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Observational Logic, ConstructorBased Logic, and their Duality
, 2002
"... Observability and reachability are important concepts for formal software development. While observability concepts are used to specify the required observable behavior of a program or system, reachability concepts are used to describe the underlying data in terms of datatype constructors. In this p ..."
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Observability and reachability are important concepts for formal software development. While observability concepts are used to specify the required observable behavior of a program or system, reachability concepts are used to describe the underlying data in terms of datatype constructors. In this paper we first reconsider the observational logic institution which provides a logical framework for dealing with observability. Then we develop in a completely analogous way the constructorbased logic institution which formalizes a novel treatment of reachability. Both institutions are tailored to capture the semantically correct realizations of a specification from either the observational or the reachability point of view. We show that there is a methodological and even formal duality between both frameworks. In particular, we establish a correspondence between observer operations and datatype constructors, observational and constructorbased algebras, fully abstract and reachable algebras, and observational and inductive consequences of specifications. The formal duality between the observability and reachability concepts is established in a categorytheoretic setting.
Logics Admitting Final Semantics
 In Foundations of Software Science and Computation Structures, volume 2303 of LNCS
, 2002
"... A logic for coalgebras is said to admit final semantics iff up to some technical requirementsall definable classes contain a fully abstract final coalgebra. It is shown that a logic admits final semantics iff the formulas of the logic are preserved under coproducts (disjoint unions) and qu ..."
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A logic for coalgebras is said to admit final semantics iff up to some technical requirementsall definable classes contain a fully abstract final coalgebra. It is shown that a logic admits final semantics iff the formulas of the logic are preserved under coproducts (disjoint unions) and quotients (homomorphic images).
CMCS’01 Preliminary Version Modal Languages for Coalgebras in a Topological Setting
"... It is well know that the solution Z of a recursive domain equation, given by an endofunctor T, is the nal Tcoalgebra. This suggests a coalgebraic approach to obtain a logical representation of the observable properties of Z. The paper considers brations of frames and (modal) logics, arising through ..."
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It is well know that the solution Z of a recursive domain equation, given by an endofunctor T, is the nal Tcoalgebra. This suggests a coalgebraic approach to obtain a logical representation of the observable properties of Z. The paper considers brations of frames and (modal) logics, arising through a set of predicate liftings. We discuss conditions, which ensure expressiveness of the resulting language (denotations of formulas determine a base of the frame over the nal coalgebra). The framework is then instantiated with categories of domains, and we establish these conditions for a large class of locally continuous endofunctors. This can be seen as a rst step towards a nal perspective on Abramsky's domain theory in logical form. 1