Definability in the monadic second-order theory of successor

Definability in the Monadic Second-Order Theory of Successor

Interpreting the Monadic Second Order Theory of One Successor in Expansions of the Real Line

We give sufficient conditions for a first order expansion of the real line to define the standard model of the monadic second order theory of one successor. Such an expansion does not satisfy any of the combinatorial tameness properties defined by Shelah, such as NIP or even NTP2. We use this to deduce the first general results about definable sets in NTP2 expansions of (R, <,+). The goal of th...

Regularity Equals Monadic Second-Order Definability for Quasi-trees

Quasi-trees generalize trees in that the unique "path" between two nodes may be infinite and have any finite or countable order type, in particular that of rational numbers. They are used to define the rank-width of a countable graph in such a way that it is the least upper-bound of the rankwidths of its finite induced subgraphs. Join-trees are the corresponding directed trees and they are also...

On the relationships between theories of time granularity and the monadic second-order theory of one successor

In this paper we explore the connections between the monadic second-order theory of one successor (MSO[<] for short) and the theories of ω-layered structures for time granularity. We first prove that the decision problem for MSO[<] and that for a suitable first-order theory of the upward unbounded layered structure are inter-reducible. Then, we show that a similar result holds for suitable chai...

The Monadic Second-Order Logic of Graphs V: On Closing the Gap Between Definability and Recognizability

Courcelle, B., The monadic second-order logic of graphs V: on closing the gap between definability and recognizability, Theoretical Computer Science 80 (1991) 153-202. Context-free graph-grammars are considered such that, in every generated graph (3, a derivation tree of G can be constructed by means of monadic second-order formulas that specify its nodes, its labels, the successors of a node e...