879 research outputs found
Order Invariance on Decomposable Structures
Order-invariant formulas access an ordering on a structure's universe, but
the model relation is independent of the used ordering. Order invariance is
frequently used for logic-based approaches in computer science. Order-invariant
formulas capture unordered problems of complexity classes and they model the
independence of the answer to a database query from low-level aspects of
databases. We study the expressive power of order-invariant monadic
second-order (MSO) and first-order (FO) logic on restricted classes of
structures that admit certain forms of tree decompositions (not necessarily of
bounded width).
While order-invariant MSO is more expressive than MSO and, even, CMSO (MSO
with modulo-counting predicates), we show that order-invariant MSO and CMSO are
equally expressive on graphs of bounded tree width and on planar graphs. This
extends an earlier result for trees due to Courcelle. Moreover, we show that
all properties definable in order-invariant FO are also definable in MSO on
these classes. These results are applications of a theorem that shows how to
lift up definability results for order-invariant logics from the bags of a
graph's tree decomposition to the graph itself.Comment: Accepted for LICS 201
Definability Equals Recognizability for -Outerplanar Graphs
One of the most famous algorithmic meta-theorems states that every graph
property that can be defined by a sentence in counting monadic second order
logic (CMSOL) can be checked in linear time for graphs of bounded treewidth,
which is known as Courcelle's Theorem. These algorithms are constructed as
finite state tree automata, and hence every CMSOL-definable graph property is
recognizable. Courcelle also conjectured that the converse holds, i.e. every
recognizable graph property is definable in CMSOL for graphs of bounded
treewidth. We prove this conjecture for -outerplanar graphs, which are known
to have treewidth at most .Comment: 40 pages, 8 figure
Meta-Kernelization using Well-Structured Modulators
Kernelization investigates exact preprocessing algorithms with performance
guarantees. The most prevalent type of parameters used in kernelization is the
solution size for optimization problems; however, also structural parameters
have been successfully used to obtain polynomial kernels for a wide range of
problems. Many of these parameters can be defined as the size of a smallest
modulator of the given graph into a fixed graph class (i.e., a set of vertices
whose deletion puts the graph into the graph class). Such parameters admit the
construction of polynomial kernels even when the solution size is large or not
applicable. This work follows up on the research on meta-kernelization
frameworks in terms of structural parameters.
We develop a class of parameters which are based on a more general view on
modulators: instead of size, the parameters employ a combination of rank-width
and split decompositions to measure structure inside the modulator. This allows
us to lift kernelization results from modulator-size to more general
parameters, hence providing smaller kernels. We show (i) how such large but
well-structured modulators can be efficiently approximated, (ii) how they can
be used to obtain polynomial kernels for any graph problem expressible in
Monadic Second Order logic, and (iii) how they allow the extension of previous
results in the area of structural meta-kernelization
Definability equals recognizability for graphs of bounded treewidth
We prove a conjecture of Courcelle, which states that a graph property is
definable in MSO with modular counting predicates on graphs of constant
treewidth if, and only if it is recognizable in the following sense:
constant-width tree decompositions of graphs satisfying the property can be
recognized by tree automata. While the forward implication is a classic fact
known as Courcelle's theorem, the converse direction remained openComment: 21 pages, an extended abstract will appear in the proceedings of LICS
201
Homogeneity in the free group
We show that any non abelian free group \F is strongly
-homogeneous, i.e. that finite tuples of elements which satisfy the
same first-order properties are in the same orbit under \Aut(\F). We give a
characterization of elements in finitely generated groups which have the same
first-order properties as a primitive element of the free group. We deduce as a
consequence that most hyperbolic surface groups are not -homogeneous.Comment: 26 page
On the Monadic Second-Order Transduction Hierarchy
We compare classes of finite relational structures via monadic second-order
transductions. More precisely, we study the preorder where we set C \subseteq K
if, and only if, there exists a transduction {\tau} such that
C\subseteq{\tau}(K). If we only consider classes of incidence structures we can
completely describe the resulting hierarchy. It is linear of order type
{\omega}+3. Each level can be characterised in terms of a suitable variant of
tree-width. Canonical representatives of the various levels are: the class of
all trees of height n, for each n \in N, of all paths, of all trees, and of all
grids
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