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Equilogical Spaces
, 1998
"... It is well known that one can build models of full higherorder dependent type theory (also called the calculus of constructions) using partial equivalence relations (PERs) and assemblies over a partial combinatory algebra (PCA). But the idea of categories of PERs and ERs (total equivalence relation ..."
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Cited by 38 (12 self)
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It is well known that one can build models of full higherorder dependent type theory (also called the calculus of constructions) using partial equivalence relations (PERs) and assemblies over a partial combinatory algebra (PCA). But the idea of categories of PERs and ERs (total equivalence relations) can be applied to other structures as well. In particular, we can easily dene the category of ERs and equivalencepreserving continuous mappings over the standard category Top 0 of topological T 0 spaces; we call these spaces (a topological space together with an ER) equilogical spaces and the resulting category Equ. We show that this categoryin contradistinction to Top 0 is a cartesian closed category. The direct proof outlined here uses the equivalence of the category Equ to the category PEqu of PERs over algebraic lattices (a full subcategory of Top 0 that is well known to be cartesian closed from domain theory). In another paper with Carboni and Rosolini (cited herein) a more abstract categorical generalization shows why many such categories are cartesian closed. The category Equ obviously contains Top 0 as a full subcategory, and it naturally contains many other well known subcategories. In particular, we show why, as a consequence of work of Ershov, Berger, and others, the KleeneKreisel hierarchy of countable functionals of nite types can be naturally constructed in Equ from the natural numbers object N by repeated use in Equ of exponentiation and binary products. We also develop for Equ notions of modest sets (a category equivalent to Equ) and assemblies to explain why a model of dependent type theory is obtained. We make some comparisons of this model to other, known models. 1
The Sreplete construction
 In CTCS 55, pages 96  116. Springer Lecture Notes in Computer Science 953
, 1995
"... this paper: (internal version) if C 1 is a quasitopos, then S ..."
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Cited by 9 (2 self)
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this paper: (internal version) if C 1 is a quasitopos, then S
Semantic Integration of Heterogeneous Formal Specifications via Transformation Systems
, 2001
"... Among the approaches to rational software systems development the viewpoint model has become predominant. That means, models of different views or aspects of a system on different levels of abstraction are built and maintained to document the design decisions passed during all development stages. Th ..."
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Among the approaches to rational software systems development the viewpoint model has become predominant. That means, models of different views or aspects of a system on different levels of abstraction are built and maintained to document the design decisions passed during all development stages. The construction of such models should be supported by adequate modelling languages or specification formalisms that allow the representation of the concerned structures, functionality, and properties directly. In order to conceive all these models as a specification of a sole system a semantic integration is required that supports the statement of correspondences of elements of di erent models and the consistency checking of groups of models. The main problem in this task is the heterogeneity of the modelling paradigms and, correspondingly, of the specification languages. The approach to the integration of heterogeneous formal specifications presented in this report is based on a formal reference model that serves as common semantic domain for the different languages. In this it way supports the comparison of models independently of the chosen languages. In the reference model formal models of dynamic entities of arbitrary granularity are defined via their static structure and their dynamic behaviour. Then development relations and composition operations are introduced and general compositionality properties are shown. In particular, structural transparency is supported, i.e., structured systems of entities can always be considered as single dynamic entities in turn. Furthermore it is shown that compatible local development steps of the same type always induce a global development step of this type containing the local ones. The applicability of the reference model is shown by the int...
A Domaintheoretic Semantics of Lax Generic Functions
"... The semantic structure of a calculus m is studied. m is a polymorphic calculus dened over a hierarchical type structure, and a function in this calculus, called a generic function, can be composed from more than one lambda expression and the ways it behaves on each type are weakly related in that it ..."
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The semantic structure of a calculus m is studied. m is a polymorphic calculus dened over a hierarchical type structure, and a function in this calculus, called a generic function, can be composed from more than one lambda expression and the ways it behaves on each type are weakly related in that it lax commutes with coercion functions. Since laxness is intermediate between adhocness and coherentness, m has syntactic properties lying between those of calculi with adhoc generic functions and coherent generic functions studied in [Tsu95]. That is, though m allows self application and thus is not normalizing, it does not have an unsolvable term. For this reason, all the semantic domains are connected by innitely many mutually recursive equations and, at the same time, they do not have the least elements. We solve them by considering opbrations and expressing the equations as one recursive equation about opbrations. We also show the adequacy theorem for m following the construct...