ECC is strongly normalizing and has other nice proof-theoretic properties. An !\GammaSet (realizability) model is described to show how the essential properties of the calculus can be captured set-theoretically.

. We present a categorical logic formulation of induction and coinduction principles for reasoning about inductively and coinductively defined types. Our main results provide sufficient criteria for the validity of such principles: in the presence of comprehension, the induction principle for initial algebras is admissible, and dually, in the presence of quotient types, the coinduction principle for terminal coalgebras is admissible. After giving an alternative formulation of induction in terms of binary relations, we combine both principles and obtain a mixed induction/coinduction principle which allows us to reason about minimal solutions X = oe(X) where X may occur both positively and negatively in the type constructor oe. We further strengthen these logical principles to deal with contexts and prove that such strengthening is valid when the (abstract) logic we consider is contextually/functionally complete. All the main results follow from a basic result about adjunc...

It is well known that one can build models of full higher-order dependent type theory (also called the calculus of constructions) using partial equivalence relations (PERs) and assemblies over a partial combinatory algebra (PCA). But the idea of categories of PERs and ERs (total equivalence relations) can be applied to other structures as well. In particular, we can easily dene the category of ERs and equivalencepreserving continuous mappings over the standard category Top 0 of topological T 0 -spaces; we call these spaces (a topological space together with an ER) equilogical spaces and the resulting category Equ. We show that this category|in contradistinction to Top 0 |is a cartesian closed category. The direct proof outlined here uses the equivalence of the category Equ to the category PEqu of PERs over algebraic lattices (a full subcategory of Top 0 that is well known to be cartesian closed from domain theory). In another paper with Carboni and Rosolini (cited herein) a more abstract categorical generalization shows why many such categories are cartesian closed. The category Equ obviously contains Top 0 as a full subcategory, and it naturally contains many other well known subcategories. In particular, we show why, as a consequence of work of Ershov, Berger, and others, the Kleene-Kreisel hierarchy of countable functionals of nite types can be naturally constructed in Equ from the natural numbers object N by repeated use in Equ of exponentiation and binary products. We also develop for Equ notions of modest sets (a category equivalent to Equ) and assemblies to explain why a model of dependent type theory is obtained. We make some comparisons of this model to other, known models. 1

The type-theoretic explanation of modules proposed to date (for programming languages like ML) is unsatisfactory, because it does not capture that evaluation of type-expressions is independent from evaluation of programexpressions. We propose a new explanation based on \programming languages as indexed categories" and illustrates how ML can be extended to support higher order modules, by developing a category-theoretic semantics for a calculus of modules with dependent types. The paper outlines also a methodology, which may lead to a modular approach in the study of programming languages. Introduction The addition of module facilities to programming languages is motivated by the need to provide a better environment for the development and maintenance of large programs. Nowadays many programming languages include such facilities. Throughout the paper Standard ML (see [Mac85, HMM86, MTH90]) is taken as representative for these languages. The implementation of module facilities has been ...

We investigate the development of theories of types and computability via realizability.

This paper describes the categorical semantics of a system of mixed intuitionistic and linear type theory (ILT). ILT was proposed by G. Plotkin and also independently by P. Wadler. The logic associated with ILT is obtained as a combination of intuitionistic logic with intuitionistic linear logic, and can be embedded in Barber and Plotkin's Dual Intuitionistic Linear Logic (DILL). However, unlike DILL, the logic for ILT lacks an explicit modality ! that translates intuitionistic proofs into linear ones. So while the semantics of DILL can be given in terms of monoidal adjunctions between symmetric monoidal closed categories and cartesian closed categories, the semantics of ILT is better presented via fibrations. These interpret double contexts, which cannot be reduced to linear ones. In order to interpret the intuitionistic and linear identity axioms acting on the same type we need fibrations satisfying the comprehension axiom.

The notion of an endofunctor having "greatest subcoalgebras" is introduced as a form of comprehension. This notion is shown to be instrumental in giving a systematic and abstract proof of the existence of limits for coalgebras -- proved earlier by Worrell and by Gumm & Schroder. These insights, in dual form, are used to reinvestigate colimits for algebras in terms of "least quotient algebras" -- leading to a uniform approach to limits of coalgebras and colimits of algebras. Finally, at an abstract level of fibrations, an equivalence is established between having greatest subcoalgebras (in a base category of types) and greatest invariants (in a total category of predicates).

In this paper we give a new presentation of ELF which is well-suited for semantic analysis. We introduce the notions of internal codability, internal definability, internal typed calculi and frame languages. These notions are central to our perspective of logical frameworks. We will argue that a logical framework is a typed calculus which formalizes the relationship between internal typed languages and frame languages. In the second half of the paper, we demonstrate the advantage of our logical framework by showing some categorical properties of it and of encodings in it. By doing so we hope to indicate a sensible model theory of encodings. Copyright c fl1993. All rights reserved. Reproduction of all or part of this work is permitted for educational or research purposes on condition that (1) this copyright notice is included, (2) proper attribution to the author or authors is made and (3) no commercial gain is involved. Technical Reports issued by the Department of Computer Sc...

) Eike Ritter Valeria de Paiva Computer Laboratory University of Cambridge 1 Introduction This paper contributes to the area of "categorical combinatory logic" or "categorical combinators", following the steps of Curien [Cur93] and Ritter [Rit92]. We provide a precise syntactic formulation of the notion of a multicategory (a recent reference is Lambek [Lam89]) and based on that we give categorical combinators for (multiplicative) Intuitionistic Linear Logic, following the general approach of Ritter for the Calculus of Constructions [Rit92]. Multicategories are usually thought of as "just like categories, except that instead of arrows A ! B one has multiarrows An ; : : : ; A 1 ! B". There is a well-known correspondence between multicategories and (rudimentary) linear logic in such a way that a multimap f : An ; : : : ; A 1 ! B above corresponds to a term denoting a proof of B from the assumptions x i : A i . But the picture of a multicategory referred to above is lacking in two accou...

This document provides an introduction to the interaction between category theory and mathematical logic which is slanted towards computer scientists.