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Rank1 modal logics are coalgebraic
 IN STACS 2007, 24TH ANNUAL SYMPOSIUM ON THEORETICAL ASPECTS OF COMPUTER SCIENCE, PROCEEDINGS
, 2007
"... Coalgebras provide a unifying semantic framework for a wide variety of modal logics. It has previously been shown that the class of coalgebras for an endofunctor can always be axiomatised in rank 1. Here we establish the converse, i.e. every rank 1 modal logic has a sound and strongly complete coal ..."
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Cited by 14 (11 self)
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Coalgebras provide a unifying semantic framework for a wide variety of modal logics. It has previously been shown that the class of coalgebras for an endofunctor can always be axiomatised in rank 1. Here we establish the converse, i.e. every rank 1 modal logic has a sound and strongly complete coalgebraic semantics. As a consequence, recent results on coalgebraic modal logic, in particular generic decision procedures and upper complexity bounds, become applicable to arbitrary rank 1 modal logics, without regard to their semantic status; we thus obtain purely syntactic versions of these results. As an extended example, we apply our framework to recently defined deontic logics.
NEIGHBOURHOOD STRUCTURES: BISIMILARITY AND BASIC MODEL THEORY
, 2009
"... Neighbourhood structures are the standard semantic tool used to reason about nonnormal modal logics. The logic of all neighbourhood models is called classical modal logic. In coalgebraic terms, a neighbourhood frame is a coalgebra for the contravariant powerset functor composed with itself, denote ..."
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Cited by 5 (1 self)
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Neighbourhood structures are the standard semantic tool used to reason about nonnormal modal logics. The logic of all neighbourhood models is called classical modal logic. In coalgebraic terms, a neighbourhood frame is a coalgebra for the contravariant powerset functor composed with itself, denoted by 2 2. We use this coalgebraic modelling to derive notions of equivalence between neighbourhood structures. 2 2bisimilarity and behavioural equivalence are well known coalgebraic concepts, and they are distinct, since 2 2 does not preserve weak pullbacks. We introduce a third, intermediate notion whose witnessing relations we call precocongruences (based on pushouts). We give backandforth style characterisations for 2 2bisimulations and precocongruences, we show that on a single coalgebra, precocongruences capture behavioural equivalence, and that between neighbourhood structures, precocongruences are a better approximation of behavioural equivalence than 2 2bisimulations. We also introduce a notion of modal saturation for neighbourhood models, and investigate its relationship with definability and imagefiniteness. We prove a HennessyMilner theorem for modally saturated and for imagefinite neighbourhood models. Our main results are an analogue of Van Benthem’s characterisation theorem and a modeltheoretic proof of Craig interpolation for classical modal logic.
Bisimulation for Neighbourhood Structures
, 2007
"... Neighbourhood structures are the standard semantic tool used to reason about nonnormal modal logics. In coalgebraic terms, a neighbourhood frame is a coalgebra for the contravariant powerset functor composed with itself, denoted by 2 2. In our paper, we investigate the coalgebraic equivalence notio ..."
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Cited by 5 (0 self)
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Neighbourhood structures are the standard semantic tool used to reason about nonnormal modal logics. In coalgebraic terms, a neighbourhood frame is a coalgebra for the contravariant powerset functor composed with itself, denoted by 2 2. In our paper, we investigate the coalgebraic equivalence notions of 2 2bisimulation, behavioural equivalence and neighbourhood bisimulation (a notion based on pushouts), with the aim of finding the logically correct notion of equivalence on neighbourhood structures. Our results include relational characterisations for 2 2bisimulation and neighbourhood bisimulation, and an analogue of Van Benthem’s characterisation theorem for all three equivalence notions. We also show that behavioural equivalence gives rise to a HennessyMilner theorem, and that this is not the case for the other two equivalence notions.
ON MINIMAL COALGEBRAS
"... Abstract. We define an outdegree for Fcoalgebras and show that the coalgebras of outdegree at most κ form a covariety. As a subcategory of all Fcoalgebras, this class has a terminal object, which for many problems can stand in for the terminal Fcoalgebra, which need not exist in general. As exam ..."
