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12
Complete Axioms for Categorical Fixedpoint Operators
 In Proceedings of 15th Annual Symposium on Logic in Computer Science
, 2000
"... We give an axiomatic treatment of fixedpoint operators in categories. A notion of iteration operator is defined, embodying the equational properties of iteration theories. We prove a general completeness theorem for iteration operators, relying on a new, purely syntactic characterisation of the fre ..."
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Cited by 29 (6 self)
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We give an axiomatic treatment of fixedpoint operators in categories. A notion of iteration operator is defined, embodying the equational properties of iteration theories. We prove a general completeness theorem for iteration operators, relying on a new, purely syntactic characterisation of the free iteration theory. We then show how iteration operators arise in axiomatic domain theory. One result derives them from the existence of sufficiently many bifree algebras (exploiting the universal property Freyd introduced in his notion of algebraic compactness) . Another result shows that, in the presence of a parameterized natural numbers object and an equational lifting monad, any uniform fixedpoint operator is necessarily an iteration operator. 1. Introduction Fixed points play a central role in domain theory. Traditionally, one works with a category such as Cppo, the category of !continuous functions between !complete pointed partial orders. This possesses a leastfixedpoint oper...
A Uniform Approach to Domain Theory in Realizability Models
 Mathematical Structures in Computer Science
, 1996
"... this paper we provide a uniform approach to modelling them in categories of modest sets. To do this, we identify appropriate structure for doing "domain theory" in such "realizability models". In Sections 2 and 3 we introduce PCAs and define the associated "realizability" categories of assemblies an ..."
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Cited by 19 (6 self)
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this paper we provide a uniform approach to modelling them in categories of modest sets. To do this, we identify appropriate structure for doing "domain theory" in such "realizability models". In Sections 2 and 3 we introduce PCAs and define the associated "realizability" categories of assemblies and modest sets. Next, in Section 4, we prepare for our development of domain theory with an analysis of nontermination. Previous approaches have used (relatively complicated) categorical formulations of partial maps for this purpose. Instead, motivated by the idea that A provides a primitive programming language, we consider a simple notion of "diverging" computation within A itself. This leads to a theory of divergences from which a notion of (computable) partial function is derived together with a lift monad classifying partial functions. The next task is to isolate a subcategory of modest sets with sufficient structure for supporting analogues of the usual domaintheoretic constructions. First, we expect to be able to interpret the standard constructions of total type theory in this category, so it should inherit cartesianclosure, coproducts and the natural numbers from modest sets. Second, it should interact well with the notion of partiality, so it should be closed under application of the lift functor. Third, it should allow the recursive definition of partial functions. This is achieved by obtaining a fixpoint object in the category, as defined in (Crole and Pitts 1992). Finally, although there is in principle no definitive list of requirements on such a category, one would like it to support more complicated constructions such as those required to interpret polymorphic and recursive types. The central part of the paper (Sections 5, 6, 7 and 9) is devoted to establish...
Relational Properties of Recursively Defined Domains
 In 8th Annual Symposium on Logic in Computer Science
, 1993
"... This paper describes a mixed induction/coinduction property of relations on recursively defined domains. We work within a general framework for relations on domains and for actions of type constructors on relations introduced by O'Hearn and Tennent [20], and draw upon Freyd's analysis [7] of recurs ..."
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Cited by 15 (2 self)
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This paper describes a mixed induction/coinduction property of relations on recursively defined domains. We work within a general framework for relations on domains and for actions of type constructors on relations introduced by O'Hearn and Tennent [20], and draw upon Freyd's analysis [7] of recursive types in terms of a simultaneous initiality/finality property. The utility of the mixed induction/coinduction property is demonstrated by deriving a number of families of proof principles from it. One instance of the relational framework yields a family of induction principles for admissible subsets of general recursively defined domains which extends the principle of structural induction for inductively defined sets. Another instance of the framework yields the coinduction principle studied by the author in [22], by which equalities between elements of recursively defined domains may be proved via `bisimulations'. 1 Introduction A characteristic feature of higherorder functional lan...
CallByPushValue: A Subsuming Paradigm
 in Proc. TLCA ’99
, 1999
"... . Callbypushvalue is a new paradigm that subsumes the callbyname and callbyvalue paradigms, in the following sense: both operational and denotational semantics for those paradigms can be seen as arising, via translations that we will provide, from similar semantics for callbypushvalue. To ..."
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Cited by 15 (0 self)
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. Callbypushvalue is a new paradigm that subsumes the callbyname and callbyvalue paradigms, in the following sense: both operational and denotational semantics for those paradigms can be seen as arising, via translations that we will provide, from similar semantics for callbypushvalue. To explain callbypushvalue, we first discuss general operational ideas, especially the distinction between values and computations, using the principle that "a value is, a computation does". Using an example program, we see that the lambdacalculus primitives can be understood as push/pop commands for an operandstack. We provide operational and denotational semantics for a range of computational effects and show their agreement. We hence obtain semantics for callbyname and callbyvalue, of which some are familiar, some are new and some were known but previously appeared mysterious. 1 Introduction 1.1 Contribution In his invited lecture at POPL '98 [32], Reynolds, surveying over 30 year...
