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Expressivity of coalgebraic modal logic: The limits and beyond
 IN FOUNDATIONS OF SOFTWARE SCIENCE AND COMPUTATION STRUCTURES, VOLUME 3441 OF LNCS
, 2005
"... Modal logic has a good claim to being the logic of choice for describing the reactive behaviour of systems modeled as coalgebras. Logics with modal operators obtained from socalled predicate liftings have been shown to be invariant under behavioral equivalence. Expressivity results stating that, c ..."
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Cited by 39 (13 self)
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Modal logic has a good claim to being the logic of choice for describing the reactive behaviour of systems modeled as coalgebras. Logics with modal operators obtained from socalled predicate liftings have been shown to be invariant under behavioral equivalence. Expressivity results stating that, conversely, logically indistinguishable states are behaviorally equivalent depend on the existence of separating sets of predicate liftings for the signature functor at hand. Here, we provide a classification result for predicate liftings which leads to an easy criterion for the existence of such separating sets, and we give simple examples of functors that fail to admit expressive normal or monotone modal logics, respectively, or in fact an expressive (unary) modal logic at all. We then move on to polyadic modal logic, where modal operators may take more than one argument formula. We show that every accessible functor admits an expressive polyadic modal logic. Moreover, expressive polyadic modal logics are, unlike unary modal logics, compositional.
A Hierarchy of Probabilistic System Types
, 2003
"... We study various notions of probabilistic bisimulation from a coalgebraic point of view, accumulating in a hierarchy of probabilistic system types. In general, a natural transformation between two Setfunctors straightforwardly gives rise to a transformation of coalgebras for the respective functors ..."
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Cited by 37 (6 self)
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We study various notions of probabilistic bisimulation from a coalgebraic point of view, accumulating in a hierarchy of probabilistic system types. In general, a natural transformation between two Setfunctors straightforwardly gives rise to a transformation of coalgebras for the respective functors. This latter transformation preserves homomorphisms and thus bisimulations. For comparison of probabilistic system types we also need reflection of bisimulation. We build the hierarchy of probabilistic systems by exploiting the new result that the transformation also reflects bisimulation in case the natural transformation is componentwise injective and the first functor preserves weak pullbacks. Additionally, we illustrate the correspondence of concrete and coalgebraic bisimulation in the case of general Segalatype systems.
Algebraiccoalgebraic specification in CoCasl
 J. LOGIC ALGEBRAIC PROGRAMMING
, 2006
"... We introduce CoCasl as a simple coalgebraic extension of the algebraic specification language Casl. CoCasl allows the nested combination of algebraic datatypes and coalgebraic process types. We show that the wellknown coalgebraic modal logic can be expressed in CoCasl. We present sufficient criter ..."
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Cited by 19 (8 self)
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We introduce CoCasl as a simple coalgebraic extension of the algebraic specification language Casl. CoCasl allows the nested combination of algebraic datatypes and coalgebraic process types. We show that the wellknown coalgebraic modal logic can be expressed in CoCasl. We present sufficient criteria for the existence of cofree models, also for several variants of nested cofree and free specifications. Moreover, we describe an extension of the existing proof support for Casl (in the shape of an encoding into higherorder logic) to CoCasl.
State Based Systems Are Coalgebras
 Cubo  Matematica Educacional 5
, 2003
"... Universal coalgebra is a mathematical theory of state based systems, which in many respects is dual to universal algebra. Equality must be replaced by indistinguishability. Coinduction replaces induction as a proof principle and maps are defined by corecursion. In this (entirely selfcontained) pap ..."
