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Modular construction of modal logics
 Concurrency Theory, CONCUR 04, volume 3170 of Lect. Notes Comput. Sci
, 2004
"... Abstract. We present a modular approach to defining logics for a wide variety of statebased systems. We use coalgebras to model the behaviour of systems, and modal logics to specify behavioural properties of systems. We show that the syntax, semantics and proof systems associated to such logics can ..."
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Cited by 26 (8 self)
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Abstract. We present a modular approach to defining logics for a wide variety of statebased systems. We use coalgebras to model the behaviour of systems, and modal logics to specify behavioural properties of systems. We show that the syntax, semantics and proof systems associated to such logics can all be derived in a modular way. Moreover, we show that the logics thus obtained inherit soundness, completeness and expressiveness properties from their building blocks. We apply these techniques to derive sound, complete and expressive logics for a wide variety of probabilistic systems. 1
Coalgebraic epistemic update without change of model
, 2007
"... We present a coalgebraic semantics for reasoning about information update in multiagent systems. The novelty is that we have one structure for both states and actions and thus our models do not involve the ”changeofmodel” phenomena that arise when using Kripke models. However, we prove that the u ..."
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Cited by 4 (0 self)
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We present a coalgebraic semantics for reasoning about information update in multiagent systems. The novelty is that we have one structure for both states and actions and thus our models do not involve the ”changeofmodel” phenomena that arise when using Kripke models. However, we prove that the usual models can be constructed from ours by categorical adjunction. The generality and abstraction of our coalgebraic model turns out to be extremely useful in proving preservation properties of update. In particular, we prove that positive knowledge is preserved and acquired as a result of epistemic update. We also prove common and nested knowledge properties of epistemic updates induced by specific epistemic actions such as public and private announcements, lying, and in particular unsafe actions of security protocols. Our model directly gives rise to a coalgebraic logic with both dynamic and epistemic modalities. We prove a soundness and completeness result for this logic, and illustrate the applicability of the logic by deriving knowledge properties of a simple security protocol.
Coalgebras, Stone Duality, Modal Logic
, 2006
"... A brief outline of the topics of the course could be as follows. Coalgebras generalise transition systems. Modal logics are the natural logics for coalgebras. Stone duality provides the relationship between coalgebras and modal logic. Furthermore, some basic category theory is needed to understand c ..."
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A brief outline of the topics of the course could be as follows. Coalgebras generalise transition systems. Modal logics are the natural logics for coalgebras. Stone duality provides the relationship between coalgebras and modal logic. Furthermore, some basic category theory is needed to understand coalgebras as well as Stone duality. So we
A Compositional Approach to Defining Logics for Coalgebras
"... We present a compositional approach to defining expressive logics for coalgebras of endofunctors on Set. This approach uses a notion of language constructor and an associated notion of semantics to capture one inductive step in the definition of a language for coalgebras and of its semantics. We sho ..."
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We present a compositional approach to defining expressive logics for coalgebras of endofunctors on Set. This approach uses a notion of language constructor and an associated notion of semantics to capture one inductive step in the definition of a language for coalgebras and of its semantics. We show that suitable choices for the language constructors and for their associated semantics yield logics which are both adequate and expressive w.r.t. behavioural equivalence. Moreover, we show that typebuilding operations give rise to corresponding operations both on language constructors and on their associated semantics, thus allowing the derivation of expressive logics for increasingly complex coalgebraic types. Our framework subsumes several existing approaches to defining logics for coalgebras, and at the same time allows the derivation of new logics, with logics for probabilistic systems being the prime example.
CMCS’04 Preliminary Version Algebraic Semantics for Coalgebraic Logics
"... With coalgebras usually being defined in terms of an endofunctor T on sets, this paper shows that modal logics for Tcoalgebras can be naturally described as functors L on boolean algebras. Building on this idea, we study soundness, completeness and expressiveness of coalgebraic logics from the pers ..."
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With coalgebras usually being defined in terms of an endofunctor T on sets, this paper shows that modal logics for Tcoalgebras can be naturally described as functors L on boolean algebras. Building on this idea, we study soundness, completeness and expressiveness of coalgebraic logics from the perspective of duality theory. That is, given a logic L for coalgebras of an endofunctor T, we construct an endofunctor L such that Lalgebras provide a sound and complete (algebraic) semantics of the logic. We show that if L is dual to T, then soundness and completeness of the algebraic semantics immediately yield the corresponding property of the coalgebraic semantics. We conclude by characterising duality between L and T in terms of the axioms of L. This provides a criterion for proving concretely given logics to be sound, complete and expressive. 1
Abstract CMCS’04 Preliminary Version On Logics for Coalgebraic Simulation
"... We investigate logics for coalgebraic simulation from a compositional perspective. Specifically, we show that the expressiveness of an inductivelydefined language for coalgebras w.r.t. a given notion of simulation comes as a consequence of an expressivity condition between the language constructor ..."
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We investigate logics for coalgebraic simulation from a compositional perspective. Specifically, we show that the expressiveness of an inductivelydefined language for coalgebras w.r.t. a given notion of simulation comes as a consequence of an expressivity condition between the language constructor used to define the language for coalgebras, and the relator used to define the notion of simulation. This result can be instantiated to obtain Baltag’s logics for coalgebraic simulation, as well as a logic which captures simulation on unlabelled probabilistic transition systems. Moreover, our approach is compositional w.r.t. coalgebraic types. This allows us to derive logics which capture other notions of simulation, including trace inclusion on labelled transition systems, and simulation on discrete Markov processes. Key words: coalgebra, simulation, modal logic 1
Abstract A Modular Approach to Defining and Characterising Notions of Simulation
"... We propose a modular approach to defining notions of simulation, and modal logics which characterise them. We use coalgebras to model statebased systems, relators to define notions of simulation for such systems, and inductive techniques to define the syntax and semantics of modal logics for coalge ..."
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We propose a modular approach to defining notions of simulation, and modal logics which characterise them. We use coalgebras to model statebased systems, relators to define notions of simulation for such systems, and inductive techniques to define the syntax and semantics of modal logics for coalgebras. We show that the expressiveness of an inductivelydefined logic for coalgebras w.r.t. a notion of simulation follows from an expressivity condition involving one step in the definition of the logic, and the relator inducing that notion of simulation. Moreover, we show that notions of simulation and associated characterising logics for increasingly complex system types can be derived by lifting the operations used to combine system types, to a relational level as well as to a logical level. We use these results to obtain Baltag’s logic for coalgebraic simulation, as well as notions of simulation and associated logics for a large class of nondeterministic and probabilistic systems. Key words: coalgebra, simulation, modal logic, probabilistic system 1
Coalgebras and Their Logics 1
, 2006
"... Some comments about the last Logic Column, on nominal logic. Pierre Lescanne points out ..."
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Some comments about the last Logic Column, on nominal logic. Pierre Lescanne points out