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Universal coalgebra: a theory of systems
, 2000
"... In the semantics of programming, nite data types such as finite lists, have traditionally been modelled by initial algebras. Later final coalgebras were used in order to deal with in finite data types. Coalgebras, which are the dual of algebras, turned out to be suited, moreover, as models for certa ..."
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Cited by 408 (42 self)
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In the semantics of programming, nite data types such as finite lists, have traditionally been modelled by initial algebras. Later final coalgebras were used in order to deal with in finite data types. Coalgebras, which are the dual of algebras, turned out to be suited, moreover, as models for certain types of automata and more generally, for (transition and dynamical) systems. An important property of initial algebras is that they satisfy the familiar principle of induction. Such a principle was missing for coalgebras until the work of Aczel (NonWellFounded sets, CSLI Leethre Notes, Vol. 14, center for the study of Languages and information, Stanford, 1988) on a theory of nonwellfounded sets, in which he introduced a proof principle nowadays called coinduction. It was formulated in terms of bisimulation, a notion originally stemming from the world of concurrent programming languages. Using the notion of coalgebra homomorphism, the definition of bisimulation on coalgebras can be shown to be formally dual to that of congruence on algebras. Thus, the three basic notions of universal algebra: algebra, homomorphism of algebras, and congruence, turn out to correspond to coalgebra, homomorphism of coalgebras, and bisimulation, respectively. In this paper, the latter are taken
A logic for metric and topology
 Journal of Symbolic Logic
, 2005
"... Abstract. We propose a logic for reasoning about metric spaces with the induced topologies. It combines the ‘qualitative ’ interior and closure operators with ‘quantitative’ operators ‘somewhere in the sphere of radius r, ’ including or excluding the boundary. We supply the logic with both the inten ..."
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Cited by 15 (12 self)
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Abstract. We propose a logic for reasoning about metric spaces with the induced topologies. It combines the ‘qualitative ’ interior and closure operators with ‘quantitative’ operators ‘somewhere in the sphere of radius r, ’ including or excluding the boundary. We supply the logic with both the intended metric space semantics and a natural relational semantics, and show that the latter (i) provides finite partial representations of (in general) infinite metric models and (ii) reduces the standard ‘εdefinitions ’ of closure and interior to simple constraints on relations. These features of the relational semantics suggest a finite axiomatisation of the logic and provide means to prove its EXPTIMEcompleteness (even if the rational numerical parameters are coded in binary). An extension with metric variables satisfying linear rational (in)equalities is proved to be decidable as well. Our logic can be regarded as a ‘wellbehaved ’ common denominator of logical systems constructed in temporal, spatial, and similaritybased quantitative and qualitative representation and reasoning. Interpreted on the real line (with its Euclidean metric), it is a natural fragment of decidable temporal logics for specification and verification of realtime systems. On the real plane, it is closely related to quantitative and qualitative formalisms for spatial
Interpolation for first order S5
 Journal of Symbolic Logic
"... web page: comet.lehman.cuny.edu/fitting ..."
AGM Belief Revision in Monotone Modal Logics
"... Classical modal logics, based on the neighborhood semantics of Scott and Montague, provide a generalization of the familiar normal systems based on Kripke semantics. This paper defines AGM revision operators on several firstorder monotonic modal correspondents, where each firstorder correspondence ..."
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Cited by 2 (1 self)
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Classical modal logics, based on the neighborhood semantics of Scott and Montague, provide a generalization of the familiar normal systems based on Kripke semantics. This paper defines AGM revision operators on several firstorder monotonic modal correspondents, where each firstorder correspondence language is defined by Marc Pauly’s version of the van Benthem characterization theorem for monotone modal logic. A revision problem expressed in a monotone modal system is translated into firstorder logic, the revision is performed, and the new belief set is translated back to the original modal system. An example is provided for the logic of Risky Knowledge that uses modal AGM contraction to construct counterfactual evidence sets in order to investigate robustness of a probability assignment given some evidence set. A proof of correctness is given. 1
Logic in Australasia
, 2008
"... Ever since the 1960s, philosophical logic has played an important part in the shaping of philosophy in Australasia, and Australasian work in philosophical logic has played its part in research in the area. In this chapter I will introduce and assess this in@luence, concentrating on two major themes ..."
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Ever since the 1960s, philosophical logic has played an important part in the shaping of philosophy in Australasia, and Australasian work in philosophical logic has played its part in research in the area. In this chapter I will introduce and assess this in@luence, concentrating on two major themes in Australasian philosophical logic, modal logic and paraconsistent logic. The discipline of logic To set the scene, I must say a little about logic as a discipline, for it does not @ind itself wholly inside the academic discipline of philosophy. Just as logic has a long history, reaching back to the work of Aristotle, it also now has a very wide intellectual geography. The discipline @inds a home in Philosophy, and philosophical logic is the theme of this chapter. However, logicians @ind a home also among mathematicians, computer scientists, and also linguists, cognitive scientists and engineers. As Robert K. Meyer (of the Australian National University) has said, logic is the Poland of the sciences. It has very many large neighbours, who sometimes harbour benign interest, and sometimes wish to colonise the territory for its own.