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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 21 (14 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 8 (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.
Ranl1 Modal Logics are Coalgebraic
"... 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 coalg ..."
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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. This is achieved by constructing for a given modal logic a canonical coalgebraic semantics, consisting of a signature functor and interpretations of modal operators, which turns out to be final among all such structures. The canonical semantics may be seen as a coalgebraic reconstruction of neighbourhood semantics, broadly construed. A finitary restriction of the canonical semantics yields a canonical weakly complete semantics which moreover enjoys the HennessyMilner property. As a consequence, the machinery of coalgebraic modal logic, in particular generic decision procedures and upper complexity bounds, becomes applicable to arbitrary rank 1 modal logics, without regard to their semantic status; we thus obtain purely syntactic versions of such results. As an extended example, we apply our framework to recently defined deontic logics. In particular, our methods lead to the new result that these logics are strongly complete.
doi:10.1017/S0960129506005706 Printed in the United Kingdom Iterative algebras at work
, 2004
"... Iterative theories, which were introduced by Calvin Elgot, formalise potentially infinite computations as unique solutions of recursive equations. One of the main results of Elgot and his coauthors is a description of a free iterative theory as the theory of all rational trees. Their algebraic proof ..."
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Iterative theories, which were introduced by Calvin Elgot, formalise potentially infinite computations as unique solutions of recursive equations. One of the main results of Elgot and his coauthors is a description of a free iterative theory as the theory of all rational trees. Their algebraic proof of this fact is extremely complicated. In our paper we show that by starting with ‘iterative algebras’, that is, algebras admitting a unique solution of all systems of flat recursive equations, a free iterative theory is obtained as the theory of free iterative algebras. The (coalgebraic) proof we present is dramatically simpler than the original algebraic one. Despite this, our result is much more general: we describe a free iterative theory on any finitary endofunctor of every locally presentable category A. Reportedly, a blow from the welterweight boxer Norman Selby, also known as Kid McCoy, left one victim proclaiming, ‘It’s the real McCoy! ’.¶ 1.
HΣX = �
, 2003
"... It is wellknown that in the category CoAlgF of Fcoalgebras for a given endofunctor F: Set − → Set the terminal object can be constructed as a limit of a certain descending chain. In case of polynomial functors F = HΣ for bounded signatures Σ, this limitobject in the corresponding category CoAlgΣ i ..."
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It is wellknown that in the category CoAlgF of Fcoalgebras for a given endofunctor F: Set − → Set the terminal object can be constructed as a limit of a certain descending chain. In case of polynomial functors F = HΣ for bounded signatures Σ, this limitobject in the corresponding category CoAlgΣ is interpreted as the set T of all Σlabelled trees with ”treedetupling ” dynamic θ in [1]. In the following we give a direct and more intuitive proof of this fact, and also a direct description of the unique homomorphism (A, α) − → (T, θ) for a Σcoalgebra (A, α). Let λ be a cardinal, considered as the set of all smaller ordinals, Σ = � n<λ Σn be a signature, bounded by λ and HΣ: Set − → Set the corresponding polynomial functor, i. e.
hyperproperties: extended version
, 2011
"... A hyperproperty is a set of sets of finite or infinite traces over some fixed alphabet and can be seen as a very generic system specification. In this work, we define the notions of holistic and incremental hyperproperties. Systems specified holistically tend to be more intuitive but difficult to re ..."
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A hyperproperty is a set of sets of finite or infinite traces over some fixed alphabet and can be seen as a very generic system specification. In this work, we define the notions of holistic and incremental hyperproperties. Systems specified holistically tend to be more intuitive but difficult to reason about, whereas incremental specifications have a straightforward verification approach. Since most interesting securityrelated hyperproperties are in the syntactic class of holistic hyperproperties, we introduce the process of incrementalization to convert holistic specifications into incremental ones. We then present three incrementalizable classes of holistic hyperproperties and a respective verification method.
Contents
, 2013
"... Abstract. In previous work with Pous, we defined a semantics for CCS which may both be viewed as an innocent presheaf semantics and as a concurrent game semantics. It is here proved that a behavioural equivalence induced by this semantics on CCS processes is fully abstract for fair testing equivalen ..."
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Abstract. In previous work with Pous, we defined a semantics for CCS which may both be viewed as an innocent presheaf semantics and as a concurrent game semantics. It is here proved that a behavioural equivalence induced by this semantics on CCS processes is fully abstract for fair testing equivalence. The proof relies on a new algebraic notion called playground, which represents the ‘rule of the game’. From any playground, two languages, equipped with labelled transition systems, are derived, as well as a strong, functional
CORECURSIVE ALGEBRAS, CORECURSIVE MONADS
, 2012
"... Vol. 10(3:19)2014, pp. 1–51 www.lmcsonline.org ..."
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SEMANTICS OF HIGHERORDER RECURSION SCHEMES
, 2010
"... Vol. 7 (1:15) 2011, pp. 1–43 ..."
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