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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 Study of Categories of Algebras and Coalgebras
, 2001
"... This thesis is intended to help develop the theory of coalgebras by, first, taking classic theorems in the theory of universal algebras and dualizing them and, second, developing an internal logic for categories of coalgebras. We begin with an introduction to the categorical approach to algebras and ..."
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Cited by 13 (5 self)
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This thesis is intended to help develop the theory of coalgebras by, first, taking classic theorems in the theory of universal algebras and dualizing them and, second, developing an internal logic for categories of coalgebras. We begin with an introduction to the categorical approach to algebras and the dual notion of coalgebras. Following this, we discuss (co)algebras for a (co)monad and develop a theory of regular subcoalgebras which will be used in the internal logic. We also prove that categories of coalgebras are complete, under reasonably weak conditions, and simultaneously prove the wellknown dual result for categories of algebras. We close the second chapter with a discussion of bisimulations in which we introduce a weaker notion of bisimulation than is current in the literature, but which is wellbehaved and reduces to the standard definition under the assumption of choice. The third chapter is a detailed look at three theorem's of G. Birkho# [Bir35, Bir44], presenting categorical proofs of the theorems which generalize the classical results and which can be easily dualized to apply to categories of coalgebras. The theorems of interest are the variety theorem, the equational completeness theorem and the subdirect product representation theorem. The duals of each of these theorems is discussed in detail, and the dual notion of "coequation" is introduced and several examples given. In the final chapter, we show that first order logic can be interpreted in categories of coalgebras and introduce two modal operators to first order logic to allow reasoning about "endomorphisminvariant" coequations and bisimulations internally. We also develop a translation of terms and formulas into the internal language of the base category, which preserves and reflects truth. La...
On Tree Coalgebras and Coalgebra Presentations
, 2002
"... For deterministic systems, expressed as coalgebras over polynomial functors, every tree t (an element of the final coalgebra) turns out to represent a new coalgebra A t . The universal property of these coalgebras, resembling freeness, is that for every state s of every system S there exists a uniqu ..."
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Cited by 7 (1 self)
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For deterministic systems, expressed as coalgebras over polynomial functors, every tree t (an element of the final coalgebra) turns out to represent a new coalgebra A t . The universal property of these coalgebras, resembling freeness, is that for every state s of every system S there exists a unique coalgebra homomorphism from a unique A t which takes the root of t to s. Moreover, the tree coalgebras are finitely presentable and form a strong generator. Thus, these categories of coalgebras are locally finitely presentable; in particular every system is a filtered colimit of finitely presentable systems.
Algebras, Coalgebras, Monads and Comonads
, 2001
"... Whilst the relationship between initial algebras and monads is wellunderstood, the relationship between nal coalgebras and comonads is less well explored. This paper shows that the problem is more subtle and that final coalgebras can just as easily form monads as comonads and dually, that initial a ..."
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Cited by 6 (3 self)
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Whilst the relationship between initial algebras and monads is wellunderstood, the relationship between nal coalgebras and comonads is less well explored. This paper shows that the problem is more subtle and that final coalgebras can just as easily form monads as comonads and dually, that initial algebras form both monads and comonads. In developing these theories we strive to provide them with an associated notion of syntax. In the case of initial algebras and monads this corresponds to the standard notion of algebraic theories consisting of signatures and equations: models of such algebraic theories are precisely the algebras of the representing monad. We attempt to emulate this result for the coalgebraic case by defining a notion cosignature and coequation and then proving the models of this syntax are precisely the coalgebras of the representing comonad.
On specification logics for algebracoalgebra structures: Reconciling reachability and observability
 Proceedings of FoSSaCS 2002, volume 2303 of LNCS
, 2002
"... Abstract. The paper builds on recent results regarding the expressiveness of modal logics for coalgebras in order to introduce a specification framework for coalgebraic structures which offers support for modular specification. An equational specification framework for algebraic structures is obtain ..."
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Cited by 5 (0 self)
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Abstract. The paper builds on recent results regarding the expressiveness of modal logics for coalgebras in order to introduce a specification framework for coalgebraic structures which offers support for modular specification. An equational specification framework for algebraic structures is obtained in a similar way. The two frameworks are then integrated in order to account for structures comprising both a coalgebraic (observational) component and an algebraic (computational) component. The integration results in logics whose sentences are either coalgebraic (modal) or algebraic (equational) in nature, but whose associated notions of satisfaction take into account both the coalgebraic and the algebraic features of the structures being specified. Each of the logics thus obtained also supports modular specification. 1
Coalgebras and modal logics for parameterised endofunctors
, 2000
"... for promotion of mathematics and computer science and their applications. SMC is sponsored by the Netherlands Organization for Scientific Research (NWO). CWI is a member of ..."
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Cited by 3 (3 self)
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for promotion of mathematics and computer science and their applications. SMC is sponsored by the Netherlands Organization for Scientific Research (NWO). CWI is a member of
The Subobject Classifier of the Category of Functional Bisimulations
, 1998
"... We show the existence of subobject classifier in the category of nondeterministic dynamical systems and functional bisimulations. Keywords: nondeterministic dynamical system, functional bisimulation, coalgebra, subobject classifier, dense, AMS Classification: 18B20, 68Q10, 18B25 1 Introduction In [ ..."
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Cited by 2 (2 self)
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We show the existence of subobject classifier in the category of nondeterministic dynamical systems and functional bisimulations. Keywords: nondeterministic dynamical system, functional bisimulation, coalgebra, subobject classifier, dense, AMS Classification: 18B20, 68Q10, 18B25 1 Introduction In [8], we studied the category NDyn of nondeterministic dynamical systems whose morphisms are functional bisimulations. 3 This work was done when the author was in Department of Mathematics, Hokkaido University. A nondeterministic dynamical system is a labelled transition system whose label set has only one element. A functional bisimulation is a map between transition systems. The main results of [8] are the following. ffl The category NDyn is an autonomous category, i.e., monoidal closed. ffl There exists a subobject classifier. The monoidal closedness was shown by constructing NDyn objects via the presheaves over the category Tree, where the Tree is a small, dense subcategory of NDyn. ...
Universality of coproducts in categories of lax algebras
 Appl. Categ. Structures
"... Abstract. Categories of lax (T, V)algebras are shown to have pullbackstable coproducts if T preserves inverse images. The general result not only gives a common proof of this property in many topological categories but also shows that important topological categories, like the ..."
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Cited by 1 (0 self)
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Abstract. Categories of lax (T, V)algebras are shown to have pullbackstable coproducts if T preserves inverse images. The general result not only gives a common proof of this property in many topological categories but also shows that important topological categories, like the