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Coalgebraic Logic
 Annals of Pure and Applied Logic
, 1999
"... We present a generalization of modal logic to logical systems which are interpreted on coalgebras of functors on sets. The leading idea is that infinitary modal logic contains characterizing formulas. That is, every modelworld pair is characterized up to bisimulation by an infinitary formula. The ..."
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We present a generalization of modal logic to logical systems which are interpreted on coalgebras of functors on sets. The leading idea is that infinitary modal logic contains characterizing formulas. That is, every modelworld pair is characterized up to bisimulation by an infinitary formula. The point of our generalization is to understand this on a deeper level. We do this by studying a frangment of infinitary modal logic which contains the characterizing formulas and is closed under infinitary conjunction and an operation called 4. This fragment generalizes to a wide range of coalgebraic logics. We then apply the characterization result to get representation theorems for final coalgebras in terms of maximal elements of ordered algebras. The end result is that the formulas of coalgebraic logics can be viewed as approximations to the elements of the final coalgebra. Keywords: infinitary modal logic, characterization theorem, functor on sets, coalgebra, greatest fixed point. 1 Intr...
A Cook’s tour of the finitary nonwellfounded sets
 Invited Lecture at BCTCS
, 1988
"... It is a great pleasure to contribute this paper to a birthday volume for Dov. Dov and I arrived at imperial College at around the same time, and soon he, Tom Maibaum and I were embarked on a joint project, the Handbook of Logic in Computer Science. We obtained a generous advance from Oxford Universi ..."
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It is a great pleasure to contribute this paper to a birthday volume for Dov. Dov and I arrived at imperial College at around the same time, and soon he, Tom Maibaum and I were embarked on a joint project, the Handbook of Logic in Computer Science. We obtained a generous advance from Oxford University Press, and a grant from the Alvey Programme, which allowed us to develop the Handbook in a rather unique, interactive way. We held regular meetings at Cosener’s House in Abingdon (a facility run by what was then the U.K. Science and Engineering Research Council), at which contributors would present their ideas and draft material for their chapters for discussion and criticism. Ideas for new chapters and the balance of the volumes were also discussed. Those were a remarkable series of meetings — a veritable education in themselves. I must confess that during this long process, I did occasionally wonder if it would ever terminate.... But the record shows that five handsome volumes were produced [6]. Moreover, I believe that the Handbook has proved to be a really valuable resource for students and researchers. It has been used as the basis for a number of summer schools. Many of the chapters have become standard references for their topics. In a field with rapidly changing fashions, most of the material has stood the test of time — thus
Bounded Hyperset Theory and Weblike Data Bases
 Computational Logic and Proof Theory, 5th Kurt Gödel Colloquium, KGC’97, Springer LNCS
, 1997
"... this paper rather abstract, \static" settheoretic view on the WorldWide Web (WWW) or, more generally, on Weblike Data Bases (WDB) and on the corresponding querying to WDB. Let us stress that it is not only about databases with an access via Web. The database itself should be organized in the same ..."
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Cited by 18 (5 self)
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this paper rather abstract, \static" settheoretic view on the WorldWide Web (WWW) or, more generally, on Weblike Data Bases (WDB) and on the corresponding querying to WDB. Let us stress that it is not only about databases with an access via Web. The database itself should be organized in the same way as Web. I.e. it must consist of hyperlinked pages distributed among the computers participating either in global network like Internet or in some local, isolated from the outside world specic network based essentially on the same principles, except globality, and called also Intranet [15].
On the Foundations of Corecursion
 Logic Journal of the IGPL
, 1997
"... We consider foundational questions related to the definition of functions by corecursion. This method is especially suited to functions into the greatest fixed point of some monotone operator, and it is most applicable in the context of nonwellfounded sets. We review the work on the Special Final C ..."
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We consider foundational questions related to the definition of functions by corecursion. This method is especially suited to functions into the greatest fixed point of some monotone operator, and it is most applicable in the context of nonwellfounded sets. We review the work on the Special Final Coalgebra Theorem of Aczel [1] and the Corecursion Theorem of Barwise and Moss [4]. We offer a condition weaker than Aczel's condition of uniformity on maps, and then we prove a result relating the operators satisfying the new condition to the smooth operators of [4]. Keywords: corecursion, coalgebra, operator on sets 1 Introduction By a stream of natural numbers we mean a pair hn; si where n 2 N and s is again a stream of natural numbers. Let f : N ! N . Consider the following function which purports to define a function from N into the streams: iter f (n) = hn; iter f f(n)i (1.1) For each n, iter f (n) is a stream, so iter f itself is a function from numbers to streams. This is an examp...
Linear ordering on graphs, antifounded sets and polynomial time computability
 THEORETICAL COMPUTER SCIENCE
, 1999
"... It is proved definability in FO+IFP of a global linear ordering on vertices of strongly extensional (SE) finitelybranching graphs. In the case of finite SE graphs this also holds for FO+LFP. This gives capturing results for PTIME computability on the latter class of graphs by FO+LFP and FO+IFP, and ..."
