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12
Coalgebraic modal logic: Soundness, completeness and decidability of local consequence
 Theoret. Comput. Sci
, 2002
"... This paper studies finitary modal logics, interpreted over coalgebras for an endofunctor, and establishes soundness, completeness and decidability results. The logics are studied within the abstract framework of coalgebraic modal logic, which can be instantiated with arbitrary endofunctors on the ca ..."
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Cited by 51 (24 self)
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This paper studies finitary modal logics, interpreted over coalgebras for an endofunctor, and establishes soundness, completeness and decidability results. The logics are studied within the abstract framework of coalgebraic modal logic, which can be instantiated with arbitrary endofunctors on the category of sets. This is achieved through the use of predicate liftings, which generalise atomic propositions and modal operators from Kripke models to arbitrary coalgebras. Predicate liftings also allow us to use induction along the terminal sequence of the underlying endofunctor as a proof principle. This induction principle is systematically exploited to establish soundness, completeness and decidability of the logics. We believe that this induction principle also opens new ways for reasoning about modal logics: Our proof of completeness does not rely on a canonical model construction, and the proof of the finite model property does not use filtrations. 1
Coalgebraic Modal Logic of Finite Rank
, 2002
"... This paper studies coalgebras from the perspective of finite observations. We introduce the notion of finite step equivalence and a corresponding category with finite step equivalencepreserving morphisms. This category always has a final object, which generalises the canonical model construction fr ..."
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Cited by 14 (8 self)
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This paper studies coalgebras from the perspective of finite observations. We introduce the notion of finite step equivalence and a corresponding category with finite step equivalencepreserving morphisms. This category always has a final object, which generalises the canonical model construction from Kripke models to coalgebras. We then turn to logics whose formulae are invariant under finite step equivalence, which we call logics of rank . For these logics, we use topological methods and give a characterisation of compact logics and definable classes of models.
A Coalgebraic Foundation for Linear Time Semantics
 In Category Theory and Computer Science
, 1999
"... We present a coalgebraic approach to trace equivalence semantics based on lifting behaviour endofunctors for deterministic action to Kleisli categories of monads for nondeterministic choice. In Set , this gives a category with ordinary transition systems as objects and with morphisms characterised ..."
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Cited by 14 (1 self)
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We present a coalgebraic approach to trace equivalence semantics based on lifting behaviour endofunctors for deterministic action to Kleisli categories of monads for nondeterministic choice. In Set , this gives a category with ordinary transition systems as objects and with morphisms characterised in terms of a linear notion of bisimulation. The final object in this category is the canonical abstract model for trace equivalence and can be obtained by extending the final coalgebra of the deterministic action behaviour to the Kleisli category of the nonempty powerset monad. The corresponding final coalgebra semantics is fully abstract with respect to trace equivalence.
An Intrinsic Characterization of Approximate Probabilistic Bisimilarity
 In: Proceedings of FOSSACS 03. LNCS
, 2003
"... Abstract. In previous work we have investigated a notion of approximate bisimilarity for labelled Markov processes. We argued that such a notion is more realistic and more feasible to compute than (exact) bisimilarity. The main technical tool used in the underlying theory was the Hutchinson metric o ..."
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Cited by 13 (2 self)
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Abstract. In previous work we have investigated a notion of approximate bisimilarity for labelled Markov processes. We argued that such a notion is more realistic and more feasible to compute than (exact) bisimilarity. The main technical tool used in the underlying theory was the Hutchinson metric on probability measures. This paper gives a more fundamental characterization of approximate bisimilarity in terms of the notion of (exact) similarity. In particular, we show that the topology of approximate bisimilarity is the Lawson topology with respect to the simulation preorder. To complement this abstract characterization we give a statistical account of similarity, and by extension, of approximate bisimilarity, in terms of the process testing formalism of Larsen and Skou. 1
A Small Final Coalgebra Theorem
 Theoretical Computer Science
, 1998
"... This paper presents an elementary and selfcontained proof of an existence theorem of final coalgebras for endofunctors on the category of sets and functions. ..."
