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38
On the Foundations of Final Coalgebra Semantics: nonwellfounded sets, partial orders, metric spaces
, 1998
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Generalized Metrics and Uniquely Determined Logic Programs
 Theoretical Computer Science
"... The introduction of negation into logic programming brings the benefit of enhanced syntax and expressibility, but creates some semantical problems. Specifically, certain operators which are monotonic in the absence of negation become nonmonotonic when it is introduced, with the result that standard ..."
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Cited by 27 (16 self)
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The introduction of negation into logic programming brings the benefit of enhanced syntax and expressibility, but creates some semantical problems. Specifically, certain operators which are monotonic in the absence of negation become nonmonotonic when it is introduced, with the result that standard approaches to denotational semantics then become inapplicable. In this paper, we show how generalized metric spaces can be used to obtain fixedpoint semantics for several classes of programs relative to the supported model semantics, and investigate relationships between the underlying spaces we employ. Our methods allow the analysis of classes of programs which include the acyclic, locally hierarchical, and acceptable programs, amongst others, and draw on fixedpoint theorems which apply to generalized ultrametric spaces and to partial metric spaces.
Generalized Metric Spaces: Completion, Topology, and Powerdomains via the Yoneda Embedding
, 1996
"... Generalized metric spaces are a common generalization of preorders and ordinary metric spaces (Lawvere 1973). Combining Lawvere's (1973) enrichedcategorical and Smyth' (1988, 1991) topological view on generalized metric spaces, it is shown how to construct 1. completion, 2. topology, and ..."
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Cited by 23 (3 self)
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Generalized metric spaces are a common generalization of preorders and ordinary metric spaces (Lawvere 1973). Combining Lawvere's (1973) enrichedcategorical and Smyth' (1988, 1991) topological view on generalized metric spaces, it is shown how to construct 1. completion, 2. topology, and 3. powerdomains for generalized metric spaces. Restricted to the special cases of preorders and ordinary metric spaces, these constructions yield, respectively: 1. chain completion and Cauchy completion; 2. the Alexandroff and the Scott topology, and the fflball topology; 3. lower, upper, and convex powerdomains, and the hyperspace of compact subsets. All constructions are formulated in terms of (a metric version of) the Yoneda (1954) embedding.
Realizability semantics of parametric polymorphism, general references, and recursive types
, 2010
"... Abstract. We present a realizability model for a callbyvalue, higherorder programming language with parametric polymorphism, general firstclass references, and recursive types. The main novelty is a relational interpretation of open types (as needed for parametricity reasoning) that include gener ..."
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Cited by 20 (13 self)
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Abstract. We present a realizability model for a callbyvalue, higherorder programming language with parametric polymorphism, general firstclass references, and recursive types. The main novelty is a relational interpretation of open types (as needed for parametricity reasoning) that include general reference types. The interpretation uses a new approach to modeling references. The universe of semantic types consists of worldindexed families of logical relations over a universal predomain. In order to model general reference types, worlds are finite maps from locations to semantic types: this introduces a circularity between semantic types and worlds that precludes a direct definition of either. Our solution is to solve a recursive equation in an appropriate category of metric spaces. In effect, types are interpreted using a Kripke logical relation over a recursively defined set of worlds. We illustrate how the model can be used to prove simple equivalences between different implementations of imperative abstract data types. 1
The FixedPoint Theorems of PriessCrampe and Ribenboim in Logic Programming
 Proceedings of the International Conference and Workshop on Valuation Theory, University of Saskatchewan in
, 1999
"... Sibylla PriessCrampe and Paulo Ribenboim recently established a general fixedpoint theorem for multivalued mappings defined on generalized ultrametric spaces, and introduced it to the area of logic programming semantics. We discuss, in this context, the applications which have been made so far of ..."
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Cited by 17 (9 self)
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Sibylla PriessCrampe and Paulo Ribenboim recently established a general fixedpoint theorem for multivalued mappings defined on generalized ultrametric spaces, and introduced it to the area of logic programming semantics. We discuss, in this context, the applications which have been made so far of this theorem and of its corollaries. In particular, we will relate these results to ScottErshov domains, familiar in programming language semantics, and to the generalized metrics of Khamsi, Kreinovich and Misane which have been applied, by these latter authors, to logic programming. Amongst other things, we will also show that a unified treatment of the fixedpoint theory of wide classes of programs can be given by means of the theorems of PriessCrampe and Ribenboim.
