31 research outputs found
Tameness in least fixed-point logic and McColm's conjecture
We investigate four model-theoretic tameness properties in the context of
least fixed-point logic over a family of finite structures. We find that each
of these properties depends only on the elementary (i.e., first-order) limit
theory, and we completely determine the valid entailments among them. In
contrast to the context of first-order logic on arbitrary structures, the order
property and independence property are equivalent in this setting. McColm
conjectured that least fixed-point definability collapses to first-order
definability exactly when proficiency fails. McColm's conjecture is known to be
false in general. However, we show that McColm's conjecture is true for any
family of finite structures whose limit theory is model-theoretically tame
On the strictness of the quantifier structure hierarchy in first-order logic
We study a natural hierarchy in first-order logic, namely the quantifier
structure hierarchy, which gives a systematic classification of first-order
formulas based on structural quantifier resource. We define a variant of
Ehrenfeucht-Fraisse games that characterizes quantifier classes and use it to
prove that this hierarchy is strict over finite structures, using strategy
compositions. Moreover, we prove that this hierarchy is strict even over
ordered finite structures, which is interesting in the context of descriptive
complexity.Comment: 38 pages, 8 figure
On the structure of random unlabelled acyclic graphs
AbstractOne can use Poisson approximation techniques to get results about the asymptotics of graphical properties on random unlabelled acyclic graphs i.e., on random unlabelled free (rootless) trees. We will use some âcoloredâ partitions to get some rough descriptions of the structure of âmostâ unlabelled acyclic graphs. In particular, we will prove that for any fixed rooted tree T, almost every sufficiently large acyclic graph has a âsubtreeâ isomorphic to T. We can use this result to get a zero-one law for Monadic Second Order queries on random unlabelled acyclic graphs
Logical limit laws for minor-closed classes of graphs
Let be an addable, minor-closed class of graphs. We prove that
the zero-one law holds in monadic second-order logic (MSO) for the random graph
drawn uniformly at random from all {\em connected} graphs in on
vertices, and the convergence law in MSO holds if we draw uniformly at
random from all graphs in on vertices. We also prove analogues
of these results for the class of graphs embeddable on a fixed surface,
provided we restrict attention to first order logic (FO). Moreover, the
limiting probability that a given FO sentence is satisfied is independent of
the surface . We also prove that the closure of the set of limiting
probabilities is always the finite union of at least two disjoint intervals,
and that it is the same for FO and MSO. For the classes of forests and planar
graphs we are able to determine the closure of the set of limiting
probabilities precisely. For planar graphs it consists of exactly 108
intervals, each of length . Finally, we analyse
examples of non-addable classes where the behaviour is quite different. For
instance, the zero-one law does not hold for the random caterpillar on
vertices, even in FO.Comment: minor changes; accepted for publication by JCT
On Sharp Thresholds in Random Geometric Graphs
We give a characterization of vertex-monotone properties with sharp thresholds in a Poisson random geometric graph or hypergraph. As an application we show that a geometric model of random k-SAT exhibits a sharp threshold for satisfiability