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Cited by 7 (2 self)
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It is proved definability in FO+IFP of a global linear ordering on vertices of strongly extensional (SE) finitelybranching graphs. In the case of finite SE graphs this also holds for FO+LFP. This gives capturing results for PTIME computability on the latter class of graphs by FO+LFP and FO+IFP, and also on the corresponding antifounded universe HFA of hereditarilyfinite sets by a language \Delta of a bounded set theory BSTA. Oracle PTIME computability over HFA is also captured by an appropriate extension of the language \Delta by predicate variables and a bounded 2recursion schema. It is also characterized the type of corresponding linear ordering on the universe HFA and on its natural extension HFA 1 consisting of hereditarilyfinite antifounded sets with possibly infinite (unlike HFA) transitive closures.
Processes and Hyperuniverses
 Proceedings of the 19th Symposium on Mathematical Foundations of Computer Science 1994, volume 841 of LNCS
, 1994
"... . We show how to define domains of processes, which arise in the denotational semantics of concurrent languages, using hypersets, i.e. nonwellfounded sets. In particular we discuss how to solve recursive equations involving settheoretic operators within hyperuniverses with atoms. Hyperuniverses ar ..."
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. We show how to define domains of processes, which arise in the denotational semantics of concurrent languages, using hypersets, i.e. nonwellfounded sets. In particular we discuss how to solve recursive equations involving settheoretic operators within hyperuniverses with atoms. Hyperuniverses are transitive sets which carry a uniform topological structure and include as a clopen subset their exponential space (i.e. the set of their closed subsets) with the exponential uniformity. This approach allows to solve many recursive domain equations of processes which cannot be even expressed in standard ZermeloFraenkel Set Theory, e.g. when the functors involved have negative occurrences of the argument. Such equations arise in the semantics of concurrrent programs in connection with function spaces and higher order assignment. Finally, we briefly compare our results to those which make use of complete metric spaces, due to de Bakker, America and Rutten. Introduction In the Semantics of ...
Partializing Stone Spaces using SFP domains (Extended Abstract)
 CAAP ’97, volume 1158 of LNCS
, 1997
"... ) F. Alessi, P. Baldan, F. Honsell Dipartimento di Matematica ed Informatica via delle Scienze 208, 33100 Udine, Italy falessi, baldan, honsellg@dimi.uniud.it Abstract. In this paper we investigate the problem of "partializing" Stone spaces by "Sequence of Finite Posets" (SFP) domains. More specif ..."
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Cited by 4 (0 self)
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) F. Alessi, P. Baldan, F. Honsell Dipartimento di Matematica ed Informatica via delle Scienze 208, 33100 Udine, Italy falessi, baldan, honsellg@dimi.uniud.it Abstract. In this paper we investigate the problem of "partializing" Stone spaces by "Sequence of Finite Posets" (SFP) domains. More specifically, we introduce a suitable subcategory SFP m of SFP which is naturally related to the special category of Stone spaces 2Stone by the functor MAX, which associates to each object of SFP m the space of its maximal elements. The category SFP m is closed under limits as well as many domain constructors, such as lifting, sum, product and Plotkin powerdomain. The functor MAX preserves limits and commutes with these constructors. Thus, SFP domains which "partialize" solutions of a vast class of domain equations in 2Stone, can be obtained by solving the corresponding equations in SFP m . Furthermore, we compare two classical partializations of the space of Milner's Synchronization Tre...
Partializing Stone Spaces using SFP domains
 TAPSOFT’97 Conference Proceedings, volume 1214 of Lecture Notes in Computer Science
"... In this paper we investigate the problem of "partializing" Stone spaces by "Sequence of Finite Posets" (SFP) domains. More specifically, we introduce a suitable subcategory SFP m of SFP which is naturally related to the special category of Stone spaces 2Stone by the functor MAX, which associates ..."
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Cited by 3 (1 self)
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In this paper we investigate the problem of "partializing" Stone spaces by "Sequence of Finite Posets" (SFP) domains. More specifically, we introduce a suitable subcategory SFP m of SFP which is naturally related to the special category of Stone spaces 2Stone by the functor MAX, which associates to each object of SFP m the space of its maximal elements. The category SFP m is closed under limits as well as many domain constructors, such as lifting, sum, product and Plotkin powerdomain. The functor MAX preserves limits and commutes with these constructors. Thus, SFP domains which "partialize" solutions of a vast class of domain equations in 2Stone, can be obtained by solving the corresponding equations in SFP m . Furthermore, we compare two classical partializations of the space of Milner's Synchronization Trees using SFP domains (see [3], [15]). Using the notion of "rigid" embedding projection pair, we show that the two domains are not isomorphic, thus providing a negative a...