L. Moss, Coalgebraic logic, Ann. Pure Appl. Logic 96 (1999), 277-317.  Home/Search   Document Details and Download   Summary   Related Articles   Check

....state based dynamical systems. Examples include automata and transition systems, but also programs (as state transformers) and classes in object oriented languages, see [13, 25, 20] Modal logic is a logic for dynamical systems. The connections between the areas of coalgebra and modal logic [19, 21, 22, 23, 16, 11, 1, 5, 4] form currently an area of active research. The following developments constitute the background of the current work. 1. The idea that the functor of a coalgebra determines a certain modal logic was rst put forward by Moss [19] He developed it for very general functors, but the idea was applied ....

....the areas of coalgebra and modal logic [19, 21, 22, 23, 16, 11, 1, 5, 4] form currently an area of active research. The following developments constitute the background of the current work. 1. The idea that the functor of a coalgebra determines a certain modal logic was rst put forward by Moss [19]. He developed it for very general functors, but the idea was applied by others (R o iger, Kurz, Jacobs, Goldblatt) mostly to a restricted class of inductively de ned polynomial functors . 2. The idea to extract coalgebraic structure from maximally consistent sets of formulas is due to R o iger ....

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L.S. Moss. Coalgebraic logic. Ann. Pure & Appl. Logic, 96(1-3):277-317, 1999.

.... weakly preserving pullbacks . In [Gum99a] the theory was generalized and extended to work with arbitrary type functors. The structure theoretic e#ect of the (weak) preservation conditions, as assumed in [AM89] and [Rut00] was characterized in [GS00a] L. Moss has introduced in [Mos99a] see also [Mos99b], a modal logic for coalgebras whose type functor weakly preserve pullbacks. The first Birkho# characterization was given in [Gum99b] the syntactical side was added in [Gum01a] ....

L.S. Moss, Erratum to "coalgebraic logic", Ann. Pure Appl. Logic 99 (1999), 241--259.

....for type functors weakly preserving pullbacks . In [Gum99a] the theory was generalized and extended to work with arbitrary type functors. The structure theoretic e#ect of the (weak) preservation conditions, as assumed in [AM89] and [Rut00] was characterized in [GS00a] L. Moss has introduced in [Mos99a], see also [Mos99b] a modal logic for coalgebras whose type functor weakly preserve pullbacks. The first Birkho# characterization was given in [Gum99b] the syntactical side was added in [Gum01a] ....

L.S. Moss, Coalgebraic logic, Ann. Pure Appl. Logic 96 (1999), 277--317.

.... ordered non determinism or determinism. In general, this underlying behaviour model can be represented by a functor B . 2 The transition system de ned by p is the structure upon which the operator is interpreted. In fact, it has been recently recognised by a number of authors (notably in [15] and [11] that a modal language associated to a T coalgebra is determined by its shape, as recorded in T. 3 This set will later be equipped with further structure to support particular interaction disciplines. For the moment just assume that actions are generated from a set L of labels, i.e. a ....

L. Moss. Coalgebraic logic. Ann. Pure & Appl. Logic, 1999.

....the previous section transfer to this context: one only has to substitute Delta for ; and L o grad 1 for L grad 1 . 5.3 Relations with Coalgebraic Logic. Givenawell behaved functor F , a general method for constructing a logic characterizing F coalgebras modulo F bisimulation is given in [5]. There it is also shown that for certain functors, the uniform ones, a single formula of the logic suffices for characterizing a pointed coalgebra. In Section 3 weproved that our functors ; and Delta are uniform and hence the results of [5] apply to our context. Moreover, one can prove that Moss ....

....F coalgebras modulo F bisimulation is given in [5] There it is also shown that for certain functors, the uniform ones, a single formula of the logic suffices for characterizing a pointed coalgebra. In Section 3 weproved that our functors ; and Delta are uniform and hence the results of [5] apply to our context. Moreover, one can prove that Moss logics relative to our functors are a fragment of the logics described in the previous sections. The method described in [5]isvery general and applies to a large class of functors, but the description of the syntax and semantics of the ....

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L. Moss. Coalgebraic Logic. Preprint

....systems and classes in object oriented languages. Temporal logic is a logic which is particularly suitable for reasoning about (reactive) state based systems, as argued for example in [Pnu77, Pnu81] via its nexttime and lasttime operators. Hence one expects a connection. It is probably Moss [Mos99] who was the rst to realise that the shape of a coalgebra (as given by its interface functor) determines a logical modal language. His emphasis lies on characterisation results, capturing bisimilarity as validity for the same formulas. This line is followed in [R o 99b, R o 99a, Kur98] Here the ....

....the de nition of Galois algebras. The next section 3 shows how coalgebras of so called polynomial functors on sets give rise to Galois algebras, making crucial use of predicate lifting . Section 4 forms an intermezzo, showing how the temporal operators can also be de ned pathwise, as in [Mos99, R o 99b, R o 99a, Kur98] and give rise to Galois algebras as well. Then, Section 5 elaborates on Galois algebras. Most of this comes directly from [Kar98] with our own notations and proofs) except for the part dealing with strict and ane lifting. The nal section 6 describes the main result ....

