6,836 research outputs found
Model Checking Spatial Logics for Closure Spaces
Spatial aspects of computation are becoming increasingly relevant in Computer
Science, especially in the field of collective adaptive systems and when
dealing with systems distributed in physical space. Traditional formal
verification techniques are well suited to analyse the temporal evolution of
programs; however, properties of space are typically not taken into account
explicitly. We present a topology-based approach to formal verification of
spatial properties depending upon physical space. We define an appropriate
logic, stemming from the tradition of topological interpretations of modal
logics, dating back to earlier logicians such as Tarski, where modalities
describe neighbourhood. We lift the topological definitions to the more general
setting of closure spaces, also encompassing discrete, graph-based structures.
We extend the framework with a spatial surrounded operator, a propagation
operator and with some collective operators. The latter are interpreted over
arbitrary sets of points instead of individual points in space. We define
efficient model checking procedures, both for the individual and the collective
spatial fragments of the logic and provide a proof-of-concept tool
Modal logic of planar polygons
We study the modal logic of the closure algebra , generated by the set
of all polygons in the Euclidean plane . We show that this logic
is finitely axiomatizable, is complete with respect to the class of frames we
call "crown" frames, is not first order definable, does not have the Craig
interpolation property, and its validity problem is PSPACE-complete
Infinitesimal and local convexity of a hypersurface in a semi-Riemannian manifold
Given a Riemannian manifold M and a hypersurface H in M, it is well known
that infinitesimal convexity on a neighborhood of a point in H implies local
convexity. We show in this note that the same result holds in a semi-Riemannian
manifold. We make some remarks for the case when only timelike, null or
spacelike geodesics are involved. The notion of geometric convexity is also
reviewed and some applications to geodesic connectedness of an open subset of a
Lorentzian manifold are given.Comment: 14 pages, AMSLaTex, 2 figures. v2: typos fixed, added one reference
and several comments, statement of last proposition correcte
Metric characterization of connectedness for topological spaces
Connectedness, path connectedness, and uniform connectedness are well-known
concepts. In the traditional presentation of these concepts there is a
substantial difference between connectedness and the other two notions, namely
connectedness is defined as the absence of disconnectedness, while path
connectedness and uniform connectedness are defined in terms of connecting
paths and connecting chains, respectively. In compact metric spaces uniform
connectedness and connectedness are well-known to coincide, thus the apparent
conceptual difference between the two notions disappears. Connectedness in
topological spaces can also be defined in terms of chains governed by open
coverings in a manner that is more reminiscent of path connectedness. We
present a unifying metric formalism for connectedness, which encompasses both
connectedness of topological spaces and uniform connectedness of uniform
spaces, and which further extends to a hierarchy of notions of connectedness
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