A. Kurz. A Co-Variety-Theorem for Modal Logic. In Proceedings of Advances in Modal Logic 2, Uppsala, 1998. Center for the Study of Language and Information, Stanford University, 2000.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....constituted a forbidden behaviour pattern . Covarieties then prove to be the same as classes of models of sets of such coequations. A coequation over a single colour covariable would appear to be an abstract analogue of the formulas in this paper that have a single state parameter s. Kurz [27, 29] formulated a covariety theorem by developing an abstract version of the notion of a modal logic, including abstract definitions of formulas, models and satisfaction. Ro]u [33] gave a characterization of equationally specifiable classes of coalgebras that is closer to the spirit of the present ....

Alexander Kurz. A covariety theorem for modal logic. In Michael Zakharyaschev, Krister Segerberg, Maarten de Rijke, and Heinrich Wansang, editors, Advances in Modal Logic, volume 2, pages 367-380. CSLI Publications, 2001.

....coalgebras are now also accepted as a semantics for classes in object oriented programming and speci cation, see [12] for the initial paper. Recently, there was a lot of work centred around the mentioned duality, especially to establish coalgebraic versions of Birkho s variety theorem, see e.g. [6], 2] 1] It is commonly seen as one of the biggest advantages of coalgebras that they deliver standard notions like bisimulation, observational equality and (path wise) modal operators for free . Also, many people see coalgebras as generalisations of transition systems. However, some parts of ....

A. Kurz. A co-variety-theorem for modal logic. In Proceedings of Advances in Modal Logic, Uppsala. CSLI,Stanford, 1998. Revised Version.

.... in this paper comes from Birkho# s famous results about definability and deducibility for universal algebras [Bir35] There has been a considerable amount of work over the last few years aimed at dualizing these results to the setting of co algebras, including [Rut00] GS01] Gum01] [Kur01], Kur00] KR02] AH00] Hug01a] Hug02] Ros00] AP01] Successful dualization often requires a reformulation at a suitable level of abstraction. Currently, the most abstract setting for Birkho# s results is given by factorization systems on appropriate categories [BH76] NS81] Birkho# s ....

....It is more natural to inherit the notions of satisfaction not from Birkho# s classical work, but rather from the dual work in categories of coalgebras. Consequently, our interpretations of the relation is inherited from the relevant definition in terms of coequations, found in [Rut00] GS01] [Kur01], Hug01b] and elsewhere. Theorem 4.2 can be considered a generalization of the quasi covariety theorem, presented as Theorem 3.6.3 in [Hug01b] Consider a fibration with a truth functor : B ## E , that is, a functor such that p and p# = id B . Given arbitrary objects E E and B B, ....

Alexander Kurz. A co-variety-theorem for modal logic. In Advances in Modal Logic 2. Center for the Study of Language and Information, Stanford University, 2001. Selected Papers from AiML 2, Uppsala, 1998.

.... in this paper comes from Birkho s famous results about de nability and deducibility for universal algebras [Bir35] There has been a considerable amount of work over the last few years aimed at dualizing these results to the setting of co algebras, including [Rut00] GS01] Gum01] [Kur01], Kur00] KR02] AH00] Hug01] Hug02] Ro s00] AP01] Successful dualization often requires a reformulation at a suitable level of abstraction. Currently, the most abstract setting for Birkho s results is given by factorization systems on appropriate categories [BH76] NS81] Birkho s ....

....It is more natural to inherit the notions of satisfaction not from Birkho s classical work, but rather from the dual work in categories of coalgebras. Consequently, our interpretations of the relation j= is inherited from the relevant de nition in terms of coequations, found in [Rut00] GS01] [Kur01], and elsewhere. Theorem 4.2 can be considered a generalization of the quasi covariety theorem, presented as Theorem 3.6.3 in [ Consider a bration with a truth functor : B ## E , that is, a functor such that p a and p = id B . Given arbitrary objects E 2 E and B 2 B , we write ....

Alexander Kurz. A co-variety-theorem for modal logic. In Advances in Modal Logic 2. Center for the Study of Language and Information, Stanford University, 2001. Selected Papers from AiML 2, Uppsala, 1998.

....algebras closed under direct products, subalgebras, and homomorphic images. A celebrated result of Garret Birkho [7] states that such varieties are precisely the equationally de nable classes of algebras. There has been a spate of papers discussing coalgebraic versions of Birkho s theorem [29, 3, 17, 20, 23, 2, 15]. This paper is concerned with covarieties that are closed under images of bisimulations, aptly named behavioural covarieties in [3] The original idea of a bisimulation [27, 26] was that of a binary relation of observational indistinguishability between states of two transition systems. States ....

Alexander Kurz. A covariety theorem for modal logic. In Michael Zakharyaschev, Krister Segerberg, Maarten de Rijke, and Heinrich Wansang, editors, Advances in Modal Logic, volume 2, pages 367-380. CSLI Publications, 2001.

....could, for instance, be used to specify systems or even to verify properties of them. In [HenR95, Jac95] equations are used to describe coalgebras. A. Corradini ( Cor97] introduces an equational calculus to describe coalgebras of certain polynomial functors. H. Gumm ( Gum98] and A. Kurz ([Kur98a]) show that covarieties are characterized by some kind of co equations (which constitutes a dual version of Birkhoff s theorem) In [Mos99] L. Moss first shows how the underlying functor determines a langugage that is based on modal logic. For a large class of functors he derives languages for ....

