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340
Metalogical Frameworks
, 1992
"... 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 me ..."
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Cited by 57 (15 self)
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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...
Types, Abstraction, and Parametric Polymorphism, Part 2
, 1991
"... The concept of relations over sets is generalized to relations over an arbitrary category, and used to investigate the abstraction (or logicalrelations) theorem, the identity extension lemma, and parametric polymorphism, for Cartesianclosedcategory models of the simply typed lambda calculus and P ..."
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Cited by 53 (1 self)
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The concept of relations over sets is generalized to relations over an arbitrary category, and used to investigate the abstraction (or logicalrelations) theorem, the identity extension lemma, and parametric polymorphism, for Cartesianclosedcategory models of the simply typed lambda calculus and PLcategory models of the polymorphic typed lambda calculus. Treatments of Kripke relations and of complete relations on domains are included.
Term Assignment for Intuitionistic Linear Logic
, 1992
"... 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. Th ..."
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Cited by 53 (9 self)
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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 welltyped). 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 cutelimination 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.
Five axioms of alphaconversion
 Ninth international Conference on Theorem Proving in Higher Order Logics TPHOL
, 1996
"... Abstract. We present five axioms of namecarrying lambdaterms identified up to alphaconversion—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 variab ..."
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Cited by 51 (0 self)
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Abstract. We present five axioms of namecarrying lambdaterms identified up to alphaconversion—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) alphaconversion itself, (4) unique existence of functions on lambdaterms defined by structural iteration, and (5) construction of lambdaabstractions given certain functions from variables to terms. By building a model from de Bruijn’s nameless lambdaterms, 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 lambdaterms, Hindley and Seldin’s substitution lemmas and
Intuitionistic Model Constructions and Normalization Proofs
, 1998
"... 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 ..."
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Cited by 44 (7 self)
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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 MartinLof's type theory as a meta language and mechanized using the A...
Categorical Models for Local Names
 LISP AND SYMBOLIC COMPUTATION
, 1996
"... This paper describes the construction of categorical models for the nucalculus, a language that combines higherorder 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. T ..."
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Cited by 39 (2 self)
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This paper describes the construction of categorical models for the nucalculus, a language that combines higherorder 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 nucalculus 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 higherorder 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 nucalculus. 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 nucalculus, and can be used to reason about observable equivalence in the language. In particular a model using logical relations is fully abstract for firstorder expressions.
Provable Isomorphisms of Types
 MATHEMATICAL STRUCTURES IN COMPUTER SCIENCE
, 1990
"... 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 ..."
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Cited by 39 (8 self)
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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.
Normalization by evaluation for typed lambda calculus with coproducts
 In LICS
, 2001
"... 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 interp ..."
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Cited by 39 (5 self)
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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
The Virtues of Etaexpansion
, 1993
"... Interpreting jconversion as an expansion rule in the simplytyped calculus maintains the confluence of reduction in a richer type structure. This use of expansions is supported by categorical models of reduction, where ficontraction, as the local counit, and jexpansion, as the local unit, are li ..."
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Cited by 36 (4 self)
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Interpreting jconversion as an expansion rule in the simplytyped calculus maintains the confluence of reduction in a richer type structure. This use of expansions is supported by categorical models of reduction, where ficontraction, as the local counit, and jexpansion, as the local unit, are linked by local triangle laws. The latter form reduction loops, but strong normalisation (to the long fijnormal forms) can be recovered by "cutting" the loops.