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Coalgebras and Monads in the Semantics of Java
 Theoretical Computer Science
, 2002
"... This paper describes the basic structures in the denotational and axiomatic semantics of sequential Java, both from a monadic and a coalgebraic perspective. This semantics is an abstraction of the one used for the verification of (sequential) Java programs using proof tools in the LOOP project at th ..."
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This paper describes the basic structures in the denotational and axiomatic semantics of sequential Java, both from a monadic and a coalgebraic perspective. This semantics is an abstraction of the one used for the verification of (sequential) Java programs using proof tools in the LOOP project at the University of Nijmegen. It is shown how the monadic perspective gives rise to the relevant computational structure in Java (composition, extension and repetition), and how the coalgebraic perspective o#ers an associated program logic (with invariants, bisimulations, and Hoare logics) for reasoning about the computational structure provided by the monad.
A linear logic model of state
 Manuscript
, 1993
"... We propose an abstract formal model of state manipulation in the framework of Girard’s linear logic. Two issues motivate this work: how to describe the semantics of higherorder imperative programming languages and how to incorporate state manipulation in functional programming languages. The centra ..."
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We propose an abstract formal model of state manipulation in the framework of Girard’s linear logic. Two issues motivate this work: how to describe the semantics of higherorder imperative programming languages and how to incorporate state manipulation in functional programming languages. The central idea is that a state is linear and “regenerative”, where the latter is the property of a value that generates a new value upon each use. Based on this, we define a type constructor for states and a “modality” type constructor for regenerative values. Just as Girard’s “of course ” modality allows him to express static values and intuitionistic logic within the framework of linear logic, our regenerative modality allows us to express dynamic values and imperative programs within the same framework. We demonstrate the expressiveness of the model by showing that a higherorder Algollike language can be embedded in it. 1
What is a Data Type?
, 1996
"... A program derivation is said to be polytypic if some of its parameters are data types. Polytypic program derivations necessitate a general, noninductive definition of `data type'. Here we propose such a definition: a data type is a relator that has membership. It is shown how this definition i ..."
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A program derivation is said to be polytypic if some of its parameters are data types. Polytypic program derivations necessitate a general, noninductive definition of `data type'. Here we propose such a definition: a data type is a relator that has membership. It is shown how this definition implies various other properties that are shared by all data types. In particular, all data types have a unique strength, and all natural transformations between data types are strong. 1 Introduction What is a data type? It is easy to list a number of examples: pairs, lists, bags, finite sets, possibly infinite sets, function spaces . . . but such a list of examples hardly makes a definition. The obvious formalisation is a definition that builds up the class of data types inductively; such an inductive definition, however, leads to cumbersome proofs if we want to prove a property of all data types. Here we aim to give a noninductive characterisation, defining a data type as a mathematical object...
An Equational Notion of Lifting Monad
 TITLE WILL BE SET BY THE PUBLISHER
, 2003
"... We introduce the notion of an equational lifting monad: a commutative strong monad satisfying one additional equation (valid for monads arising from partial map classifiers). We prove that any equational lifting monad has a representation by a partial map classifier such that the Kleisli category ..."
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We introduce the notion of an equational lifting monad: a commutative strong monad satisfying one additional equation (valid for monads arising from partial map classifiers). We prove that any equational lifting monad has a representation by a partial map classifier such that the Kleisli category of the former fully embeds in the partial category of the latter. Thus equational lifting monads precisely capture the equational properties of partial maps as induced by partial map classifiers. The representation theorem also provides a tool for transferring nonequational properties of partial map classifiers to equational lifting monads. It is proved using a direct axiomatization of Kleisli categories of equational lifting monads. This axiomatization is of interest in its own right. 1
A Semantic Formulation of ⊤⊤lifting and Logical Predicates for Computational Metalanguage
 In Proc. CSL 2005. LNCS 3634
, 2005
"... Abstract. A semantic formulation of Lindley and Stark’s ⊤⊤lifting is given. We first illustrate our semantic formulation of the ⊤⊤lifting in Set with several examples, and apply it to the logical predicates for Moggi’s computational metalanguage. We then abstract the semantic ⊤⊤lifting as the lif ..."
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Abstract. A semantic formulation of Lindley and Stark’s ⊤⊤lifting is given. We first illustrate our semantic formulation of the ⊤⊤lifting in Set with several examples, and apply it to the logical predicates for Moggi’s computational metalanguage. We then abstract the semantic ⊤⊤lifting as the lifting of strong monads across bifibrations with lifted symmetric monoidal closed structures. 1
Covariant Types
 Theoretical Computer Science
, 1997
"... The covariant type system is an impredicative system that is rich enough to represent some polymorphism on inductive types, such as lists and trees, and yet is simple enough to have a settheoretic semantics. Its chief novelty is to replace function types by transformation types, which denote parame ..."
