5,158 research outputs found
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
Undecidable First-Order Theories of Affine Geometries
Tarski initiated a logic-based approach to formal geometry that studies
first-order structures with a ternary betweenness relation (\beta) and a
quaternary equidistance relation (\equiv). Tarski established, inter alia, that
the first-order (FO) theory of (R^2,\beta,\equiv) is decidable. Aiello and van
Benthem (2002) conjectured that the FO-theory of expansions of (R^2,\beta) with
unary predicates is decidable. We refute this conjecture by showing that for
all n>1, the FO-theory of monadic expansions of (R^2,\beta) is \Pi^1_1-hard and
therefore not even arithmetical. We also define a natural and comprehensive
class C of geometric structures (T,\beta), where T is a subset of R^2, and show
that for each structure (T,\beta) in C, the FO-theory of the class of monadic
expansions of (T,\beta) is undecidable. We then consider classes of expansions
of structures (T,\beta) with restricted unary predicates, for example finite
predicates, and establish a variety of related undecidability results. In
addition to decidability questions, we briefly study the expressivity of
universal MSO and weak universal MSO over expansions of (R^n,\beta). While the
logics are incomparable in general, over expansions of (R^n,\beta), formulae of
weak universal MSO translate into equivalent formulae of universal MSO.
This is an extended version of a publication in the proceedings of the 21st
EACSL Annual Conferences on Computer Science Logic (CSL 2012).Comment: 21 pages, 3 figure
Topological Models of Columnar Vagueness
This paper intends to further the understanding of the formal properties of (higher-order) vagueness by connecting theories of (higher-order) vagueness with more recent work in topology. First, we provide a “translation” of Bobzien's account of columnar higher-order vagueness into the logic of topological spaces. Since columnar vagueness is an essential ingredient of her solution to the Sorites paradox, a central problem of any theory of vagueness comes into contact with the modern mathematical theory of topology. Second, Rumfitt’s recent topological reconstruction of Sainsbury’s theory of prototypically defined concepts is shown to lead to the same class of spaces that characterize Bobzien’s account of columnar vagueness, namely, weakly scattered spaces. Rumfitt calls these spaces polar spaces. They turn out to be closely related to Gärdenfors’ conceptual spaces, which have come to play an ever more important role in cognitive science and related disciplines. Finally, Williamson’s “logic of clarity” is explicated in terms of a generalized topology (“locology”) that can be considered an alternative to standard topology. Arguably, locology has some conceptual advantages over topology with respect to the conceptualization of a boundary and a borderline. Moreover, in Williamson’s logic of clarity, vague concepts with respect to a notion of a locologically inspired notion of a “slim boundary” are (stably) columnar. Thus, Williamson’s logic of clarity also exhibits a certain affinity for columnar vagueness. In sum, a topological perspective is useful for a conceptual elucidation and unification of central aspects of a variety of contemporary accounts of vagueness
Unprovability of the Logical Characterization of Bisimulation
We quickly review labelled Markov processes (LMP) and provide a
counterexample showing that in general measurable spaces, event bisimilarity
and state bisimilarity differ in LMP. This shows that the logic in Desharnais
[*] does not characterize state bisimulation in non-analytic measurable spaces.
Furthermore we show that, under current foundations of Mathematics, such
logical characterization is unprovable for spaces that are projections of a
coanalytic set. Underlying this construction there is a proof that stationary
Markov processes over general measurable spaces do not have semi-pullbacks.
([*] J. Desharnais, Labelled Markov Processes. School of Computer Science.
McGill University, Montr\'eal (1999))Comment: Extended introduction and comments; extra section on semi-pullbacks;
11 pages Some background details added; extra example on the non-locality of
state bisimilarity; 14 page
Invariance and Logicality in Perspective
Although the invariance criterion of logicality first emerged as a criterion of a purely mathematical interest, it has developed into a criterion of considerable linguistic and philosophical interest. In this paper I compare two different perspectives on this criterion. The first is the perspective of natural language. Here, the invariance criterion is measured by its success in capturing our linguistic intuitions about logicality and explaining our logical behavior in natural-linguistic settings. The second perspective is more theoretical. Here, the invariance criterion is used as a tool for developing a theoretical foundation of logic, focused on a critical examination, explanation, and justification of its veridicality and modal force
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
Combining Spatial and Temporal Logics: Expressiveness vs. Complexity
In this paper, we construct and investigate a hierarchy of spatio-temporal
formalisms that result from various combinations of propositional spatial and
temporal logics such as the propositional temporal logic PTL, the spatial
logics RCC-8, BRCC-8, S4u and their fragments. The obtained results give a
clear picture of the trade-off between expressiveness and computational
realisability within the hierarchy. We demonstrate how different combining
principles as well as spatial and temporal primitives can produce NP-, PSPACE-,
EXPSPACE-, 2EXPSPACE-complete, and even undecidable spatio-temporal logics out
of components that are at most NP- or PSPACE-complete
Indeterminateness and `The' Universe of Sets: Multiversism, Potentialism, and Pluralism
In this article, I survey some philosophical attitudes to talk concerning `the' universe of sets. I separate out four different strands of the debate, namely: (i) Universism, (ii) Multiversism, (iii) Potentialism, and (iv) Pluralism. I discuss standard arguments and counterarguments concerning the positions and some of the natural mathematical programmes that are suggested by the various views
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