6 research outputs found
Successor-Invariant First-Order Logic on Graphs with Excluded Topological Subgraphs
We show that the model-checking problem for successor-invariant first-order
logic is fixed-parameter tractable on graphs with excluded topological
subgraphs when parameterised by both the size of the input formula and the size
of the exluded topological subgraph. Furthermore, we show that model-checking
for order-invariant first-order logic is tractable on coloured posets of
bounded width, parameterised by both the size of the input formula and the
width of the poset.
Our result for successor-invariant FO extends previous results for this logic
on planar graphs (Engelmann et al., LICS 2012) and graphs with excluded minors
(Eickmeyer et al., LICS 2013), further narrowing the gap between what is known
for FO and what is known for successor-invariant FO. The proof uses Grohe and
Marx's structure theorem for graphs with excluded topological subgraphs. For
order-invariant FO we show that Gajarsk\'y et al.'s recent result for FO
carries over to order-invariant FO
Model Checking Lower Bounds for Simple Graphs
A well-known result by Frick and Grohe shows that deciding FO logic on trees
involves a parameter dependence that is a tower of exponentials. Though this
lower bound is tight for Courcelle's theorem, it has been evaded by a series of
recent meta-theorems for other graph classes. Here we provide some additional
non-elementary lower bound results, which are in some senses stronger. Our goal
is to explain common traits in these recent meta-theorems and identify barriers
to further progress. More specifically, first, we show that on the class of
threshold graphs, and therefore also on any union and complement-closed class,
there is no model-checking algorithm with elementary parameter dependence even
for FO logic. Second, we show that there is no model-checking algorithm with
elementary parameter dependence for MSO logic even restricted to paths (or
equivalently to unary strings), unless E=NE. As a corollary, we resolve an open
problem on the complexity of MSO model-checking on graphs of bounded max-leaf
number. Finally, we look at MSO on the class of colored trees of depth d. We
show that, assuming the ETH, for every fixed d>=1 at least d+1 levels of
exponentiation are necessary for this problem, thus showing that the (d+1)-fold
exponential algorithm recently given by Gajarsk\`{y} and Hlin\u{e}n\`{y} is
essentially optimal
FO model checking of interval graphs
We study the computational complexity of the FO model checking problem on interval graphs, i.e., intersection graphs of intervals on the real line. The main positive result is that FO model checking and successor-invariant FO model checking can be solved in time O(n log n) for n-vertex interval graphs with representations containing only intervals with lengths from a prescribed finite set. We complement this result by showing that the same is not true if the lengths are restricted to any set that is dense in an open subset, e.g. in the set (1, 1 + ε)
Model-Checking on Ordered Structures
We study the model-checking problem for first- and monadic second-order logic
on finite relational structures. The problem of verifying whether a formula of
these logics is true on a given structure is considered intractable in general,
but it does become tractable on interesting classes of structures, such as on
classes whose Gaifman graphs have bounded treewidth. In this paper we continue
this line of research and study model-checking for first- and monadic
second-order logic in the presence of an ordering on the input structure. We do
so in two settings: the general ordered case, where the input structures are
equipped with a fixed order or successor relation, and the order invariant
case, where the formulas may resort to an ordering, but their truth must be
independent of the particular choice of order. In the first setting we show
very strong intractability results for most interesting classes of structures.
In contrast, in the order invariant case we obtain tractability results for
order-invariant monadic second-order formulas on the same classes of graphs as
in the unordered case. For first-order logic, we obtain tractability of
successor-invariant formulas on classes whose Gaifman graphs have bounded
expansion. Furthermore, we show that model-checking for order-invariant
first-order formulas is tractable on coloured posets of bounded width.Comment: arXiv admin note: substantial text overlap with arXiv:1701.0851
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