9 research outputs found
On the model-checking of monadic second-order formulas with edge set quantifications
AbstractWe extend clique-width to graphs with multiple edges. We obtain fixed-parameter tractable model-checking algorithms for certain monadic second-order graph properties that depend on the multiplicities of edges, with respect to this “new” clique-width. We define special tree-width, the variant of tree-width relative to tree-decompositions such that the boxes that contain a vertex are on a path originating from some fixed node. We study its main properties. This definition is motivated by the construction of finite automata associated with monadic second-order formulas using edge set quantifications. These automata yield fixed-parameter linear algorithms with respect to tree-width for the model-checking of these formulas. Their construction is much simpler for special tree-width than for tree-width, for reasons that we explain
Fly-automata for checking MSO 2 graph properties
A more descriptive but too long title would be : Constructing fly-automata to
check properties of graphs of bounded tree-width expressed by monadic
second-order formulas written with edge quantifications. Such properties are
called MSO2 in short. Fly-automata (FA) run bottom-up on terms denoting graphs
and compute "on the fly" the necessary states and transitions instead of
looking into huge, actually unimplementable tables. In previous works, we have
constructed FA that process terms denoting graphs of bounded clique-width, in
order to check their monadic second-order (MSO) properties (expressed by
formulas without edge quan-tifications). Here, we adapt these FA to incidence
graphs, so that they can check MSO2 properties of graphs of bounded tree-width.
This is possible because: (1) an MSO2 property of a graph is nothing but an MSO
property of its incidence graph and (2) the clique-width of the incidence graph
of a graph is linearly bounded in terms of its tree-width. Our constructions
are actually implementable and usable. We detail concrete constructions of
automata in this perspective.Comment: Submitted for publication in December 201
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
Computations by fly-automata beyond monadic second-order logic
We present logically based methods for constructing XP and FPT graph
algorithms, parametrized by tree-width or clique-width. We will use
fly-automata introduced in a previous article. They make possible to check
properties that are not monadic second-order expressible because their states
may include counters, so that their sets of states may be infinite. We equip
these automata with output functions, so that they can compute values
associated with terms or graphs. Rather than new algorithmic results we present
tools for constructing easily certain dynamic programming algorithms by
combining predefined automata for basic functions and properties.Comment: Accepted for publication in Theoretical Computer Scienc