A Cook's Tour of the Finitary Non−Well−Founded Sets

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Coalgebras of topological types

In This work, we focus on developing the basic theory of coalgebras over the category Top (the category of topological spaces with continuous maps). We argue that, besides Set, the category Top is an interesting base category for coalgebras. We study some endofunctors on Top, in particular, Vietoris functor and P-Vietoris Functor (where P is a set of propositional letters) that due to Hofmann et. al. [42] can be considered as the topological versions of the powerset functor and P-Kripke functor, respectively. We define the notion of compact Kripke structures and we prove that Kripke homomorphisms preserve compactness. Our definition of "compact Kripke structure" coincides with the notion of "modally saturated structures" introduced in Fine [27]. We prove that the class of compact Kripke structures has Hennessy-Milner property. As a consequence we show that in this class of Kripke structures, bihavioral equivalence, modal equivalence and Kripke bisimilarity all coincide.Furthermore, we generalize the notion of descriptive structures defined in Venema et. al. [11] by introducing a notion Vietoris models. We identify Vietoris models as coalgebras for the P-Vietoris functor on the category Top. One can see that each compact Kripke model can be modified to a Vietoris model. This yields an adjunction between the category of Vietoris structures (VS) and the category of compact Kripke structurs (CKS). Moreover, we will prove that the category of Vietoris models has a terminal object. We study the concept of a Vietoris bisimulation between Vietoris models, and we will prove that the closure of a Kripke bisimulation between underlying Kripke models of two Vietoris models is a Vietoris bisimulation. In the end, it will be shown that in the class of Vietoris models, Vietoris bisimilarity, bihavioral equivalence, modal equivalence, all coincide

Proceedings of Monterey Workshop 2001 Engineering Automation for Sofware Intensive System Integration

The 2001 Monterey Workshop on Engineering Automation for Software Intensive System Integration was sponsored by the Office of Naval Research, Air Force Office of Scientific Research, Army Research Office and the Defense Advance Research Projects Agency. It is our pleasure to thank the workshop advisory and sponsors for their vision of a principled engineering solution for software and for their many-year tireless effort in supporting a series of workshops to bring everyone together.This workshop is the 8 in a series of International workshops. The workshop was held in Monterey Beach Hotel, Monterey, California during June 18-22, 2001. The general theme of the workshop has been to present and discuss research works that aims at increasing the practical impact of formal methods for software and systems engineering. The particular focus of this workshop was "Engineering Automation for Software Intensive System Integration". Previous workshops have been focused on issues including, "Real-time & Concurrent Systems", "Software Merging and Slicing", "Software Evolution", "Software Architecture", "Requirements Targeting Software" and "Modeling Software System Structures in a fastly moving scenario".Office of Naval ResearchAir Force Office of Scientific Research Army Research OfficeDefense Advanced Research Projects AgencyApproved for public release, distribution unlimite

A category of compositional domain-models for separable Stone spaces

In this paper we introduce SFPM, a category of SFP domains which provides very satisfactory domain-models, i.e. "partializations", of separable Stone spaces (2-Stone spaces). More specifically, SFPM is a subcategory of SFPep, closed under direct limits as well as many constructors, such as lifting, sum, product and Plotkin powerdomain (with the notable exception of the function space constructor). SFPM is "structurally well behaved", in the sense that the functor MAX, which associates to each object of SFPM the Stone space of its maximal elements, is compositional with respect to the constructors above, and \u3c9-continuous. A correspondence can be established between these constructors over SFPM and appropriate constructors on Stone spaces, whereby SFPM domain-models of Stone spaces defined as solutions of a vast class of recursive equations in SFPM, can be obtained simply by solving the corresponding equations in SFPM. Moreover any continuous function between two 2-Stone spaces can be extended to a continuous function between any two SFPM domain-models of the original spaces. The category SFPM does not include all the SFP's with a 2-Stone space of maximal elements (CSFP's). We show that the CSFP's can be characterized precisely as suitable retracts of SFPM objects. Then the results proved for SFPM easily extends to the wider category having CSFP's as objects. Using SFPM we can provide a plethora of "partializations" of the space of finitary hypersets (the hyperuniverse N\u3c9 (Ann. New York Acad. Sci. 806 (1996) 140). These includes the classical ones proposed in Abramsky (A Cook's tour of the finitary non-well-founded sets unpublished manuscript, 1988; Inform. Comput. 92(2) (1991) 161) and Mislove et al. (Inform. Comput. 93(1) (1991) 16), which are also shown to be non-isomorphic, thus providing a negative answer to a problem raised in Mislove et al