9 research outputs found
Faster Existential FO Model Checking on Posets
We prove that the model checking problem for the existential fragment of
first-order (FO) logic on partially ordered sets is fixed-parameter tractable
(FPT) with respect to the formula and the width of a poset (the maximum size of
an antichain). While there is a long line of research into FO model checking on
graphs, the study of this problem on posets has been initiated just recently by
Bova, Ganian and Szeider (CSL-LICS 2014), who proved that the existential
fragment of FO has an FPT algorithm for a poset of fixed width. We improve upon
their result in two ways: (1) the runtime of our algorithm is
O(f(|{\phi}|,w).n^2) on n-element posets of width w, compared to O(g(|{\phi}|).
n^{h(w)}) of Bova et al., and (2) our proofs are simpler and easier to follow.
We complement this result by showing that, under a certain
complexity-theoretical assumption, the existential FO model checking problem
does not have a polynomial kernel.Comment: Paper as accepted to the LMCS journal. An extended abstract of an
earlier version of this paper has appeared at ISAAC'14. Main changes to the
previous version are improvements in the Multicoloured Clique part (Section
4
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
Quantified Conjunctive Queries on Partially Ordered Sets
We study the computational problem of checking whether a quantified
conjunctive query (a first-order sentence built using only conjunction as
Boolean connective) is true in a finite poset (a reflexive, antisymmetric, and
transitive directed graph). We prove that the problem is already NP-hard on a
certain fixed poset, and investigate structural properties of posets yielding
fixed-parameter tractability when the problem is parameterized by the query.
Our main algorithmic result is that model checking quantified conjunctive
queries on posets of bounded width is fixed-parameter tractable (the width of a
poset is the maximum size of a subset of pairwise incomparable elements). We
complement our algorithmic result by complexity results with respect to classes
of finite posets in a hierarchy of natural poset invariants, establishing its
tightness in this sense.Comment: Accepted at IPEC 201
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