35 research outputs found
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
Nonconvergence, undecidability, and intractability in asymptotic problems
Results delimiting the logical and effective content of asymptotic combinatorics are presented. For the class of binary relations with an underlying linear order, and the class of binary functions, there are properties, given by first-order sentences, without asymptotic probabilities; every first-order asymptotic problem (i.e., set of first-order sentences with asymptotic probabilities bounded by a given rational number between zero and one) for these two classes is undecidable. For the class of pairs of unary functions or permutations, there are monadic second-order properties without asymptotic probabilities; every monadic second-order asymptotic problem for this class is undecidable. No first-order asymptotic problem for the class of unary functions is elementary recursive.Peer Reviewedhttp://deepblue.lib.umich.edu/bitstream/2027.42/26912/1/0000478.pd