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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
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
"... In previous work I introduced a generalised notion of coalgebra that is capable of modelling binary methods as they occur in objectoriented programming. An important problem with this generalisation is that bisimulations are not closed under union and that a greatest bisimulation does not exists in ..."
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In previous work I introduced a generalised notion of coalgebra that is capable of modelling binary methods as they occur in objectoriented programming. An important problem with this generalisation is that bisimulations are not closed under union and that a greatest bisimulation does not exists in general. There are two possible approaches to improve this situation: First, to strengthen the definition of bisimulation, and second, to place constraints on the coalgebras (i.e., on the behaviour of the binary methods). In this paper I combine both approaches to show that (under reasonable assumptions) the greatest bisimulation does exist for all coalgebras of extended polynomial functors.
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
2.2 Relative Interpretations....................... 5
, 2006
"... In this paper, we provide basic facts about the category INT of interpretations. E.g., we give a characterization of its epimorphisms and we show that, modulo a small detail, its opposite category is regular and ..."
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In this paper, we provide basic facts about the category INT of interpretations. E.g., we give a characterization of its epimorphisms and we show that, modulo a small detail, its opposite category is regular and
A Flexible Semantic Framework for Effects
"... Effects are a powerful and convenient component of programming. They enable programmers to interact with the user, take advantage of efficient stateful memory, throw exceptions, and nondeterministically execute programs in parallel. However, they also complicate every aspect of reasoning about a pro ..."
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Effects are a powerful and convenient component of programming. They enable programmers to interact with the user, take advantage of efficient stateful memory, throw exceptions, and nondeterministically execute programs in parallel. However, they also complicate every aspect of reasoning about a program or language, and as a result it is crucially important to have a good understanding of what effects are and how they work. In this paper we present a new framework for formalizing the semantics of effects that is more general and thorough than previous techniques while clarifying many of the important concepts. By returning to the categorytheoretic roots of monads, our framework is rich enough to describe the semantics of effects for a large class of languages including common imperative and functional languages. It is also capable of capturing more expressive, precise, and practical effect systems than previous approaches. Finally, our framework enables one to reason about effects in a languageindependent manner, and so can be applied to many stages of language design and implementation in order to create more broadly applicable tools for programming languages. 1.