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With coalgebras usually being defined in terms of an endofunctor T on sets, this paper shows that modal logics for T-coalgebras 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… (More)

In this paper we argue that the category of Stone spaces forms an interesting base category for coalgebras, in particular, if one considers the Vietoris functor as an analogue to the power set functor on the category of sets. We prove that the so-called descriptive general frames, which play a fundamental role in the semantics of modal logics, can be seen… (More)

We show how to use duality theory to construct minimized versions of a wide class of automata. We work out three cases in detail: (a variant of) ordinary au-tomata, weighted automata and probabilistic automata. The basic idea is that instead of constructing a maximal quotient we go to the dual and look for a minimal subalge-bra and then return to the… (More)

Answering conjunctive queries (CQs) over a set of facts extended with existential rules is a prominent problem in knowledge representation and databases. This problem can be solved using the chase algorithm, which extends the given set of facts with fresh facts in order to satisfy the rules. If the chase terminates, then CQs can be evaluated directly in the… (More)

- Bernardo Cuenca Grau, Ian Horrocks, Markus Krötzsch, Clemens Kupke, Despoina Magka, Boris Motik +1 other
- KR
- 2012

Answering conjunctive queries (CQs) over a set of facts extended with existential rules is a key problem in knowledge representation and databases. This problem can be solved using the chase (aka materialisation) algorithm; however, CQ answering is undecidable for general existential rules, so the chase is not guaranteed to terminate. Several acyclicity… (More)

It has been recognised that the expressivity of description logics benefits from the introduction of non-standard modal operators beyond existential and number restrictions. Such operators support notions such as uncertainty, defaults, agency, obligation , or evidence, whose semantics often lies outside the realm of relational structures. Coalgebraic hybrid… (More)

Coalgebras can be seen as a natural abstraction of Kripke frames. In the same sense, coalge-braic logics are generalised modal logics. In this paper, we give an overview of the basic tools, techniques and results that connect coalgebras and modal logic. We argue that coalgebras unify the semantics of a large range of different modal logics (such as… (More)

We present a (co)algebraic treatment of iteration-free dynamic modal logics such as Propositional Dynamic Logic (PDL) and Game Logic (GL), both without star. The main observation is that the program/game constructs of PDL/GL arise from monad structure, and the axioms of these logics correspond to certain compatibilty requirements between the modalities and… (More)

We generalize some of the central results in automata theory to the abstraction level of coalgebras and thus lay out the foundations of a universal theory of automata operating on infinite objects. Let F be any set functor that preserves weak pullbacks. We show that the class of recognizable languages of F-coalgebras is closed under taking unions,… (More)