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Cited by 7 (0 self)
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This paper presents an elementary and selfcontained proof of an existence theorem of final coalgebras for endofunctors on the category of sets and functions.
On iterable endofunctors
 Category Theory and Computer Science 2002, number 69 in Elect. Notes in Theor. Comp. Sci
, 2003
"... Completely iterative monads of Elgot et al. are the monads such that every guarded iterative equation has a unique solution. Free completely iterative monads are known to exist on every iteratable endofunctor H, i. e., one with final coalgebras of all functors H ( ) + X. We show that conversely, if ..."
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Cited by 3 (0 self)
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Completely iterative monads of Elgot et al. are the monads such that every guarded iterative equation has a unique solution. Free completely iterative monads are known to exist on every iteratable endofunctor H, i. e., one with final coalgebras of all functors H ( ) + X. We show that conversely, if H generates a free completely iterative monad, then it is iteratable. Key words: monad, completely iterative, iterable 1
Functorial coalgebraic logic: The case of manysorted varieties
 Electron. Notes Theor. Comput. Sci
"... Following earlier work, a modal logic for Tcoalgebras is a functor L on a suitable variety. Syntax and proof system of the logic are given by presentations of the functor. This paper makes two contributions. First, a previous result characterizing those functors that have presentations is generaliz ..."
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Cited by 3 (2 self)
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Following earlier work, a modal logic for Tcoalgebras is a functor L on a suitable variety. Syntax and proof system of the logic are given by presentations of the functor. This paper makes two contributions. First, a previous result characterizing those functors that have presentations is generalized from endofunctors on onesorted varieties to functors between manysorted varieties. This yields an equational logic for the presheaf semantics of higherorder abstract syntax. As another application, we show how the move to functors between manysorted varieties allows to modularly combine syntax and proof systems of different logics. Second, we show how to associate to any setfunctor T a complete (finitary) logic L consisting of modal operators and Boolean connectives.
Completeness of the finitary Moss logic
 In Areces and Goldblatt [3
"... abstract. We give a sound and complete derivation system for the valid formulas in the finitary version of Moss ’ coalgebraic logic, for coalgebras of arbitrary type. ..."
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Cited by 2 (1 self)
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abstract. We give a sound and complete derivation system for the valid formulas in the finitary version of Moss ’ coalgebraic logic, for coalgebras of arbitrary type.
Talk I: Final Coalgebras
, 2003
"... For the remainder of this exposition assume that T: Set! Set is an endofunctor. Unless otherwise stated, the results and proofs of the material presented is taken from Rutten [10]. 1 Preliminaries Deo/nition 1.1. (i) A Tcoalgebra (Tsystem) is pair (C; fl) where C is a set and fl: C! T C. (ii) If ( ..."
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For the remainder of this exposition assume that T: Set! Set is an endofunctor. Unless otherwise stated, the results and proofs of the material presented is taken from Rutten [10]. 1 Preliminaries Deo/nition 1.1. (i) A Tcoalgebra (Tsystem) is pair (C; fl) where C is a set and fl: C! T C. (ii) If (C; fl) and (D; ffi) are Tsystems, then f: C! D is a Tcoalgebra homomorphism (Tmorphism), if ffi ffi f = T f ffi fl, that is, if the diagram C fl fflffl f // D
Semantic Principles in the . . .
, 2001
"... Coalgebras for a functor on the category of sets subsume many formulations of the notion of transition system, including labelled transition systems, Kripke models, Kripke frames and many types of automata. This paper presents a multimodal language which is bisimulation invariant and (under a natur ..."
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Coalgebras for a functor on the category of sets subsume many formulations of the notion of transition system, including labelled transition systems, Kripke models, Kripke frames and many types of automata. This paper presents a multimodal language which is bisimulation invariant and (under a natural completeness condition) expressive enough to characterise elements of the underlying state space up to bisimulation. Like Moss' coalgebraic logic, the theory can be applied to an arbitrary signature functor on the category of sets. Also, an upper bound for the size of conjunctions and disjunctions needed to obtain characteristic formulas is given.