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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 298 (31 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
"... ..."
Functors for Coalgebras
 Algebra Universalis
"... . Functors preserving weak pullbacks provide the basis for a rich structure theory of coalgebras. We give an easy to use criterion to check whether a functor preserves weak pullbacks. We apply the characterization to the functor F which associates a set X with the set F(X) of all filters on X. It t ..."
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Cited by 19 (5 self)
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. Functors preserving weak pullbacks provide the basis for a rich structure theory of coalgebras. We give an easy to use criterion to check whether a functor preserves weak pullbacks. We apply the characterization to the functor F which associates a set X with the set F(X) of all filters on X. It turns out that this functor preserves weak pullbacks, yet does not preserve weak generalized pullbacks. Since topological spaces can be considered as F coalgebras, in fact they constitute a covariety, we find that the intersection of subcoalgebras need not be a coalgebra, and 1generated Fcoalgebras need not exist. 1. Introduction Coalgebras have been introduced by Aczel and Mendler [AM89] to model various types of transition systems. Reichel [Rei95], and Jacobs [Jac96] show that coalgebras are well suited for modeling object oriented programmming and for program verification. In [Rut96], J.J.M.M. Rutten develops the a fundamental theory of "universal coalgebra" along the lines of univers...
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 { namely, ..."
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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 9 (4 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 Birkho and of Mal’cev
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.
Coalgebras in Specification and Verification for ObjectOriented Languages
, 1999
"... The aim of this short note is to give an impression of the use of coalgebras in specification and verification for objectoriented languages. Particular emphasis will be given to the rôle of coalgebraic operations in describing statebased systems. At the end some active research topics in coalgebra ..."
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Cited by 3 (0 self)
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The aim of this short note is to give an impression of the use of coalgebras in specification and verification for objectoriented languages. Particular emphasis will be given to the rôle of coalgebraic operations in describing statebased systems. At the end some active research topics in coalgebra will be sketched, together with pointers to the literature.
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).
Modal Operators for Coequations
, 2001
"... this paper, we develop the theory of coequations from a logical viewpoint. To clarify, let G = #G, #, ## be a comonad on E , where G preserves regular monos and E is "coBirkho #" (see Definition 2.1). A coequation # over a set C of colors is a regular subobject of GC, the carrier of the cofree ..."
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this paper, we develop the theory of coequations from a logical viewpoint. To clarify, let G = #G, #, ## be a comonad on E , where G preserves regular monos and E is "coBirkho #" (see Definition 2.1). A coequation # over a set C of colors is a regular subobject of GC, the carrier of the cofree coalgebra # C : GC ## G 2 C over C. Hence, we can view # as a predicate over GC. In particular, we can form new coequations out of old by means of the logical connectives #, #, etc. Furthermore, we have available a modal operator taking a coequation # to the (carrier of the) largest subcoalgebra # contained in the coequation. As we will see, arises as the formal dual of a familiar operation on sets of equations in categories of algebras. Explicitly, the operator is dual to the closure operation taking a set E of equations over X (i.e., E # UFX UFX , where UFX is the carrier of the free algebra over X) to the least congruence containing E. Hence, is dual to the closure of sets of equations under the first four rules of inference of Birkho#'s equational logic (Birkho#, 1935). Thus, we see that closure under these rules of inferences is dual to the "coalgebra interior" of a set of elements. We introduce a modal operator that is dual to closure under Birkho#'s fifth rule of inference, i.e., substitution of terms for variables. We confirm that is an S4 operator and show that, under certain conditions, commutes with . We then prove the invariance theorem in terms of and . In this way, we develop the coequationsaspredicates view by augmenting the predicates over GC with two modal operators and and show that the partial order of covarieties definable by arbitrary coequations over C is isomorphic to the partial order of predicates # over GC such th...