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 “colored” partitions to get some rough descriptions of the structure of “most” unlabelled acyclic graphs. In particular, we will prove that for any fixed rooted tree T, almost every sufficiently large acyclic graph has a “subtree” isomorphic to T. We can use this result to get a zero-one law for Monadic Second Order queries on random unlabelled acyclic graphs

Parametrization over inductive relations of a bounded number of variables

AbstractWe present a Parametrization Theorem for (positive elementary) inductions that use a bounded number of variables. We investigate associated halting problem(s) on classes of finite structures and on solitary ‘unreasonable’ structures. These results involve the complexity of the inductive relations—and the complexity of the structure or class of structures on which these relations live. We also apply this Parametrization Theorem to Moschovakis closure ordinals, to determine when the closure ordinal is greater than ω, and to investigate the closure ordinals of unreasonable structures