In computer science we speak of implementing a logic; this is done in a programming language, such as Lisp, called here the implementation language. We also reason about the logic, as in understanding how to search for proofs; these arguments are expressed in the metalanguage and conducted in the metalogic of the object language being implemented. We also reason about the implementation itself, say to know it is correct; this is done in a programming logic. How do all these logics relate? This paper considers that question and more. We show that by taking the view that the metalogic is primary, these other parts are related in standard ways. The metalogic should be suitably rich so that the object logic can be presented as an abstract data type, and it must be suitably computational (or constructive) so that an instance of that type is an implementation. The data type abstractly encodes all that is relevant for metareasoning, i.e., not only the term constructing functions but also the...

The concept of relations over sets is generalized to relations over an arbitrary category, and used to investigate the abstraction (or logical-relations) theorem, the identity extension lemma, and parametric polymorphism, for Cartesian-closed-category models of the simply typed lambda calculus and PL-category models of the polymorphic typed lambda calculus. Treatments of Kripke relations and of complete relations on domains are included.

In this paper we consider the problem of deriving a term assignment system for Girard's Intuitionistic Linear Logic for both the sequent calculus and natural deduction proof systems. Our system differs from previous calculi (e.g. that of Abramsky) and has two important properties which they lack. These are the substitution property (the set of valid deductions is closed under substitution) and subject reduction (reduction on terms is well-typed). We define a simple (but more general than previous proposals) categorical model for Intuitionistic Linear Logic and show how this can be used to derive the term assignment system. We also consider term reduction arising from cut-elimination in the sequent calculus and normalisation in natural deduction. We explore the relationship between these, as well as with the equations which follow from our categorical model.

Abstract. We present five axioms of name-carrying lambda-terms identified up to alpha-conversion—that is, up to renaming of bound variables. We assume constructors for constants, variables, application and lambdaabstraction. Other constants represent a function Fv that returns the set of free variables in a term and a function that substitutes a term for a variable free in another term. Our axioms are (1) equations relating Fv and each constructor, (2) equations relating substitution and each constructor, (3) alpha-conversion itself, (4) unique existence of functions on lambda-terms defined by structural iteration, and (5) construction of lambda-abstractions given certain functions from variables to terms. By building a model from de Bruijn’s nameless lambda-terms, we show that our five axioms are a conservative extension of HOL. Theorems provable from the axioms include distinctness, injectivity and an exhaustion principle for the constructors, principles of structural induction and primitive recursion on lambda-terms, Hindley and Seldin’s substitution lemmas and

This thesis contains a study of the proof theory of classical logic and addresses the prob-lem of giving a computational interpretation to classical proofs. This interpretation aims to capture features of computation that go beyond what can be expressed in intuitionisticlogic. We introduce several strongly normalising cut-elimination procedures for classicallogic. Our procedures are less restrictive than previous strongly normalising procedures, while at the same time retaining the strong normalisation property, which various standardcut-elimination procedures lack. In order to apply proof techniques from term rewriting, including symmetric reducibility candidates and recursive path ordering, we develop termannotations for sequent proofs of classical logic. We then present a sequence-conclusion natural deduction calculus for classical logicand study the correspondence between cut-elimination and normalisation. In contrast to earlier work, which analysed this correspondence in various fragments of intuitionisticlogic, we establish the correspondence in classical logic. Finally, we study applications of cut-elimination. In particular, we analyse severalclassical proofs with respect to their behaviour under cut-elimination. Because our cutelimination procedures impose fewer constraints than previous procedures, we are ableto show how a fragment of classical logic can be seen as a typing system for the simplytyped lambda calculus extended with an erratic choice operator. As a pleasing conse-quence, we can give a simple computational interpretation to Lafont's example.

We investigate semantical normalization proofs for typed combinatory logic and weak -calculus. One builds a model and a function `quote' which inverts the interpretation function. A normalization function is then obtained by composing quote with the interpretation function. Our models are just like the intended model, except that the function space includes a syntactic component as well as a semantic one. We call this a `glued' model because of its similarity with the glueing construction in category theory. Other basic type constructors are interpreted as in the intended model. In this way we can also treat inductively defined types such as natural numbers and Brouwer ordinals. We also discuss how to formalize -terms, and show how one model construction can be used to yield normalization proofs for two different typed -calculi -- one with explicit and one with implicit substitution. The proofs are formalized using Martin-Lof's type theory as a meta language and mechanized using the A...

A constructive characterization is given of the isomorphisms which must hold in all models of the typed lambda calculus with surjective pairing. By the close relation between closed Cartesian categories and models of these calculi, we also produce a characterization of those isomorphisms which hold in all CCC's. By the correspondence between these calculi and proofs in intuitionistic positive propositional logic, we thus provide a characterization of equivalent formulae of this logic, where the definition of equivalence of terms depends on having "invertible" proofs between the two terms. Rittri (1989), on types as search keys in program libraries, provides an interesting example of use of these characterizations.

This paper describes the construction of categorical models for the nu-calculus, a language that combines higher-order functions with dynamically created names. Names are created with local scope, they can be compared with each other and passed around through function application, but that is all. The intent behind this language is to examine one aspect of the imperative character of Standard ML: the use of local state by dynamic creation of references. The nu-calculus is equivalent to a certain fragment of ML, omitting side effects, exceptions, datatypes and recursion. Even without all these features, the interaction of name creation with higher-order functions can be complex and subtle; it is particularly difficult to characterise the observable behaviour of expressions. Categorical monads, in the style of Moggi, are used to build denotational models for the nu-calculus. An intermediate stage is the use of a computational metalanguage, which distinguishes in the type system between values and computations. The general requirements for a categorical model are presented, and two specific examples described in detail. These provide a sound denotational semantics for the nu-calculus, and can be used to reason about observable equivalence in the language. In particular a model using logical relations is fully abstract for first-order expressions.

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Interpreting j-conversion as an expansion rule in the simply-typed -calculus maintains the confluence of reduction in a richer type structure. This use of expansions is supported by categorical models of reduction, where fi-contraction, as the local counit, and j-expansion, as the local unit, are linked by local triangle laws. The latter form reduction loops, but strong normalisation (to the long fij-normal forms) can be recovered by "cutting" the loops.