6,701 research outputs found
Verifying Recursive Active Documents with Positive Data Tree Rewriting
This paper proposes a data tree-rewriting framework for modeling evolving
documents. The framework is close to Guarded Active XML, a platform used for
handling XML repositories evolving through web services. We focus on automatic
verification of properties of evolving documents that can contain data from an
infinite domain. We establish the boundaries of decidability, and show that
verification of a {\em positive} fragment that can handle recursive service
calls is decidable. We also consider bounded model-checking in our data
tree-rewriting framework and show that it is \nexptime-complete
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
Tree-width for first order formulae
We introduce tree-width for first order formulae \phi, fotw(\phi). We show
that computing fotw is fixed-parameter tractable with parameter fotw. Moreover,
we show that on classes of formulae of bounded fotw, model checking is fixed
parameter tractable, with parameter the length of the formula. This is done by
translating a formula \phi\ with fotw(\phi)<k into a formula of the k-variable
fragment L^k of first order logic. For fixed k, the question whether a given
first order formula is equivalent to an L^k formula is undecidable. In
contrast, the classes of first order formulae with bounded fotw are fragments
of first order logic for which the equivalence is decidable.
Our notion of tree-width generalises tree-width of conjunctive queries to
arbitrary formulae of first order logic by taking into account the quantifier
interaction in a formula. Moreover, it is more powerful than the notion of
elimination-width of quantified constraint formulae, defined by Chen and Dalmau
(CSL 2005): for quantified constraint formulae, both bounded elimination-width
and bounded fotw allow for model checking in polynomial time. We prove that
fotw of a quantified constraint formula \phi\ is bounded by the
elimination-width of \phi, and we exhibit a class of quantified constraint
formulae with bounded fotw, that has unbounded elimination-width. A similar
comparison holds for strict tree-width of non-recursive stratified datalog as
defined by Flum, Frick, and Grohe (JACM 49, 2002).
Finally, we show that fotw has a characterization in terms of a cops and
robbers game without monotonicity cost
The Logic of Counting Query Answers
We consider the problem of counting the number of answers to a first-order
formula on a finite structure. We present and study an extension of first-order
logic in which algorithms for this counting problem can be naturally and
conveniently expressed, in senses that are made precise and that are motivated
by the wish to understand tractable cases of the counting problem
Solving Problems on Graphs of High Rank-Width
A modulator of a graph G to a specified graph class H is a set of vertices
whose deletion puts G into H. The cardinality of a modulator to various
tractable graph classes has long been used as a structural parameter which can
be exploited to obtain FPT algorithms for a range of hard problems. Here we
investigate what happens when a graph contains a modulator which is large but
"well-structured" (in the sense of having bounded rank-width). Can such
modulators still be exploited to obtain efficient algorithms? And is it even
possible to find such modulators efficiently?
We first show that the parameters derived from such well-structured
modulators are strictly more general than the cardinality of modulators and
rank-width itself. Then, we develop an FPT algorithm for finding such
well-structured modulators to any graph class which can be characterized by a
finite set of forbidden induced subgraphs. We proceed by showing how
well-structured modulators can be used to obtain efficient parameterized
algorithms for Minimum Vertex Cover and Maximum Clique. Finally, we use
well-structured modulators to develop an algorithmic meta-theorem for deciding
problems expressible in Monadic Second Order (MSO) logic, and prove that this
result is tight in the sense that it cannot be generalized to LinEMSO problems.Comment: Accepted at WADS 201
Time and Parallelizability Results for Parity Games with Bounded Tree and DAG Width
Parity games are a much researched class of games in NP intersect CoNP that
are not known to be in P. Consequently, researchers have considered specialised
algorithms for the case where certain graph parameters are small. In this
paper, we study parity games on graphs with bounded treewidth, and graphs with
bounded DAG width. We show that parity games with bounded DAG width can be
solved in O(n^(k+3) k^(k + 2) (d + 1)^(3k + 2)) time, where n, k, and d are the
size, treewidth, and number of priorities in the parity game. This is an
improvement over the previous best algorithm, given by Berwanger et al., which
runs in n^O(k^2) time. We also show that, if a tree decomposition is provided,
then parity games with bounded treewidth can be solved in O(n k^(k + 5) (d +
1)^(3k + 5)) time. This improves over previous best algorithm, given by
Obdrzalek, which runs in O(n d^(2(k+1)^2)) time. Our techniques can also be
adapted to show that the problem of solving parity games with bounded treewidth
lies in the complexity class NC^2, which is the class of problems that can be
efficiently parallelized. This is in stark contrast to the general parity game
problem, which is known to be P-hard, and thus unlikely to be contained in NC
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