A. Kurz, A Co-variety-theorem for modal logic, Proceedings of Advances in Modal Logic, Uppsala, CSLI, Stanford, 1998. 32 Coalgebras and Modal Logic

.... systems, Mooreand Mealy automata and deterministic systems, see e.g. Rutten [28] The research on modal logics as speci cation languages for coalgebras began with Moss [20] and was taken up in e.g. 17, 27, 25, 8, 9] The relationship between modal logic and coalgebras has been explained in [15] as follows. If Z denotes the carrier of the nal coalgebra, we can consider the semantics of a modal formula # as the subset [ #] Z of states which satisfy #. Intuitively, the elements of Z are behaviours, and every modal formula # determines a set of behaviours which satisfy #. In case the ....

....behaviours, and every modal formula # determines a set of behaviours which satisfy #. In case the logic is fully expressive in the sense that it allows to de ne all subsets of Z, we can identify modal formulae with subsets of Z, resulting in an algebraic approach to investigate modal logics, see [15, 16]. In general however, nitary modal logics are not fully expressive. It is the main issue of this paper to present a semantic representation, which ts nitary logics as nicely as the representation as subsets of the nal model suits fully expressive logics. We use the so called terminal ....

Alexander Kurz. A co-variety-theorem for modal logic. In M. Zakharyaschev, K. Segerberg, M. de Rijke, and H. Wansing, editors, Advances in Modal Logic 2. Center for the Study of Language and Information, Stanford University, 2001.

.... systems, Moore and Mealy automata and deterministic systems, see e.g. Rutten [28] The re search on modal logics as specification languages for coalgebras began with Moss [201 and was taken up in e.g. 17, 27, 25, 8, 9] The relationship between modal logic and coalgebras has been explained in [15] as follows. If Z denotes the carrier of the final coalgebra, we can consider the semantics of a modal formula qo as the subset Iqol C Z of states which satisfy qo. Intuitively, the elements of Z are behaviours, and every modal formula qo determines a set of behaviours which satisfy qo. In case ....

....and every modal formula qo determines a set of behaviours which satisfy qo. In case the logic is fully expressive in the sense that it allows to define all subsets of Z, we can identify modal formulae with subsets of Z, resulting in an algebraic approach to investigate modal logics, see [15, 16]. In general however, finitary modal logics are not fully expressive. It is the main issue of this paper to present a semantic representation, which fits finitary logics as nicely as the representation as subsets of the final model suits fully expressive logics. We use the so called terminal ....

Alexander Kurz. A co-variety-theorem for modal logic. In M. Za- kharyaschev, K. Segerberg, M. de Rijke, and H. Wansing, editors, Advances in Modal Logic 2. Center for the Study of Language and Information, Stanford University, 2001.

....propositional variables add indeed expressiveness. On the other hand, formulas with propositional variables are still preserved under subcoalgebras, that is, definable classes are covarieties. Conversely, every covariety is definable by an (infinitary) modal logic with propositional variables (see [12]) We show now how to build logics whose formulas are not necessarily preserved under subcoalgebras. A logic for coalgebras, possibly with propositional variables, can be strengthened by adding rules. Given two formulas ; we call = a rule and extend the satisfaction relation via X j= ....

Alexander Kurz. A co-variety-theorem for modal logic. In Advances in Modal Logic

.... systems, Moore and Mealy automata and deterministic systems, see e.g. Rutten [22] The research on modal logics as speci cation languages for coalgebras began with Moss [15] and was taken up in e.g. 14,21,20,8,9] The relationship between modal logic and coalgebras has been explained in [12] as follows. Denoting by Z the carrier of the nal coalgebra, we can Email: kurz cwi.nl Email: pattinso informatik.uni muenchen.de c 2002 Published by Elsevier Science B. V. consider as the semantics of a modal formula the subset [ Z satisfying . The intuition here is: The ....

.... is the property (i.e. set) of behaviours de ned by . In case that the logics we are interested in are fully expressive in the sense that they allow to de ne all subsets of Z, we can identify modal formulae and subsets of Z, resulting in an approach to algebraically investigate modal logics, see [12,13]. Unfortunately, modal logics given by a nitary syntax are in general not fully expressive. The reason for this is simply that not all properties of behaviours can be described in a nitary language. One of the main topics of this paper is the quest for a semantics of modal logic that suits ....

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Alexander Kurz. A co-variety-theorem for modal logic. In Advances in Modal Logic 2. Center for the Study of Language and Information, Stanford University, 2001.

.... in Electronic Notes in Theoretical Computer Science URL: www.elsevier.nl locate entcs The research on modal logics as speci cation languages for coalgebras began with Moss [15] and was taken up in e.g. 14,21,20,8,9] The relationship between modal logic and coalgebras has been explained in [12] as follows. Denoting by Z the carrier of the nal coalgebra, we can consider as the semantics of a modal formula the subset [ Z satisfying . The intuition here is: The elements of Z are the behaviours and the meaning of a modal formula is the property (i.e. set) of behaviours de ned by ....

....formula is the property (i.e. set) of behaviours de ned by . In case that the logics we are interested in are expressive in the sense that they allow to de ne all subsets of Z, we can identify modal formulae and subsets of Z, resulting in a new way to algebraically investigate modal logics, see [12,13]. Unfortunately, modal logics given by a nitary syntax are in general not expressive. The reason for this is simply that not all properties of behaviours can be described in a nite language. One of the main topics of this paper is the quest for a semantics of modal logic that suits nitary ....

[Article contains additional citation context not shown here]

Alexander Kurz. A co-variety-theorem for modal logic. In Advances in Modal Logic 2. Center for the Study of Language and Information, Stanford University,

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A. Kurz. A Co-Variety-Theorem for Modal Logic. In Proceedings of Advances in Modal Logic 2, Uppsala, 1998. Center for the Study of Language and Information, Stanford University, 2000.

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Alexander Kurz. A co-variety-theorem for modal logic. Proceedings of Advances in Modal Logic, 1998.

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