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Abstract. We define an outdegree for Fcoalgebras and show that the coalgebras of outdegree at most κ form a covariety. As a subcategory of all Fcoalgebras, this class has a terminal object, which for many problems can stand in for the terminal Fcoalgebra, which need not exist in general. As examples, we derive structure theoretic results about minimal coalgebras, showing that, for instance minimization of coalgebras is functorial, that products of finitely many minimal coalgebras exist and are given by their largest common subcoalgebra, that minimal subcoalgebras have no inner endomorphisms and show how minimal subcoalgebras can be constructed from Mooreautomata. Since the elements of minimal subcoalgebras must correspond uniquely to the formulae of any logic characterizing observational equivalence, we give in the last section a straightforward and selfcontained account of the coalgebraic logic of D. Pattinson and L. Schröder, which we believe is simpler and more direct than the original exposition. For every automaton A there exists a minimal automaton ∇(A), which displays
Notes on coalgebras, cofibrations and concurrency
 CMCS’2000
, 2000
"... We consider categories of coalgebras as (co)fibred over a base category of parameters and analyse categorical constructions in the total category of deterministic and nondeterministic coalgebras. ..."
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Cited by 3 (0 self)
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We consider categories of coalgebras as (co)fibred over a base category of parameters and analyse categorical constructions in the total category of deterministic and nondeterministic coalgebras.
Distributivity of Categories of Coalgebras
"... We prove that for any F the category of F coalgebras is distributive if F preserves preimages, i.e. pullbacks along an injective map, and that the converse is also true whenever has finite products. ..."
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Cited by 2 (2 self)
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We prove that for any F the category of F coalgebras is distributive if F preserves preimages, i.e. pullbacks along an injective map, and that the converse is also true whenever has finite products.
A Companion to Coalgebraic Weak Bisimulation for ActionType Systems
, 2009
"... We propose a coalgebraic definition of weak bisimulation for classes of coalgebras obtained from bifunctors in the category Set. Weak bisimilarity for a system is obtained as strong bisimilarity of a transformed system. The particular transformation consists of two steps: First, the behavior on acti ..."
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We propose a coalgebraic definition of weak bisimulation for classes of coalgebras obtained from bifunctors in the category Set. Weak bisimilarity for a system is obtained as strong bisimilarity of a transformed system. The particular transformation consists of two steps: First, the behavior on actions is lifted to behavior on finite words. Second, the behavior on finite words is taken modulo the hiding of internal or invisible actions, yielding behavior on equivalence classes of words closed under silent steps. The coalgebraic definition is validated by two correspondence results: one for the classical notion of weak bisimulation of Milner, another for the notion of weak bisimulation for generative probabilistic transition systems as advocated by Baier and Hermanns.
Presentation of set functors: a coalgebraic perspective
"... Abstract. Accessible set functors can be presented by signatures and equations as quotients of polynomial functors. We determine how preservation of pullbacks and other related properties (often applied in coalgebra) are re ected in the structure of the system of equations. 1. ..."
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Cited by 1 (1 self)
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Abstract. Accessible set functors can be presented by signatures and equations as quotients of polynomial functors. We determine how preservation of pullbacks and other related properties (often applied in coalgebra) are re ected in the structure of the system of equations. 1.
Comprehension for Coalgebras
 SIAM J. Matrix Anal. Appl
, 2002
"... The notion of an endofunctor having "greatest subcoalgebras" is introduced as a form of comprehension. This notion is shown to be instrumental in giving a systematic and abstract proof of the existence of limits for coalgebras  proved earlier by Worrell and by Gumm & Schroder. These ..."
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The notion of an endofunctor having "greatest subcoalgebras" is introduced as a form of comprehension. This notion is shown to be instrumental in giving a systematic and abstract proof of the existence of limits for coalgebras  proved earlier by Worrell and by Gumm & Schroder. These insights, in dual form, are used to reinvestigate colimits for algebras in terms of "least quotient algebras"  leading to a uniform approach to limits of coalgebras and colimits of algebras. Finally, at an abstract level of fibrations, an equivalence is established between having greatest subcoalgebras (in a base category of types) and greatest invariants (in a total category of predicates).