[Article contains additional citation context not shown here]

L.S. Moss. Coalgebraic logic. Ann. Pure & Appl. Logic, 1999. To appear.

....section transfer to this context: one only has to substitute Delta for Gamma and L o grad 1 for L grad 1 . 5.3 Relations with Coalgebraic Logic. Given a well behaved functor F , a general method for constructing a logic characterizing F coalgebras modulo F bisimulation is given in [5]. There it is also shown that for certain functors, the uniform ones, a single formula of the logic suffices for characterizing a pointed coalgebra. In Section 3 we proved that our functors Gamma and Delta are uniform and hence the results of [5] apply to our context. Moreover, one can prove ....

....F coalgebras modulo F bisimulation is given in [5] There it is also shown that for certain functors, the uniform ones, a single formula of the logic suffices for characterizing a pointed coalgebra. In Section 3 we proved that our functors Gamma and Delta are uniform and hence the results of [5] apply to our context. Moreover, one can prove that Moss logics relative to our functors are a fragment of the logics described in the previous sections. The method described in [5] is very general and applies to a large class of functors, but the description of the syntax and semantics of the ....

[Article contains additional citation context not shown here]

L. Moss. Coalgebraic Logic. Preprint

....important for the study of such dynamical systems. In this eld one naturally reasons in terms of invariance and bisimilarity. Indeed, these notions are fundamental in the theory of coalgebras 1 . Further, a recent development is the close connection between coalgebras and temporal logic, see [26, 14]. The temporal operators for henceforth and for eventually can be de ned easily in a coalgebraic setting, in terms of invariants. Their use is quite natural in a state based setting, namely for reasoning about all some future states in safety progress formulas. In this paper we illustrate the ....

L.S. Moss. Coalgebraic logic. Ann. Pure & Appl. Logic, 1999. To appear.

....systems and classes in object oriented languages. Temporal logic is a logic which is particularly suitable for reasoning about (reactive) state based systems, as argued for example in [Pnu77, Pnu81] via its nexttime and lasttime operators. Hence one expects a connection. It is probably Moss [Mos99] who was the rst to realise that the shape of a coalgebra (as given by its functor) determines a logical modal language. His emphasis lies on characterisation results, capturing bisimilarity as validity for the same formulas. This line is followed in [R o 99b, R o 99a, Kur98] Here the emphasis ....

....the de nition of Galois algebras. The next section 3 shows how coalgebras of so called polynomial functors on sets give rise to Galois algebras, making crucial use of predicate li ng . Section 4 forms an intermezzo, showing how the temporal operators can also be de ned pathwise, as in [Mos99, R o 99b, R o 99a, Kur98] and give rise to Galois algebras as well. Then, Section 5 elaborates on Galois algebras. Most of this comes directly from [Kar98] with our own notations and proofs) except for the part dealing with strict and a ne lifting. The nal section 6 describes the main result ....

[Article contains additional citation context not shown here]

L.S. Moss. Coalgebraic logic. Ann. Pure & Appl. Logic, 1999. To appear.

....Also, probabilistic bisimulation for Markov processes may be studied from a coalgebraic perspective [VR98, DEP98] Coalgebras and modal logic Modal logic [Eme90, Gol92] may be seen as the logic of dynamics. Therefore, it combines very well with coalgebras, as witnessed by several recent studies [Mos99, R o 99, Kur98] Coalgebras and non well founded set theory In the original monograph [Acz88] on non well founded sets, coalgebras are already prominently present, since they are fundamental to the idea of a non well founded set. Further investigations occur in [BM96, TR98] Coalgebras and ....

L.S. Moss. Coalgebraic logic. Ann. Pure & Appl. Logic, 1999. To appear.

....hypotheses on the category. This is a key contribution of the paper. It is natural to ask whether all of the final coalgebra interpretations are in some sense reducible to other kinds of interpretations, especially those involving dcpo s. This question was asked in a more precise way in our paper [10]. We showed that for essentially all functors on sets which arise in practice, the final coalgebra interpretations could indeed be obtained as the maximal elements of some dcpo. However, that work involved assuming hypotheses on the functors that go beyond what we needed here (but which ....

Lawrence S. Moss, "Coalgebraic logic," in Festschrift on the occasion of Professor Rohit Parikh's 60th birthday. Annals of Pure Applied Logic 96 (1999), no. 1-3, 277--317.

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L. Moss, Coalgebraic logic, Ann. Pure Appl. Logic 96 (1999), 277-317.

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Lawrence Moss. Coalgebraic logic. unpublished, 1997.

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L.S. Moss. Coalgebraic logic. Ann. Pure Appl. Logic, 96:277--317, 1999.

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Larry Moss, Coalgebraic Logic, 1997, to appear.

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L.S. Moss. Coalgebraic logic. Ann. Pure & Appl. Logic, 1999. To appear.

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