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Cited by 2 (0 self)
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Universal coalgebra is a mathematical theory of state based systems, which in many respects is dual to universal algebra. Equality must be replaced by indistinguishability. Coinduction replaces induction as a proof principle and maps are defined by corecursion. In this (entirely selfcontained) paper we give a first glimpse at the general theory and focus on some applications in Computer Science. 1. State based systems State based systems can be found everywhere in our environment  from simple appliances like alarm clocks and answering machines to sophisticated computing devices. Typically, such systems receive some input and, as a result, produce some output. In contrast to purely algebraic systems, however, the output is not only determined by the input received, but also by some modifiable "internal state". Internal states are usually not directly observable, so there may as well be di#erent states that cannot be distinguished from the inputoutput behavior of the system. A simple example of a state based system is a digital watch with several buttons and a display. Clearly, the buttons that are pressed do not by themselves determine the output  it also depends on the internal state, which might include the current time, the mode (time/alarm/stopwatch), and perhaps the information which buttons have been pressed previously. The user of a system is normally not interested in knowing precisely, what the internal states of the system are, nor how they are represented. Of course, he might try to infer all possible states by testing various inputoutput combinations and attribute di#erent behaviors to di#erent states. Some states might not be distinguishable by their outside behavior. It is therefore natural to define an appropriate indistinguishability relation "#...
Equational Coalgebraic Logic
 MFPS
, 2009
"... Coalgebra develops a general theory of transition systems, parametric in a functor T; the functor T specifies the possible onestep behaviours of the system. A fundamental question in this area is how to obtain, for an arbitrary functor T, a logic for Tcoalgebras. We compare two existing proposals, ..."
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Cited by 1 (1 self)
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Coalgebra develops a general theory of transition systems, parametric in a functor T; the functor T specifies the possible onestep behaviours of the system. A fundamental question in this area is how to obtain, for an arbitrary functor T, a logic for Tcoalgebras. We compare two existing proposals, Moss’s coalgebraic logic and the logic of all predicate liftings, by providing onestep translations between them, extending the results in [21] by making systematic use of Stone duality. Our main contribution then is a novel coalgebraic logic, which can be seen as an equational axiomatization of Moss’s logic. The three logics are equivalent for a natural but restricted class of functors. We give examples showing that the logics fall apart in general. Finally, we argue that the quest for a generic logic for Tcoalgebras is still open in the general case.
UNIVERSAL COALGEBRAS AND THEIR LOGICS
, 2009
"... ABSTRACT. We survey coalgebras as models of state based systems together with their global and local logics. We convey some useful intuition regarding Setfunctors which leads naturally to coalgebraic modal logic where modalities are validity patterns for the successor object of a state. 1. ..."
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ABSTRACT. We survey coalgebras as models of state based systems together with their global and local logics. We convey some useful intuition regarding Setfunctors which leads naturally to coalgebraic modal logic where modalities are validity patterns for the successor object of a state. 1.
Expressiveness of Positive Coalgebraic Logic
"... From the point of view of modal logic, coalgebraic logic over posets is the natural coalgebraic generalisation of positive modal logic. From the point of view of coalgebra, posets arise if one is interested in simulations as opposed to bisimulations. From a categorical point of view, one moves from ..."
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From the point of view of modal logic, coalgebraic logic over posets is the natural coalgebraic generalisation of positive modal logic. From the point of view of coalgebra, posets arise if one is interested in simulations as opposed to bisimulations. From a categorical point of view, one moves from ordinary categories to enriched categories. We show that the basic setup of coalgebraic logic extends to this more general setting and that every finitary functor on posets has a logic that is expressive, that is, has the HennessyMilner property. Keywords: Coalgebra, Modal Logic, Poset
COINDUCTIVE PROPERTIES OF LIPSCHITZ FUNCTIONS ON STREAMS
, 809
"... Abstract. A simple hierarchical structure is imposed on the set of Lipschitz functions on streams (i.e. sequences over a fixed alphabet set) under the standard metric. We prove that sets of nonexpanding and contractive functions are closed under a certain coiterative construction. The closure prope ..."
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Abstract. A simple hierarchical structure is imposed on the set of Lipschitz functions on streams (i.e. sequences over a fixed alphabet set) under the standard metric. We prove that sets of nonexpanding and contractive functions are closed under a certain coiterative construction. The closure property is used to construct new final stream coalgebras over finite alphabets. For an example, we show that the 2adic extension of the Collatz function and certain variants yield final bitstream coalgebras. 1.