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43
Coalgebras for Binary Methods
, 2000
"... Coalgebras for endofunctors C > C can be used to model classes of object oriented languages. However, binary methods do not fit directly into this approach. This paper proposes an extension of the coalgebraic framework, namely the use of extended polynomial functors C^op x C > C . This ext ..."
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Cited by 9 (2 self)
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Coalgebras for endofunctors C > C can be used to model classes of object oriented languages. However, binary methods do not fit directly into this approach. This paper proposes an extension of the coalgebraic framework, namely the use of extended polynomial functors C^op x C > C . This extension allows the incorporation of binary methods into coalgebraic class specifications. The paper also discusses how to define bisimulation for coalgebras of extended polynomial functors and proves some standard results.
Representing Nested Inductive Types Using Wtypes
"... We show that strictly positive inductive types, constructed from polynomial functors, constant exponentiation and arbitrarily nested inductive ..."
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We show that strictly positive inductive types, constructed from polynomial functors, constant exponentiation and arbitrarily nested inductive
Presheaf Models for CCSlike Languages
 THEORETICAL COMPUTER SCIENCE
, 1999
"... The aim of this paper is to harness the mathematical machinery around presheaves for the purposes of process calculi. Joyal, Nielsen and Winskel proposed a general definition of bisimulation from open maps. Here we show that openmap bisimulations within a range of presheaf models are congruences ..."
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Cited by 8 (2 self)
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The aim of this paper is to harness the mathematical machinery around presheaves for the purposes of process calculi. Joyal, Nielsen and Winskel proposed a general definition of bisimulation from open maps. Here we show that openmap bisimulations within a range of presheaf models are congruences for a general process language, in which CCS and related languages are easily encoded. The results are then transferred to traditional models for processes. By first establishing the congruence results for presheaf models, abstract, general proofs of congruence properties can be provided and the awkwardness caused through traditional models not always possessing the cartesian liftings, used in the breakdown of process operations, are sidestepped. The abstract results are applied to show that hereditary historypreserving bisimulation is a congruence for CCSlike languages to which is added a refinement operator on event structures as proposed by van Glabbeek and Goltz.
Categories of Theories and Interpretations
 Logic in Tehran. Proceedings of the workshop and conference on Logic, Algebra and Arithmetic, held October 18–22
"... In this paper we study categories of theories and interpretations. In these categories, notions of sameness of theories, like synonymy, biinterpretability and mutual interpretability, take the form of isomorphism. We study the usual notions like monomorphism and product in the various theories. We ..."
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In this paper we study categories of theories and interpretations. In these categories, notions of sameness of theories, like synonymy, biinterpretability and mutual interpretability, take the form of isomorphism. We study the usual notions like monomorphism and product in the various theories. We provide some examples to separate notions across categories. In contrast, we show that, in some cases, notions in different categories do coincide. E.g., we can, under suchandsuch conditions, infer synonymity of two theories from their being equivalent in the sense of a coarser equivalence relation. We illustrate that the categories offer an appropriate framework for conceptual analysis of notions. For example, we provide a ‘coordinate free ’ explication of the notion of axiom scheme. Also we give a closer analysis of the objectlanguage / metalanguage distinction. Our basic category can be enriched with a form of 2structure. We use
When is a type refinement an inductive type
 In FOSSACS, volume 6604 of Lecture Notes in Computer Science
, 2011
"... Abstract. Dependently typed programming languages allow sophisticated properties of data to be expressed within the type system. Of particular use in dependently typed programming are indexed types that refine data by computationally useful information. For example, the Nindexed type of vectors ref ..."
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Abstract. Dependently typed programming languages allow sophisticated properties of data to be expressed within the type system. Of particular use in dependently typed programming are indexed types that refine data by computationally useful information. For example, the Nindexed type of vectors refines lists by their lengths. Other data types may be refined in similar ways, but programmers must produce purposespecific refinements on an ad hoc basis, developers must anticipate which refinements to include in libraries, and implementations often store redundant information about data and their refinements. This paper shows how to generically derive inductive characterisations of refinements of inductive types, and argues that these characterisations can alleviate some of the aforementioned difficulties associated with ad hoc refinements. These characterisations also ensure that standard techniques for programming with and reasoning about inductive types are applicable to refinements, and that refinements can themselves be further refined. 1
Greatest Bisimulations for Binary Methods
 IN PROCEEDINGS OF CMCS’02, VOLUME 65(1) OF ENTCS
, 2002
"... ..."
Monads and eects
 Lecture Notes in Computer Science
, 2002
"... Abstract. A tension in language design has been between simple semantics on the one hand, and rich possibilities for sideeects, exception handling and so on on the other. The introduction of monads has made a large step towards reconciling these alternatives. First proposed by Moggi as a way of st ..."
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Abstract. A tension in language design has been between simple semantics on the one hand, and rich possibilities for sideeects, exception handling and so on on the other. The introduction of monads has made a large step towards reconciling these alternatives. First proposed by Moggi as a way of structuring semantic descriptions, they were adopted by Wadler to structure Haskell programs, and now oer a general technique for delimiting the scope of eects, thus reconciling referential transparency and imperative operations within one programming language. Monads have been used to solve longstanding problems such as adding pointers and assignment, interlanguage working, and exception handling to Haskell, without compromising its purely functional semantics. The course will introduce monads, eects and related notions, and exemplify their applications in programming (Haskell) and in compilation (MLj). The course will present typed metalanguages for monads and related categorical notions, and describe how they can be further rened by introducing eects.
Guarded dependent type theory with coinductive types, 2015, submitted for publication. Available online at http://userscs.au.dk/birke/papers/gdttconf.pdf
"... We present guarded dependent type theory, gDTT, an extensional dependent type theory with a ‘later ’ modality and clock quantifiers for programming and proving with guarded recursive and coinductive types. The later modality is used to ensure the productivity of recursive definitions in a modular, ..."
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We present guarded dependent type theory, gDTT, an extensional dependent type theory with a ‘later ’ modality and clock quantifiers for programming and proving with guarded recursive and coinductive types. The later modality is used to ensure the productivity of recursive definitions in a modular, type based, way. Clock quantifiers are used for controlled elimination of the later modality and for encoding coinductive types using guarded recursive types. We demonstrate the expressiveness of gDTT via a range of examples. Key to the development of gDTT are novel type and term formers involving what we call ‘delayed substitutions’. These generalise the applicative functor rules for the later modality considered in earlier work, and are crucial for programming and proving with dependent types. We show soundness of the type theory using a denotational model.
Linear realizability
, 2007
"... Abstract. We define a notion of relational linear combinatory algebra (rLCA) which is a generalization of a linear combinatory algebra defined by Abramsky, Haghverdi and Scott. We also define a category of assemblies as well as a category of modest sets which are realized by rLCA. This is a linear s ..."
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Abstract. We define a notion of relational linear combinatory algebra (rLCA) which is a generalization of a linear combinatory algebra defined by Abramsky, Haghverdi and Scott. We also define a category of assemblies as well as a category of modest sets which are realized by rLCA. This is a linear style of realizability in a way that duplicating and discarding of realizers is allowed in a controlled way. Both categories form linearnonlinear models and their coKleisli categories have a natural number object. We construct some examples of rLCA’s which have some relations to well known PCA’s. 1