A Semantic Foundation for Hidden State
"... We present the first complete soundness proof of the antiframe rule, a recently proposed proof rule for capturing information hiding in the presence of higherorder store. Our proof involves solving a nontrivial recursive domain equation. It helps identify some of the key ingredients for soundness, ..."
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Cited by 16 (10 self)
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We present the first complete soundness proof of the antiframe rule, a recently proposed proof rule for capturing information hiding in the presence of higherorder store. Our proof involves solving a nontrivial recursive domain equation. It helps identify some of the key ingredients for soundness, and thereby suggests how one might hope to relax some of the restrictions imposed by the rule.
Generalized Ultrametric Spaces: Completion, Topology, and Powerdomains via the Yoneda Embedding
, 1995
"... Generalized ultrametric spaces are a common generalization of preorders and ordinary ultrametric spaces (Lawvere 1973, Rutten 1995). Combining Lawvere's (1973) enrichedcategorical and Smyth' (1987, 1991) topological view on generalized (ultra)metric spaces, it is shown how to construct 1. ..."
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Cited by 15 (5 self)
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Generalized ultrametric spaces are a common generalization of preorders and ordinary ultrametric spaces (Lawvere 1973, Rutten 1995). Combining Lawvere's (1973) enrichedcategorical and Smyth' (1987, 1991) topological view on generalized (ultra)metric spaces, it is shown how to construct 1. completion, 2. topology, and 3. powerdomains for generalized ultrametric spaces. Restricted to the special cases of preorders and ordinary ultrametric spaces, these constructions yield, respectively: 1. chain completion and Cauchy completion; 2. the Alexandroff and the Scott topology, and the fflball topology; 3. lower, upper, and convex powerdomains, and the powerdomain of compact subsets. Interestingly, all constructions are formulated in terms of (an ultrametric version of) the Yoneda (1954) lemma.
A Semantic Model for Graphical User Interfaces
, 2011
"... We give a denotational model for graphical user interface (GUI) programming in terms of the cartesian closed category of ultrametric spaces. The metric structure allows us to capture natural restrictions on reactive systems, such as causality, while still allowing recursively defined values. We capt ..."
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Cited by 8 (1 self)
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We give a denotational model for graphical user interface (GUI) programming in terms of the cartesian closed category of ultrametric spaces. The metric structure allows us to capture natural restrictions on reactive systems, such as causality, while still allowing recursively defined values. We capture the arbitrariness of user input (e.g., a user gets to decide the stream of clicks she sends to a program) by making use of the fact that the closed subsets of a metric space themselves form a metric space under the Hausdorff metric, allowing us to interpret nondeterminism with a “powerspace ” monad on ultrametric spaces. The powerspace monad is commutative, and hence gives rise to a model of linear logic. We exploit this fact by constructing a mixed linear/nonlinear domainspecific language for GUI programming. The linear sublanguage naturally captures the usage constraints on the various linear objects in GUIs, such as the elements of a DOM or scene graph. We have implemented this DSL as an extension to OCaml, and give examples demonstrating that programs in this style can be short and readable.
Some Issues Concerning Fixed Points in Computational Logic: QuasiMetrics, Multivalued Mappings and the KnasterTarski Theorem
, 2000
"... Many questions concerning the semantics of disjunctive databases and of logic programming systems depend on the fixed points of various multivalued mappings and operators determined by the database or program. We discuss known versions, for multivalued mappings, of the KnasterTarski theorem and of ..."
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Cited by 8 (7 self)
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Many questions concerning the semantics of disjunctive databases and of logic programming systems depend on the fixed points of various multivalued mappings and operators determined by the database or program. We discuss known versions, for multivalued mappings, of the KnasterTarski theorem and of the Banach contraction mapping theorem, and formulate a version of the classical fixedpoint theorem (sometimes attributed to Kleene) which is new. All these results have applications to the semantics of disjunctive logic programs, and we will describe a class of programs to which the new theorem can be applied. We also show that a unification of the latter two theorems is possible, using quasimetrics, which parallels the wellknown unification of Rutten and Smyth in the case of conventional programming language semantics.