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Inductive and Coinductive types with Iteration and Recursion
 Proceedings of the 1992 Workshop on Types for Proofs and Programs, Bastad
, 1992
"... We study (extensions of) simply and polymorphically typed lambda calculus from a point of view of how iterative and recursive functions on inductive types are represented. The inductive types can usually be understood as initial algebras in a certain category and then recursion can be defined in ter ..."
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We study (extensions of) simply and polymorphically typed lambda calculus from a point of view of how iterative and recursive functions on inductive types are represented. The inductive types can usually be understood as initial algebras in a certain category and then recursion can be defined in terms of iteration. However, in the syntax we often have only weak initiality, which makes the definition of recursion in terms of iteration inefficient or just impossible. We propose a categorical notion of (primitive) recursion which can easily be added as computation rule to a typed lambda calculus and gives us a clear view on what the dual of recursion, corecursion, on coinductive types is. (The same notion has, independently, been proposed by [Mendler 1991].) We look at how these syntactic notions work out in the simply typed lambda calculus and the polymorphic lambda calculus. It will turn out that in the syntax, recursion can be defined in terms of corecursion and vice versa using polymo...
Deliverables: A Categorical Approach to Program Development in Type Theory
, 1992
"... This thesis considers the problem of program correctness within a rich theory of dependent types, the Extended Calculus of Constructions (ECC). This system contains a powerful programming language of higherorder primitive recursion and higherorder intuitionistic logic. It is supported by Pollack&a ..."
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Cited by 24 (1 self)
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This thesis considers the problem of program correctness within a rich theory of dependent types, the Extended Calculus of Constructions (ECC). This system contains a powerful programming language of higherorder primitive recursion and higherorder intuitionistic logic. It is supported by Pollack's versatile LEGO implementation, which I use extensively to develop the mathematical constructions studied here. I systematically investigate Burstall's notion of deliverable, that is, a program paired with a proof of correctness. This approach separates the concerns of programming and logic, since I want a simple program extraction mechanism. The \Sigmatypes of the calculus enable us to achieve this. There are many similarities with the subset interpretation of MartinLof type theory. I show that deliverables have a rich categorical structure, so that correctness proofs may be decomposed in a principled way. The categorical combinators which I define in the system package up much logical bo...
From proof nets to the free * autonomous category
 Logical Methods in Computer Science, 2(4:3):1–44, 2006. Available from: http://arxiv.org/abs/cs/0605054. [McK05] Richard McKinley. Classical categories and deep inference. In Structures and Deduction 2005 (Satellite Workshop of ICALP’05
, 2005
"... Vol. 2 (4:3) 2006, pp. 1–44 www.lmcsonline.org ..."
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Monads And Interpolads In Bicategories
, 1997
"... . Given a bicategory, Y , with stable local coequalizers, we construct a bicategory of monads Y mnd by using lax functors from the generic 0cell, 1cell and 2cell, respectively, into Y . Any lax functor into Y factors through Y mnd and the 1cells turn out to be the familiar bimodules. The local ..."
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. Given a bicategory, Y , with stable local coequalizers, we construct a bicategory of monads Y mnd by using lax functors from the generic 0cell, 1cell and 2cell, respectively, into Y . Any lax functor into Y factors through Y mnd and the 1cells turn out to be the familiar bimodules. The locally ordered bicategory rel and its bicategory of monads both fail to be Cauchycomplete, but have a wellknown Cauchycompletion in common. This prompts us to formulate a concept of Cauchycompleteness for bicategories that are not locally ordered and suggests a weakening of the notion of monad. For this purpose, we develop a calculus of general modules between unstructured endo1cells. These behave well with respect to composition, but in general fail to have identities. To overcome this problem, we do not need to impose the full structure of a monad on endo1cells. We show that associative coequalizing multiplications suffice and call the resulting structures interpolads. Together with str...
A Categorytheoretic characterization of functional completeness
, 1990
"... . Functional languages are based on the notion of application: programs may be applied to data or programs. By application one may define algebraic functions; and a programming language is functionally complete when any algebraic function f(x 1 ,...,x n ) is representable (i.e. there is a constant a ..."
