We consider the problem of simulation preorder/equivalence between innnite-state processes and nite-state ones. We prove that simulation preorder (in both directions) and simulation equivalence are intractable between all major classes of innnite-state systems and nite-state ones. This result is obtained by showing that the problem whether a BPA (or BPP) process simulates a nite-state one is PS...

We characterize the existence of a nonnegative, sublinear and continuous order-preserving function for a not necessarily complete preorder on a real convex cone in an arbitrary topological real vector space. As a corollary of the main result, we present necessary and sufficient conditions for the existence of such an order-preserving function for a complete preorder.

A preorder consists of linearly ordered equivalence classes called blocks, and an alignment is a sequence of cycles on n labelled elements. We investigate the block structure of a random preorder chosen uniformly at random among all preorders on n elements, and also the distribution of cycles in a random alignment chosen uniformly at random among all alignments on n elements, as n →∞.

Let 1R be the preorder of embeddability between countable linear orders colored with elements of Rado’s partial order (a standard example of a wqo which is not a bqo). We show that 1R has fairly high complexity with respect to Borel reducibility (e.g. if P is a Borel preorder then P ≤B 1R), although its exact classification remains open.

We introduce the notion of doubly rooted plane trees and give a decomposition of these trees, called the butterfly decomposition which turns out to have many applications. From the butterfly decomposition we obtain a oneto-one correspondence between doubly rooted plane trees and free Dyck paths, which implies a simple derivation of a relation between the Catalan numbers and the central binomial...