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10
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
Coalgebras and Modal Logic
 Coalgebraic Methods in Computer Science, Volume 33 in Electronic Notes in Theoretical Computer Science
, 2000
"... Coalgebras are of growing importance in theoretical computer science. To develop languages for them is significant for the specification and verification of systems modelled with them. Modal logic has proved to be suitable for this purpose. So far, most approaches have presented a language to descri ..."
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Cited by 33 (0 self)
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Coalgebras are of growing importance in theoretical computer science. To develop languages for them is significant for the specification and verification of systems modelled with them. Modal logic has proved to be suitable for this purpose. So far, most approaches have presented a language to describe only deterministic coalgebras. The present paper introduces a generalization that also covers nondeterministic systems. As a special case, we obtain the "usual" modal logic for Kripkestructures. Models for our modal language L F are Fcoalgebras where the functor F is inductively constructed from constant sets and the identity functor using product, coproduct, exponentiation, and the power set functor. We define a language L F and show that it embeds into L F . We prove that, for imagefinite coalgebras, L F is expressive enough to distinguish elements up to bisimilarity and therefore L F does so, too. Moreover, we also give a complete calculus for L F in case the constants...
Incompleteness of Behavioral Logics
, 2000
"... Incompleteness results for behavioral logics are investigated. We show that there is a basic finite behavioral specification for which the behavioral satisfaction problem is not recursively enumerable, which means that there are no automatic methods for proving all true statements; in particular, be ..."
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Cited by 25 (6 self)
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Incompleteness results for behavioral logics are investigated. We show that there is a basic finite behavioral specification for which the behavioral satisfaction problem is not recursively enumerable, which means that there are no automatic methods for proving all true statements; in particular, behavioral logics do not admit complete deduction systems. This holds for all of the behavioral logics of which we are aware. We also prove that the behavioral satisfaction problem is not corecursively enumerable, which means that there is no automatic way to refute false statements in behavioral logics. In fact we show stronger results, that all behavioral logics are # 0 2 hard, and that, for some data algebras, the complexity of behavioral satisfaction is not even arithmetic; matching upper bounds are established for some behavioral logics. In addition, we show for the fixeddata case that if operations mayhave more than one hidden argument, then final models need not exist, so that the coalgebraic flavor of behavioral logic is lost.
What is the Coalgebraic Analogue of Birkhoff's Variety Theorem?
 THEORETICAL COMPUTER SCIENCE
, 2000
"... Logical definability is investigated for certain classes of coalgebras related to statetransition systems, hidden algebras and Kripke models. The filter enlargement of a coalgebra A is introduced as a new coalgebra A + whose states are special "observationally rich" filters on the state set of A. T ..."
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Cited by 7 (4 self)
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Logical definability is investigated for certain classes of coalgebras related to statetransition systems, hidden algebras and Kripke models. The filter enlargement of a coalgebra A is introduced as a new coalgebra A + whose states are special "observationally rich" filters on the state set of A. The ultrafilter enlargement is the subcoalgebra A of A + whose states are ultrafilters. Boolean combinations of equations between terms of observable (or output) type are identified as a natural class of formulas for specifying properties of coalgebras. These observable formulas are permitted to have a single state variable, and form a language in which modalities describing the effects of state transitions are implicitly present. A and A + validate the same observable formulas. It is shown that a class of coalgebras is de nable by observable formulas iff the class is closed under disjoint unions, images of bisimulations, and (ultra) lter enlargements. (Closure under images of bisimulations is equivalent to closure under images and domains of coalgebraic morphisms.) Moreover, every set of observable formulas has the same models as some set of conditional equations. Examples are
Themes in Final Semantics
 Dipartimento di Informatica, Università di
, 1998
"... C'era una volta un re seduto in canap`e, che disse alla regina raccontami una storia. La regina cominci`o: "C'era una volta un re seduto in canap`e ..."
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Cited by 5 (2 self)
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C'era una volta un re seduto in canap`e, che disse alla regina raccontami una storia. La regina cominci`o: "C'era una volta un re seduto in canap`e
A Birkhofflike Axiomatizability Result for Hidden Algebra and Coalgebra
, 2000
"... A characterization result for behaviorally definable classes of hidden algebras shows that a class of hidden algebras is behaviorally definable by equations if and only if it is closed under coproducts, quotients, morphisms and representative inclusions. The second part of the paper categorically ge ..."
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Cited by 4 (0 self)
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A characterization result for behaviorally definable classes of hidden algebras shows that a class of hidden algebras is behaviorally definable by equations if and only if it is closed under coproducts, quotients, morphisms and representative inclusions. The second part of the paper categorically generalizes this result to a framework of any category with coproducts, a final object and an inclusion system; this is general enough to include all coalgebra categories of interest. As a technical issue, the notions of equation and satisfaction are axiomatized in order to include the different approaches in the literature.
Coalgebras and modal logics for parameterised endofunctors
, 2000
"... for promotion of mathematics and computer science and their applications. SMC is sponsored by the Netherlands Organization for Scientific Research (NWO). CWI is a member of ..."
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Cited by 3 (3 self)
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for promotion of mathematics and computer science and their applications. SMC is sponsored by the Netherlands Organization for Scientific Research (NWO). CWI is a member of
A Birkholike axiomatizability result for hidden algebra and coalgebra
 In Proceedings of CMCS'98, volume 11 of ENTCS
, 1998
"... Acharacterization result for behaviorally de nable classes of hidden algebras shows that a class of hidden algebras is behaviorally de nable by equations if and only if it is closed under coproducts, quotients, morphisms and representative inclusions. The second part of the paper categorically gener ..."
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Cited by 1 (1 self)
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Acharacterization result for behaviorally de nable classes of hidden algebras shows that a class of hidden algebras is behaviorally de nable by equations if and only if it is closed under coproducts, quotients, morphisms and representative inclusions. The second part of the paper categorically generalizes this result to a framework of any category with coproducts, a nal object and an inclusion system; this is general enough to include all coalgebra categories of interest. As a technical issue, the notions of equation and satisfaction are axiomatized in order to include the di erent approaches in the literature. 1
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...