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12
ManySorted Coalgebraic Modal Logic: a Modeltheoretic Study
 Theoretical Informatics and Applications
, 2001
"... This paper gives a semantical underpinning for a manysorted modal logic associated with certain dynamical systems, like transition systems, automata or classes in objectoriented languages. These systems will be described as coalgebras of socalled polynomial functors, built up from constants an ..."
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Cited by 53 (3 self)
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This paper gives a semantical underpinning for a manysorted modal logic associated with certain dynamical systems, like transition systems, automata or classes in objectoriented languages. These systems will be described as coalgebras of socalled polynomial functors, built up from constants and identities, using products, coproducts and powersets. The semantical account involves Boolean algebras with operators indexed by polynomial functors, called MBAOs, for Manysorted Boolean Algebras with Operators, combining standard (categorical) models of modal logic and of manysorted predicate logic.
The Temporal Logic of Coalgebras via Galois Algebras
, 1999
"... This paper introduces a temporal logic for coalgebras. Nexttime and lasttime operators are dened for a coalgebra, acting on predicates on the state space. They give rise to what is called a Galois algebra. Galois algebras form models of temporal logics like Linear Temporal Logic (LTL) and Computatio ..."
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Cited by 33 (7 self)
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This paper introduces a temporal logic for coalgebras. Nexttime and lasttime operators are dened for a coalgebra, acting on predicates on the state space. They give rise to what is called a Galois algebra. Galois algebras form models of temporal logics like Linear Temporal Logic (LTL) and Computation Tree Logic (CTL). The mapping from coalgebras to Galois algebras turns out to be functorial, yielding indexed categorical structures. This gives many examples, for coalgebras of polynomial functors on sets. Additionally, it will be shown how \fuzzy" predicates on metric spaces, and predicates on presheaves, yield indexed Galois algebras, in basically the same coalgebraic manner. Keywords: Temporal logic, coalgebra, Galois connection, fuzzy predicate, presheaf Classication: 68Q60, 03G05, 03G25, 03G30 (AMS'91); D.2.4, F.3.1, F.4.1 (CR'98). 1 Introduction This paper combines the areas of coalgebra and of temporal logic. Coalgebras are simple mathematical structures (similar, but dual, to...
Towards a Duality Result in the Modal Logic of Coalgebras
 In Coalgebraic Methods in Computer Science, volume 33 of ENTCS
, 2000
"... This paper forms a step in the development of the recently emerged connection between coalgebra and modal logic. It introduces (backandforth) transformations between coalgebras of simple polynomial functors and certain Boolean algebras with operators (BAOs). Categorically, these transformations ta ..."
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Cited by 21 (0 self)
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This paper forms a step in the development of the recently emerged connection between coalgebra and modal logic. It introduces (backandforth) transformations between coalgebras of simple polynomial functors and certain Boolean algebras with operators (BAOs). Categorically, these transformations take the form of an adjunction. The BAO associated with a coalgebra can be used for specification, e.g. of classes in objectoriented languages.
Reasoning about Java classes
 OOPSLAâ€™98, ACM SIGPLAN Notices
, 1998
"... We present the first results of a project called LOOP, on formal methods for the objectoriented language Java. It aims at verification of program properties, with support of modern tools. We use our own frontend tool (which is still partly under construction) for translating Java classes into logi ..."
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Cited by 18 (0 self)
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We present the first results of a project called LOOP, on formal methods for the objectoriented language Java. It aims at verification of program properties, with support of modern tools. We use our own frontend tool (which is still partly under construction) for translating Java classes into logic, and a backend theorem prover (namely PVS, developed at SRI) for reasoning. In several examples we will demonstrate how nontrivial properties of Java programs and classes can be proved following this twostep approach.
Exercises in Coalgebraic Specification
, 1999
"... An introduction to coalgebraic specification is presented via examples. A coalgebraic specification describes a collection of coalgebras satisfying certain assertions. It is thus an axiomatic description of a particular class of mathematical structures. Such specifications are especially suitable fo ..."
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Cited by 16 (2 self)
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An introduction to coalgebraic specification is presented via examples. A coalgebraic specification describes a collection of coalgebras satisfying certain assertions. It is thus an axiomatic description of a particular class of mathematical structures. Such specifications are especially suitable for statebased dynamical systems in general, and for classes in objectoriented programming languages in particular. This paper will gradually introduce the notions of bisimilarity, invariance, component classes, temporal logic and refinement in a coalgebraic setting. Besides the running example of the coalgebraic specification of (possibly infinite) binary trees, a specification of Peterson's mutual exclusion algorithm is elaborated in detail.
Coalgebras for Binary Methods
, 2000
"... Coalgebras for endofunctors C > C can be used to model classes of object oriented languages. However, binary methods do not fit directly into this approach. This paper proposes an extension of the coalgebraic framework, namely the use of extended polynomial functors C^op x C > C . This extension ..."
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Cited by 8 (2 self)
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Coalgebras for endofunctors C > C can be used to model classes of object oriented languages. However, binary methods do not fit directly into this approach. This paper proposes an extension of the coalgebraic framework, namely the use of extended polynomial functors C^op x C > C . This extension allows the incorporation of binary methods into coalgebraic class specifications. The paper also discusses how to define bisimulation for coalgebras of extended polynomial functors and proves some standard results.
Some properties of Fib as a fibred 2category
, 1997
"... We consider some basic properties of the 2category Fib of fibrations over arbitrary bases, exploiting the fact that it is fibred over Cat. We show a factorisation property for adjunctions in Fib, which has direct consequences for fibrations, e.g. a characterisation of limits and colimits for them. ..."
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Cited by 6 (0 self)
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We consider some basic properties of the 2category Fib of fibrations over arbitrary bases, exploiting the fact that it is fibred over Cat. We show a factorisation property for adjunctions in Fib, which has direct consequences for fibrations, e.g. a characterisation of limits and colimits for them. We also consider oplax colimits in Fib, with the construction of Kleisli objects as a particular example. All our constructions are based on an elementary characterisation of Fib as a fibration.
Induction, Coinduction, and Adjoints
, 2002
"... We investigate the reasons for which the existence of certain right adjoints implies the existence of some nal coalgebras, and viceversa. In particular we prove and discuss the following theorem which has been partially available in the literature: let F a G be a pair of adjoint functors, and supp ..."
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Cited by 3 (2 self)
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We investigate the reasons for which the existence of certain right adjoints implies the existence of some nal coalgebras, and viceversa. In particular we prove and discuss the following theorem which has been partially available in the literature: let F a G be a pair of adjoint functors, and suppose that an initial algebra F (X) of the functor H(Y ) = X + F (Y ) exists; then a right adjoint G(X) to F (X) exists if and only if a nal coalgebra G(X) of the functor K(Y ) = X G(Y ) exists. Motivated by the problem of understanding the structures that arise from initial algebras, we show the following: if F is a left adjoint with a certain commutativity property, then an initial algebra of H(Y ) = X + F (Y ) generates a subcategory of functors with inductive types where the functorial composition is constrained to be a Cartesian product.
A Categorical Outlook on Relational Modalities and Simulations
, 2002
"... We characterise bicategories of spans, relations and partial maps universally in terms of factorisations involving maps . We apply this characterisation to show that the standard modalities 2 and arise canonically as the extension of a predicate logic from functions to (abstract) relations . ..."
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Cited by 3 (1 self)
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We characterise bicategories of spans, relations and partial maps universally in terms of factorisations involving maps . We apply this characterisation to show that the standard modalities 2 and arise canonically as the extension of a predicate logic from functions to (abstract) relations .