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
Successor-Invariant First-Order Logic on Classes of Bounded Degree
We study the expressive power of successor-invariant first-order logic, which
is an extension of first-order logic where the usage of an additional successor
relation on the structure is allowed, as long as the validity of formulas is
independent on the choice of a particular successor. We show that when the
degree is bounded, successor-invariant first-order logic is no more expressive
than first-order logic
Succinctness in subsystems of the spatial mu-calculus
In this paper we systematically explore questions of succinctness in modal
logics employed in spatial reasoning. We show that the closure operator,
despite being less expressive, is exponentially more succinct than the
limit-point operator, and that the -calculus is exponentially more
succinct than the equally-expressive tangled limit operator. These results hold
for any class of spaces containing at least one crowded metric space or
containing all spaces based on ordinals below , with the usual
limit operator. We also show that these results continue to hold even if we
enrich the less succinct language with the universal modality
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
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
Expressivity and Succinctness of Order-Invariant Logics on Depth-Bounded Structures
We study the expressive power and succinctness of order-invariant sentences of first-order (FO) and monadic second-order (MSO) logic on graphs of bounded tree-depth. Order-invariance is undecidable in general and, therefore, in finite model theory, one strives for logics with a decidable syntax that have the same expressive power as order-invariant sentences. We show that on graphs of bounded tree-depth, order-invariant FO has the same expressive power as FO, and order-invariant MSO has the same expressive power as the extension of FO with modulo-counting quantifiers. Our proof techniques allow for a fine-grained analysis of the succinctness of these translations. We show that for every order-invariant FO sentence there exists an FO sentence whose size is elementary in the size of the original sentence, and whose number of quantifier alternations is linear in the tree-depth. Our techniques can be adapted to obtain a similar quantitative variant of a known result that the expressive power of MSO and FO coincides on graphs of bounded tree-depth