The Girard Translation Extended with Recursion
 In Proceedings of Computer Science Logic
, 1995
"... This paper extends CurryHoward interpretations of Intuitionistic Logic (IL) and Intuitionistic Linear Logic (ILL) with rules for recursion. The resulting term languages, the rec calculus and the linear rec calculus respectively, are given sound categorical interpretations. The embedding of ..."
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Cited by 11 (0 self)
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This paper extends CurryHoward interpretations of Intuitionistic Logic (IL) and Intuitionistic Linear Logic (ILL) with rules for recursion. The resulting term languages, the rec calculus and the linear rec calculus respectively, are given sound categorical interpretations. The embedding of proofs of IL into proofs of ILL given by the Girard Translation is extended with the rules for recursion, such that an embedding of terms of the rec calculus into terms of the linear rec calculus is induced via the extended CurryHoward isomorphisms. This embedding is shown to be sound with respect to the categorical interpretations. Full version of paper to appear in Proceedings of CSL '94, LNCS 933, 1995. y Basic Research in Computer Science, Centre of the Danish National Research Foundation. Contents 1 Introduction 4 2 The Categorical Picture 6 2.1 Previous Work and Related Results : : : : : : : : : : : : : : : : : : : : : : 6 2.2 How to deal with parameters : : : : : : : ...
Recursive Types in Kleisli Categories
 Preprint 2004. MFPS Tutorial, April 2007 Classical Domain Theory 75/75
, 1992
"... We show that an enriched version of Freyd's principle of versality holds in the Kleisli category of a commutative strong monad with fixedpoint object. This gives a general categorical setting in which it is possible to model recursive types involving the usual datatype constructors. ..."
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Cited by 7 (2 self)
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We show that an enriched version of Freyd's principle of versality holds in the Kleisli category of a commutative strong monad with fixedpoint object. This gives a general categorical setting in which it is possible to model recursive types involving the usual datatype constructors.
Traced Premonoidal Categories
, 1999
"... Motivated by some examples from functional programming, we propose a generalization of the notion of trace to symmetric premonoidal categories and of Conway operators to Freyd categories. We show that in a Freyd category, these notions are equivalent, generalizing a wellknown theorem relating trace ..."
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Cited by 7 (0 self)
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Motivated by some examples from functional programming, we propose a generalization of the notion of trace to symmetric premonoidal categories and of Conway operators to Freyd categories. We show that in a Freyd category, these notions are equivalent, generalizing a wellknown theorem relating traces and Conway operators in cartesian categories.
The Recursion Scheme from the Cofree Recursive Comonad
"... We instantiate the general comonadbased construction of recursion schemes for the initial algebra of a functor F to the cofree recursive comonad on F. Differently from the scheme based on the cofree comonad on F in a similar fashion, this scheme allows not only recursive calls on elements structura ..."
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Cited by 2 (1 self)
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We instantiate the general comonadbased construction of recursion schemes for the initial algebra of a functor F to the cofree recursive comonad on F. Differently from the scheme based on the cofree comonad on F in a similar fashion, this scheme allows not only recursive calls on elements structurally smaller than the given argument, but also subsidiary recursions. We develop a Mendler formulation of the scheme via a generalized Yoneda lemma for initial algebras involving strong dinaturality and hint a relation to circular proofs à la Cockett, Santocanale.
Categories and Types for Axiomatic Domain Theory
, 2003
"... Domain Theory provides a denotational semantics for programming languages and calculi containing fixed point combinators and other socalled paradoxical combinators. This dissertation presents results in the category theory and type theory of Axiomatic Domain Theory. Prompted by the adjunctions of D ..."
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Domain Theory provides a denotational semantics for programming languages and calculi containing fixed point combinators and other socalled paradoxical combinators. This dissertation presents results in the category theory and type theory of Axiomatic Domain Theory. Prompted by the adjunctions of Domain Theory, we extend Benton’s linear/nonlinear dualsequent calculus to include recursive linear types and define a class of models by adding Freyd’s notion of algebraic compactness to the monoidal adjunctions that model Benton’s calculus. We observe that algebraic compactness is better behaved in the context of categories with structural actions than in the usual context of enriched categories. We establish a theory of structural algebraic compactness that allows us to describe our models without reference to enrichment. We develop a 2categorical perspective on structural actions, including a presentation of monoidal categories that leads directly to Kelly’s reduced coherence conditions. We observe that Benton’s adjoint type constructors can be treated individually, semantically as well as syntactically, using free representations of distributors. We type various of fixed point combinators using recursive types and function types, which