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Categorical Logic
 A CHAPTER IN THE FORTHCOMING VOLUME VI OF HANDBOOK OF LOGIC IN COMPUTER SCIENCE
, 1995
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HasCASL: Towards Integrated Specification and Development of Functional Programs
, 2002
"... The development of programs in modern functional languages such as Haskell calls for a widespectrum specification formalism that supports the type system of such languages, in particular higher order types, type constructors, and parametric polymorphism, and contains a functional language as an exe ..."
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Cited by 25 (11 self)
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The development of programs in modern functional languages such as Haskell calls for a widespectrum specification formalism that supports the type system of such languages, in particular higher order types, type constructors, and parametric polymorphism, and contains a functional language as an executable subset in order to facilitate rapid prototyping. We lay out the design of HasCasl, a higher order extension of the algebraic specification language Casl that is geared towards precisely this purpose. Its semantics is tuned to allow program development by specification refinement, while at the same time staying close to the settheoretic semantics of first order Casl. The number of primitive concepts in the logic has been kept as small as possible; we demonstrate how various extensions to the logic, in particular general recursion, can be formulated within the language itself.
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 
The maximality of the typed lambda calculus and of cartesian closed categories
 Publ. Inst. Math. (N.S
"... From the analogue of Böhm’s Theorem proved for the typed lambda calculus, without product types and with them, it is inferred that every cartesian closed category that satisfies an equality between arrows not satisfied in free cartesian closed categories must be a preorder. A new proof is given here ..."
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Cited by 17 (2 self)
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From the analogue of Böhm’s Theorem proved for the typed lambda calculus, without product types and with them, it is inferred that every cartesian closed category that satisfies an equality between arrows not satisfied in free cartesian closed categories must be a preorder. A new proof is given here of these results, which were obtained previously by Richard Statman and Alex K. Simpson.
Normal Forms and CutFree Proofs as Natural Transformations
 in : Logic From Computer Science, Mathematical Science Research Institute Publications 21
, 1992
"... What equations can we guarantee that simple functional programs must satisfy, irrespective of their obvious defining equations? Equivalently, what nontrivial identifications must hold between lambda terms, thoughtof as encoding appropriate natural deduction proofs ? We show that the usual syntax g ..."
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Cited by 12 (4 self)
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What equations can we guarantee that simple functional programs must satisfy, irrespective of their obvious defining equations? Equivalently, what nontrivial identifications must hold between lambda terms, thoughtof as encoding appropriate natural deduction proofs ? We show that the usual syntax guarantees that certain naturality equations from category theory are necessarily provable. At the same time, our categorical approach addresses an equational meaning of cutelimination and asymmetrical interpretations of cutfree proofs. This viewpoint is connected to Reynolds' relational interpretation of parametricity ([27], [2]), and to the KellyLambekMac LaneMints approach to coherence problems in category theory. 1 Introduction In the past several years, there has been renewed interest and research into the interconnections of proof theory, typed lambda calculus (as a functional programming paradigm) and category theory. Some of these connections can be surprisingly subtle. Here we a...
Nontrivial Power Types can't be Subtypes of Polymorphic Types
 in 4th Annual Symposium on Logic in Computer Science
, 1989
"... This paper establishes a new, limitative relation between the polymorphic lambda calculus and the kind of higherorder type theory which is embodied in the logic of toposes. It is shown that any embedding in a topos of the cartesian closed category of (closed) types of a model of the polymorphic lam ..."
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Cited by 6 (0 self)
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This paper establishes a new, limitative relation between the polymorphic lambda calculus and the kind of higherorder type theory which is embodied in the logic of toposes. It is shown that any embedding in a topos of the cartesian closed category of (closed) types of a model of the polymorphic lambda calculus must place the polymorphic types well away from the powertypes oe !\Omega of the topos, in the sense that oe !\Omega is a subtype of a polymorphic type only in the case that oe is empty (and hence oe !\Omega is terminal) . As corollaries, we obtain strengthenings of Reynolds' result on the nonexistence of settheoretic models of polymorphism. Introduction The results reported in this paper have their origin in Reynolds' discovery that the standard settheoretic model of the simply typed lambda calulus cannot be extended to model the polymorphic, or secondorder, typed lambda calculus. In [9] Reynolds speculated that there might be a model of polymorphism in which the typ...
Nonlocal dependencies via variable contexts
 Workshop on New Directions in TypeTheoretic Grammar. ESSLLI07
, 2007
"... This paper is an interim report on my efforts since around the turn of the millenium to develop a grammar framework that combines the advantages of derivational/prooftheoretic approaches such as Principles and Parameters (P&P) and Categorial Grammar (CG) with those of constraintbased/modeltheoret ..."
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Cited by 5 (3 self)
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This paper is an interim report on my efforts since around the turn of the millenium to develop a grammar framework that combines the advantages of derivational/prooftheoretic approaches such as Principles and Parameters (P&P) and Categorial Grammar (CG) with those of constraintbased/modeltheoretic
HigherOrder Categorical Grammars
 Proceedings of Categorial Grammars 04
"... into two principal paradigms: modeltheoretic syntax (MTS), which ..."
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Cited by 4 (1 self)
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into two principal paradigms: modeltheoretic syntax (MTS), which