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On the Foundations of Final Semantics: NonStandard Sets, Metric Spaces, Partial Orders
 PROCEEDINGS OF THE REX WORKSHOP ON SEMANTICS: FOUNDATIONS AND APPLICATIONS, VOLUME 666 OF LECTURE NOTES IN COMPUTER SCIENCE
, 1998
"... Canonical solutions of domain equations are shown to be final coalgebras, not only in a category of nonstandard sets (as already known), but also in categories of metric spaces and partial orders. Coalgebras are simple categorical structures generalizing the notion of postfixed point. They are ..."
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Cited by 48 (10 self)
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Canonical solutions of domain equations are shown to be final coalgebras, not only in a category of nonstandard sets (as already known), but also in categories of metric spaces and partial orders. Coalgebras are simple categorical structures generalizing the notion of postfixed point. They are also used here for giving a new comprehensive presentation of the (still) nonstandard theory of nonwellfounded sets (as nonstandard sets are usually called). This paper is meant to provide a basis to a more general project aiming at a full exploitation of the finality of the domains in the semantics of programming languages  concurrent ones among them. Such a final semantics enjoys uniformity and generality. For instance, semantic observational equivalences like bisimulation can be derived as instances of a single `coalgebraic' definition (introduced elsewhere), which is parametric of the functor appearing in the domain equation. Some properties of this general form of equivalence are also studied in this paper.
Fixpoint Semantics and Simulation
 Theor. Comp. Sci
, 2000
"... A general functorial framework for recursive definitions is presented in which simulation of a definition scheme by another one implies an ordering between the values defined by these schemes in an arbitrary model. Under mild conditions on the functor involved, the converse implication also holds: a ..."
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Cited by 5 (1 self)
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A general functorial framework for recursive definitions is presented in which simulation of a definition scheme by another one implies an ordering between the values defined by these schemes in an arbitrary model. Under mild conditions on the functor involved, the converse implication also holds: a model is constructed such that, if the values defined are ordered, there is a simulation between the definition schemes. The theory is illustrated by applications to contextfree grammars, recursive procedures in imperative languages, and simulation and bisimulation of processes. Keywords: simulation, fixpoint semantics, recursion, model 1 Introduction The ideas we present here, came up in the search for rules to prove refinement and equivalence between recursive procedures in imperative programming languages, cf. [Hes92]. In such a language, one may consider two procedures, p0 and p1, both defined by mutual recursion, and ask whether p0 refines (i.e., implements) p1. One of the ways to p...
Metric Predicate Transformers: Towards a Notion of Refinement for Concurrency
, 1994
"... For two parallel languages with recursion a compositional weakest precondition semantics is given using two new metric resumption domains. The underlying domains are characterized by domain equations involving functors that deliver `observable' and `safety' predicate transformers. Further ..."
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For two parallel languages with recursion a compositional weakest precondition semantics is given using two new metric resumption domains. The underlying domains are characterized by domain equations involving functors that deliver `observable' and `safety' predicate transformers. Further a refinement relation is defined for this domains and illustrated by rules dealing with concurrent composition. It turns out, by extending the classical duality of predicate vs. state transformers, that the weakest precondition semantics for the parallel languages is isomorphic to the standard metric state transformers semantics. Moreover, the proposed refinement relation on the predicate transformer domain will correspond to the familiar notion of simulation in the state transformer domain. Contents 1 Introduction 1 2 Mathematical Preliminaries 3 3 Four Languages with Recursion 5 4 Domains for Predicate Transformers 8 5 Predicate Transformer Semantics 14 6 Refinement, Simulation and State Transforme...