61 research outputs found
Moss' logic for ordered coalgebras
We present a finitary coalgebraic logic for -coalgebras, where is a
locally monotone endofunctor of the category of posets and monotone maps that
preserves exact squares and finite intersections. The logic uses a single cover
modality whose arity is given by the dual of the coalgebra functor , and the
semantics of the modality is given by relation lifting. For the finitary
setting to work, we need to develop a notion of a base for subobjects of .
This in particular allows us to talk about a finite poset of subformulas for a
given formula, and of a finite poset of successors for a given state in a
coalgebra. The notion of a base is introduced generally for a category equipped
with a suitable factorisation system.
We prove that the resulting logic has the Hennessy-Milner property for the
notion of similarity based on the notion of relation lifting. We define a
sequent proof system for the logic and prove its completeness
Graded Monads and Graded Logics for the Linear Time - Branching Time Spectrum
State-based models of concurrent systems are traditionally considered under a variety of notions of process equivalence. In the case of labelled transition systems, these equivalences range from trace equivalence to (strong) bisimilarity, and are organized in what is known as the linear time - branching time spectrum. A combination of universal coalgebra and graded monads provides a generic framework in which the semantics of concurrency can be parametrized both over the branching type of the underlying transition systems and over the granularity of process equivalence. We show in the present paper that this framework of graded semantics does subsume the most important equivalences from the linear time - branching time spectrum. An important feature of graded semantics is that it allows for the principled extraction of characteristic modal logics. We have established invariance of these graded logics under the given graded semantics in earlier work; in the present paper, we extend the logical framework with an explicit propositional layer and provide a generic expressiveness criterion that generalizes the classical Hennessy-Milner theorem to coarser notions of process equivalence. We extract graded logics for a range of graded semantics on labelled transition systems and probabilistic systems, and give exemplary proofs of their expressiveness based on our generic criterion
Coalgebraic Geometric Logic: Basic Theory
Using the theory of coalgebra, we introduce a uniform framework for adding
modalities to the language of propositional geometric logic. Models for this
logic are based on coalgebras for an endofunctor on some full subcategory of
the category of topological spaces and continuous functions. We investigate
derivation systems, soundness and completeness for such geometric modal logics,
and we we specify a method of lifting an endofunctor on Set, accompanied by a
collection of predicate liftings, to an endofunctor on the category of
topological spaces, again accompanied by a collection of (open) predicate
liftings. Furthermore, we compare the notions of modal equivalence, behavioural
equivalence and bisimulation on the resulting class of models, and we provide a
final object for the corresponding category
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