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. Functional languages are based on the notion of application: programs may be applied to data or programs. By application one may define algebraic functions; and a programming language is functionally complete when any algebraic function f(x 1 ,...,x n ) is representable (i.e. there is a constant a such that f(x 1 ,...,x n ) = (a . x 1 . ... . x n ). Combinatory Logic is the simplest typefree language which is functionally complete. In a sound categorytheoretic framework the constant a above may be considered as an "abstract gödelnumber" for f, when gödelnumberings are generalized to "principal morphisms", in suitable categories. By this, models of Combinatory Logic are categorically characterized and their relation is given to lambdacalculus models within Cartesian Closed Categories. Finally, the partial recursive functionals in any finite higher type are shown to yield models of Combinatory Logic. ________________ (+) Theoretical Computer Science, 70 (2), 1990, pp.193211. A p...
Orthomodular lattices, Foulis semigroups and dagger kernel categories
 Logical Methods in Comp. Sci., 2009
"... This paper is a sequel to [19] and continues the study of quantum logic via dagger kernel categories. It develops the relation between these categories and both orthomodular lattices and Foulis semigroups. The relation between the latter two notions has been uncovered in the 1960s. The current categ ..."
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This paper is a sequel to [19] and continues the study of quantum logic via dagger kernel categories. It develops the relation between these categories and both orthomodular lattices and Foulis semigroups. The relation between the latter two notions has been uncovered in the 1960s. The current categorical perspective gives a broader context and reconstructs this relationship between orthomodular lattices and Foulis semigroups as special instance. 1
A NOTE ON WEAK ALGEBRAIC THEORIES
"... Abstract. In our PhD thesis ([4]), we showed that for the study of denotational semantics of linear logic ([6]), it is crucial to generalize the standard notion of monad on a category. An earlier generalization was already given by Hoofman ([9]): semimonads. But it was not suitable for our problem. ..."
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Abstract. In our PhD thesis ([4]), we showed that for the study of denotational semantics of linear logic ([6]), it is crucial to generalize the standard notion of monad on a category. An earlier generalization was already given by Hoofman ([9]): semimonads. But it was not suitable for our problem. This is why we introduced another generalization: weak monads. In this note, we present this new notion and give some examples.
Int Construction and Semibiproducts
, 2009
"... We study a relationship between the Int construction of Joyal et al. and a weakening of biproducts called semibiproducts. We then provide an application of geometry of interaction interpretation for the multiplicative additive linear logic (MALL for short) of Girard. We consider not biproducts but s ..."
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We study a relationship between the Int construction of Joyal et al. and a weakening of biproducts called semibiproducts. We then provide an application of geometry of interaction interpretation for the multiplicative additive linear logic (MALL for short) of Girard. We consider not biproducts but semibiproducts because in general the Int construction does not preserve biproducts. We show that Int construction is left biadjoint to the forgetful functor from the 2category of compact closed categories with semibiproducts to the 2category of traced symmetric monoidal categories with semibiproducts. We then illustrate a traced distributive symmetric monoidal category with biproducts B(Pfn) and relate the interpretation of MALL in Int(B(Pfn)) to token machines defined over weighted MALL proofs.
A Note on SemiAdjuctions
, 1990
"... We consider two methods to generalise categorical notions involving functots to sem/functors. It turns out that the two methods give in general the same results. In particular, the notion of semiadjunction is studied. Semiadjunctions in our sense prove to be exactly the normal semiadjunctions ..."
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We consider two methods to generalise categorical notions involving functots to sem/functors. It turns out that the two methods give in general the same results. In particular, the notion of semiadjunction is studied. Semiadjunctions in our sense prove to be exactly the normal semiadjunctions of Hayashi [2].
semifunCtors
, 1990
"... Girard categories (GC's) were defined in [14] as categorical models for linear logic. It was shown that the Kleisli category of a GC is Cartesian closed. ..."
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Girard categories (GC's) were defined in [14] as categorical models for linear logic. It was shown that the Kleisli category of a GC is Cartesian closed.