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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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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
HOMOMORPHISMS FOR TOPOLOGICAL COALGEBRAS
"... Topological spaces are coalgebras for the filter functor [3]. However, standard coalgebra homomorphisms correspond to continuous and open maps. In this paper, we show that a suitably weakened homomorphism concept yields the correct topological space homomorphisms, ..."
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Topological spaces are coalgebras for the filter functor [3]. However, standard coalgebra homomorphisms correspond to continuous and open maps. In this paper, we show that a suitably weakened homomorphism concept yields the correct topological space homomorphisms,
On Tree Coalgebras and Coalgebra Presentations
"... For deterministic systems, expressed as coalgebras over polynomial functors, every tree t (an element of the final coalgebra) turns out to represent a new coalgebra At. The universal property of these coalgebras, resembling freeness, is that for every state s of every system S there exists a unique ..."
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coalgebra homomorphism from a unique At which takes the root of t to s. Moreover, the tree coalgebras are finitely presentable and form a strong generator. Thus, these categories of coalgebras are locally finitely presentable; in particular every system is a filtered colimit of finitely presentable systems
A Calculus of Transition Systems (towards Universal Coalgebra)
 IN ALBAN PONSE, MAARTEN DE RIJKE, AND YDE VENEMA, EDITORS, MODAL LOGIC AND PROCESS ALGEBRA, CSLI LECTURE NOTES NO
, 1995
"... By representing transition systems as coalgebras, the three main ingredients of their theory: coalgebra, homomorphism, and bisimulation, can be seen to be in a precise correspondence to the basic notions of universal algebra: \Sigmaalgebra, homomorphism, and substitutive relation (or congruence). ..."
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Cited by 30 (1 self)
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By representing transition systems as coalgebras, the three main ingredients of their theory: coalgebra, homomorphism, and bisimulation, can be seen to be in a precise correspondence to the basic notions of universal algebra: \Sigmaalgebra, homomorphism, and substitutive relation (or congruence
Automata and coinduction (an exercise in coalgebra
 LNCS
, 1998
"... The classical theory of deterministic automata is presented in terms of the notions of homomorphism and bisimulation, which are the cornerstones of the theory of (universal) coalgebra. This leads to a transparent and uniform presentation of automata theory and yields some new insights, amongst which ..."
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Cited by 86 (19 self)
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The classical theory of deterministic automata is presented in terms of the notions of homomorphism and bisimulation, which are the cornerstones of the theory of (universal) coalgebra. This leads to a transparent and uniform presentation of automata theory and yields some new insights, amongst
Recursive coalgebras of finitary functors
 Department of Computer Science, University of Wales Swansea
, 2005
"... Abstract For finitary set functors preserving inverse images several concepts of coalgebras A are proved to be equivalent: (i) A has a homomorphism into the initial algebra, (ii) A is recursive, i.e., A has a unique coalgebratoalgebra morphism into any algebra, and (iii) A is parametrically recurs ..."
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Cited by 15 (0 self)
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Abstract For finitary set functors preserving inverse images several concepts of coalgebras A are proved to be equivalent: (i) A has a homomorphism into the initial algebra, (ii) A is recursive, i.e., A has a unique coalgebratoalgebra morphism into any algebra, and (iii) A is parametrically
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
ContextFree Languages, Coalgebraically
, 2011
"... We give a coalgebraic account of contextfree languages using the functor D(X) =2 × X A for deterministic automata over an alphabet A, in three different but equivalent ways: (i) by viewing contextfree grammars as Dcoalgebras; (ii) by defining a format for behavioural differential equations (w.r. ..."
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Cited by 11 (9 self)
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.r.t. D) for which the unique solutions are precisely the contextfree languages; and (iii) as the Dcoalgebra of generalized regular expressions in which the Kleene star is replaced by a unique fixed point operator. In all cases, semantics is defined by the unique homomorphism into the final coalgebra
Coalgebraic semantics for logic programming
 18th Worshop on (Constraint) Logic Programming, WLP 2004, March 0406
, 2004
"... www.dis.uniroma1.it / ¢ majkic/ Abstract. General logic programs with negation have the 3valued minimal Herbrand models based on the Kripke’s fixpoint knowledge revision operator and on Clark’s completion. Based on these results we deifine a new algebra £¥ ¤, (with the relational algebra embedded ..."
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Cited by 7 (1 self)
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in it), and present an algorithmic transformation of logic programs into the system of tuplevariable equations which is a £ ¤coalgebra. The solution of any such system of equations (a £ ¤coalgebra) corresponds to the unique homomorphism from this £¦ ¤coalgebra into the final £ ¤coalgebra, which
Observational Ultrapowers of Polynomial Coalgebras
, 2001
"... Coalgebras of polynomial functors constructed from set of observable elements have been found useful in modelling various kinds of data types and statetransition systems. This paper continues the study of equational logic and model theory for polynomial coalgebras begun in [6], where it was shown t ..."
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Cited by 1 (1 self)
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a structural characterisation of classes of coalgebras definable by observable formulas. This is an analogue for polynomial coalgebras of Birkhoff's celebrated characterisation of equationally definable classes of abstract algebras as being those closed under homomorphic images, subalgebras
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