G. Mints,T. Zhang
Propositional logic of continuous transformations
Dynamical topological logic studies models of the form (X,T), where is
a topological space, T a transformation on X. Propositional formulas
are constructed from variables (atomic formulas) by Boolean
connectives, necessity \Box and a monadic operation o. Variables are
interpreted by subset of X, Boolean connectives act in a natural way,
\Box is the interior and o is the pre-image under operation T. Under
this interpretation the axiom schema o\Box A->\Box oA called (C)
expresses continuity of T. J. Davoren proved completeness of S4+C for
arbitrary topological spaces, in particular for finite spaces derived
from Kripke models.
We prove completeness of S4+C for Cantor space. The proof uses a
continuous and open map W from the Cantor space onto a suitable Kripke
model (K,T) [with T continuous in order topology on K] and a
continuous map S on Cantor space satisfying condition:
WS=TW.
P. Kremer pointed out that the real line is not complete for S4+C. A
counterexample constructed by J. van Benthem will probably be
presented at NASSLI.
Ian Pratt-Hartmann
Where Spatial Reasoning is (or should be) going
For most of the previous century, a small but tenacious band of
authors argued that a parsimonious and conceptually elegant treatment
of space can be obtained by taking the position that regions, not
points, are the primitive spatial entities. These arguments resulted
in the development of various calculi for qualitative spatial
reasoning in which regions are taken as primary, an enterprise which
we might reasonably call `mereogeometry'.
In the past decade, the Computer Science community in particular has
produced a plethora of new technical results in mereogeometry,
especially concerning the decidability and computational complexity of
region-based geometrical representation languages. Unfortunately, this
work has often lacked a coordinating sense of direction; and what has
emerged is not so much a unified discipline as a motley of formal
systems which are difficult to compare and to evaluate. Yet buried in
the confusion lies much that is of value. The aim of this talk is to
provide enough of an overview to assess where the field is (or
perhaps, should be) going, and to identify its immediate future
challenges.
Dave Randell
Occlusion Calculi: A Cognitive Robotics Perspective
This talk motivates and describes two first-order axiomatic
theories originally developed to model viewpoint dependent (spatial)
occlusion relations between arbitrary shaped bodies. Both calculi
respectively assume a region-based ontology and embed the spatial theories:
RCC-5 and RCC-8. The ontology and set of defined relations is sufficient
for modelling domains where changing, but discrete, modes of overlap between
images of objects coincide with changing lines of sight. When applied to
mobile robots with machine vision, occlusion calculi can be used to model
and assist reasoning about such tasks as: robot location, scene
interpretation, navigation and route-planning, and general map-building.
The talk concludes by highlighting the challenges faced when implementing
such spatial logics on real world robots; and raises the interesting
question of what, if anything, can Cognitive Robotics tell us about Spatial
Reasoning?
Darko Sarenac
Modal Logics for Products of Topologies
Joint work with Johan van Benthem and Guram Bezhanishvili
Arguably the most basic logic for spatial reasoning is S4 interpreted over topological spaces. Our primary goal in this paper is to study a minimal augmentation of this logic. We will study multimodal languages of products of topologies where the modality of each component of the product is preserved into the product. This work generalizes some of the results of D.M. Gabbay and V.B. Shehtman on products of modal logics to the topological setting.
Thus in the most general case, each component modality will have S4 as a complete axiomatization. The interesting question, of course, is how the modalities of various dimensions interact. The expansion of the language enables us to `track dimensions' in the product and this in turn will add some expressive power to the language.
Chris Steinsvold
Towards A Topology Of Knowledge And Belief
KD45 is often taken as a logic of justified belief.
Interpreting the diamond of KD45 as the derived
set operator (AKA set of limit points), a topological
completeness result is available. Thus, the
dual of the derived set operator acts like
justified belief (given certain restrictions
on the space). Also, if we take the interior
operator to represent knowledge, we get
a philosophically defensible logic of
knowledge and belief. We'll discuss the
possibility of using the topological semantics
as an alternative semantics for a reasoner.
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