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The covariant type system is an impredicative system that is rich enough to represent some polymorphism on inductive types, such as lists and trees, and yet is simple enough to have a settheoretic semantics. Its chief novelty is to replace function types by transformation types, which denote parametric functions. Their free type variables are all in positive positions, and so can be modelled by covariant functors. Similarly, terms denote natural transformations. There is a translation from the covariant type system to system F which preserves nontrivial reductions. It follows that covariant reduction is strongly normalising and confluent. This work suggests a new approach to the semantics of system F, and new ways of basing type systems on the categorical notions of functor and natural transformation. Keywords covariant types, polymorphism, parametricity, transformation types. 1 Introduction The pros and cons of typing programs are already well known. In brief, static typecheckin...
Equational Systems and Free Constructions (Extended Abstract)
"... Abstract. The purpose of this paper is threefold: to present a general abstract, yet practical, notion of equational system; to investigate and develop a theory of free constructions for such equational systems; and to illustrate the use of equational systems as needed in modern applications, specif ..."
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Abstract. The purpose of this paper is threefold: to present a general abstract, yet practical, notion of equational system; to investigate and develop a theory of free constructions for such equational systems; and to illustrate the use of equational systems as needed in modern applications, specifically to the theory of substitution in the presence of variable binding and to models of namepassing process calculi. 1
Calculating invariants as coreflexive bisimulations
, 2008
"... Invariants, bisimulations and assertions are the main ingredients of coalgebra theory applied to computer systems engineering. In this paper we reduce the first to a particular case of the second and show how both together pave the way to a theory of coalgebras which regards invariant predicates as ..."
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Invariants, bisimulations and assertions are the main ingredients of coalgebra theory applied to computer systems engineering. In this paper we reduce the first to a particular case of the second and show how both together pave the way to a theory of coalgebras which regards invariant predicates as types. An outcome of such a theory is a calculus of invariants ’ proof obligation discharge, a fragment of which is presented in the paper. The approach has two main ingredients: one is that of adopting relations as “first class citizens” in a pointfree reasoning style; the other lies on a synergy found between a relational construct, Reynolds ’ relation on functions involved in the abstraction theorem on parametric polymorphism and the coalgebraic account of bisimulation and invariants. In this process, we provide an elegant proof of the equivalence between two different definitions of bisimulation found in coalgebra literature (due to B. Jacobs and Aczel & Mendler, respectively) and their instantiation to the classical ParkMilner definition popular in process algebra.
COMMUTATIVE MONADS AS A THEORY OF DISTRIBUTIONS
"... Abstract. It is shown how the theory of commutative monads provides an axiomatic framework for several aspects of distribution theory in a broad sense, including probability distributions, physical extensive quantities, and Schwartz distributions of compact support. Among the particular aspects cons ..."
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Abstract. It is shown how the theory of commutative monads provides an axiomatic framework for several aspects of distribution theory in a broad sense, including probability distributions, physical extensive quantities, and Schwartz distributions of compact support. Among the particular aspects considered here are the notions of convolution, density, expectation, and conditional probability.
A.,Some problems and results in synthetic functional analysis
 in Category Theoretic Methods in Geometry, Proceedings Aarhus 1983, Aarhus Various Publication Series No
, 1983
"... This somewhat tentative note aims at making a status about “functional analysis ” in certain ringed toposes E, R, in particular, duality theory for Rmodules in E. This is an area where knowledge still seems fragmentary, but where some of the known facts indicate the importance of such theory. The f ..."
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This somewhat tentative note aims at making a status about “functional analysis ” in certain ringed toposes E, R, in particular, duality theory for Rmodules in E. This is an area where knowledge still seems fragmentary, but where some of the known facts indicate the importance of such theory. The first §, however, does not deal with specific toposes, but is of very general nature, making contact with the general theory of monads on closed categories, [3][6]. Most of the results and problems in this note were presented at the workshop. Closely related viewpoints were contained in Lawvere’s workshopcontribution on Intensive and Extensive Quantities, which have also influenced the presentation given in the following pages. 1 Restricted double dualization monads Let R be a ring in a topos E and let RMod denote the Ecategory of Rmodules (left Rmodules, say). The Evalued homfunctor for it is denoted Hom R(−, −). If X ∈ E and V ∈ RMod, there is a natural structure of Rmodule on V X = ΠXV, and for any U ∈ RMod Hom R(U, V X) ∼ = (Hom R(U, V)) X, naturally in X, U, and V. This expresses that V X is a cotensor of V with X (cf. e.g. [2]). Equivalently, it expresses that, for fixed V, we have a Estrong left adjoint V (−) : E → (RMod) op to the functor Hom R (−, V). Thus we