334 research outputs found
Non-homogeneous polygonal Markov fields in the plane: graphical representations and geometry of higher order correlations
We consider polygonal Markov fields originally introduced by Arak and
Surgailis (1989). Our attention is focused on fields with nodes of order two,
which can be regarded as continuum ensembles of non-intersecting contours in
the plane, sharing a number of features with the two-dimensional Ising model.
We introduce non-homogeneous version of polygonal fields in anisotropic
enviroment. For these fields we provide a class of new graphical constructions
and random dynamics. These include a generalised dynamic representation,
generalised and defective disagreement loop dynamics as well as a generalised
contour birth and death dynamics. Next, we use these constructions as tools to
obtain new exact results on the geometry of higher order correlations of
polygonal Markov fields in their consistent regime.Comment: 54 page
Decidability Results for the Boundedness Problem
We prove decidability of the boundedness problem for monadic least
fixed-point recursion based on positive monadic second-order (MSO) formulae
over trees. Given an MSO-formula phi(X,x) that is positive in X, it is
decidable whether the fixed-point recursion based on phi is spurious over the
class of all trees in the sense that there is some uniform finite bound for the
number of iterations phi takes to reach its least fixed point, uniformly across
all trees. We also identify the exact complexity of this problem. The proof
uses automata-theoretic techniques. This key result extends, by means of
model-theoretic interpretations, to show decidability of the boundedness
problem for MSO and guarded second-order logic (GSO) over the classes of
structures of fixed finite tree-width. Further model-theoretic transfer
arguments allow us to derive major known decidability results for boundedness
for fragments of first-order logic as well as new ones
Courcelle's Theorem - A Game-Theoretic Approach
Courcelle's Theorem states that every problem definable in Monadic
Second-Order logic can be solved in linear time on structures of bounded
treewidth, for example, by constructing a tree automaton that recognizes or
rejects a tree decomposition of the structure. Existing, optimized software
like the MONA tool can be used to build the corresponding tree automata, which
for bounded treewidth are of constant size. Unfortunately, the constants
involved can become extremely large - every quantifier alternation requires a
power set construction for the automaton. Here, the required space can become a
problem in practical applications.
In this paper, we present a novel, direct approach based on model checking
games, which avoids the expensive power set construction. Experiments with an
implementation are promising, and we can solve problems on graphs where the
automata-theoretic approach fails in practice.Comment: submitte
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