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13
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 302 (32 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
Abstract behavior types: A foundation model for components and their composition
 SCIENCE OF COMPUTER PROGRAMMING
, 2003
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Products of coalgebras
, 2001
"... We prove that the category of Fcoalgebras is complete, that is products and equalizers exist, provided that the type functor F is bounded or preserves mono sources. This generalizes and simplifies a result of Worrell ([Wor98]). We also describe the relationship between the product A × B and the lar ..."
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Cited by 19 (5 self)
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We prove that the category of Fcoalgebras is complete, that is products and equalizers exist, provided that the type functor F is bounded or preserves mono sources. This generalizes and simplifies a result of Worrell ([Wor98]). We also describe the relationship between the product A × B and the largest bisimulation ∼ A,B between A and B and find an example of two finite coalgebras whose product is infinite.
The Coalgebraic Dual Of Birkhoff's Variety Theorem
, 2000
"... We prove an abstract dual of Birkho's variety theorem for categories E of coalgebras, given suitable assumptions on the underlying category E and suitable : E ## E . We also discuss covarieties closed under bisimulations and show that they are denable by a trivial kind of coequation { nam ..."
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Cited by 11 (0 self)
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We prove an abstract dual of Birkho's variety theorem for categories E of coalgebras, given suitable assumptions on the underlying category E and suitable : E ## E . We also discuss covarieties closed under bisimulations and show that they are denable by a trivial kind of coequation { namely, over one "color". We end with an example of a covariety which is not closed under bisimulations. This research is part of the Logic of Types and Computation project at Carnegie Mellon University under the direction of Dana Scott.
Equational and implicational classes of coalgebras
, 2001
"... If F: Set → Set is a functor which is bounded and preserves weak generalized pullbacks then a class of Fcoalgebras is a covariety, i.e., closed under H (homomorphic images), S (subcoalgebras) and � (sums), if and only if it can be de ned by a set of “coequations”. Similarly, quasicovarieties, i.e ..."
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Cited by 8 (3 self)
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If F: Set → Set is a functor which is bounded and preserves weak generalized pullbacks then a class of Fcoalgebras is a covariety, i.e., closed under H (homomorphic images), S (subcoalgebras) and � (sums), if and only if it can be de ned by a set of “coequations”. Similarly, quasicovarieties, i.e., classes closed under H and � , can be characterized by implications of coequations. These results are analogous to the theorems of Birkhoff and of Mal’cev in classical
Coalgebras of Bounded Type
 UNDER CONSIDERATION FOR PUBLICATION IN MATH. STRUCT. IN COMP. SCIENCE
, 2001
"... Using results of Trnková, we first show that subcoalgebras are always closed under finite intersections. Assuming that the type functor F is bounded, we obtain a concrete representation of the terminal Fcoalgebra. Several equivalent characterizations of boundedness are provided. ..."
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Cited by 8 (4 self)
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Using results of Trnková, we first show that subcoalgebras are always closed under finite intersections. Assuming that the type functor F is bounded, we obtain a concrete representation of the terminal Fcoalgebra. Several equivalent characterizations of boundedness are provided.
Observational Ultraproducts of Polynomial Coalgebras
, 2002
"... Coalgebras of polynomial functors constructed from sets of observable elements have been found useful in modelling various kinds of data types and statetransition systems. ..."
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Cited by 6 (3 self)
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Coalgebras of polynomial functors constructed from sets of observable elements have been found useful in modelling various kinds of data types and statetransition systems.
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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Cited by 1 (0 self)
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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).