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Solving Recursive Domain Equations with Enriched Categories
, 1994
"... Both preorders and metric spaces have been used at various times as a foundation for the solution of recursive domain equations in the area of denotational semantics. In both cases the central theorem states that a `converging' sequence of `complete' domains/spaces with `continuous' retraction pair ..."
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Both preorders and metric spaces have been used at various times as a foundation for the solution of recursive domain equations in the area of denotational semantics. In both cases the central theorem states that a `converging' sequence of `complete' domains/spaces with `continuous' retraction pairs between them has a limit in the category of complete domains/spaces with retraction pairs as morphisms. The preorder version was discovered first by Scott in 1969, and is referred to as Scott's inverse limit theorem. The metric version was mainly developed by de Bakker and Zucker and refined and generalized by America and Rutten. The theorem in both its versions provides the main tool for solving recursive domain equations. The proofs of the two versions of the theorem look astonishingly similar, but until now the preconditions for the preorder and the metric versions have seemed to be fundamentally different. In this thesis we establish a more general theory of domains based on the noti...
First Order Linear Logic in Symmetric Monoidal Closed Categories
, 1991
"... There has recently been considerable interest in the development of `logical frameworks ' which can represent many of the logics arising in computer science in a uniform way. Within the Edinburgh LF project, this concept is split into two components; the first being a general proof theoretic encodin ..."
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There has recently been considerable interest in the development of `logical frameworks ' which can represent many of the logics arising in computer science in a uniform way. Within the Edinburgh LF project, this concept is split into two components; the first being a general proof theoretic encoding of logics, and the second a uniform treatment of their model theory. This thesis forms a case study for the work on model theory. The models of many first and higher order logics can be represented as fibred or indexed categories with certain extra structure, and this has been suggested as a general paradigm. The aim of the thesis is to test the strength and flexibility of this paradigm by studying the specific case of Girard's linear logic. It should be noted that the exact form of this logic in the first order case is not entirely certain, and the system treated here is significantly different to that considered by Girard.
Sheaves over Right Sided Idempotent Quantales
 Journal of the IGPL
, 1998
"... We present a discussion of sheaves and presheaves over a right sided idempotent quantale in a fashion that is similar to the way that these objects are conceived over complete Heyting algebras by Fourman and Scott in [5]. Keywords : quantales, Qsets, sheaf, characteristic maps, firstorder quantifi ..."
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We present a discussion of sheaves and presheaves over a right sided idempotent quantale in a fashion that is similar to the way that these objects are conceived over complete Heyting algebras by Fourman and Scott in [5]. Keywords : quantales, Qsets, sheaf, characteristic maps, firstorder quantifiers The idea of a quantale originated with C.J.Mulvey ([6]) as an attempt to code a lattice theoretic construct that might be appropriate to obtain, for non commutative C # algebras, an analogue of the classical duality between commutative C # algebras and compact Hausdor# spaces. We cite [8] as a general reference on the circle of ideas connected with quantales. A number of authors have studied the possibility of extending the notions of sheaf and presheaf over a complete Heyting algebra (cHa) or frame to this new context ([7], [2], [1], [4]). It should be mentioned that an exposition of the content of [2] can be found in [8]. We consider [5] as a basic reference for presheaves and sheaves...