The MSO+U theory of (N, <) is undecidable

We consider the logic MSO+U, which is monadic second-order logic extended with the unbounding quantifier. The unbounding quantifier is used to say that a property of finite sets holds for sets of arbitrarily large size. We prove that the logic is undecidable on infinite words, i.e. the MSO+U theory of (N,<) is undecidable. This settles an open problem about the logic, and improves a previous undecidability result, which used infinite trees and additional axioms from set theory.Comment: 9 pages, with 2 figure

Stable graphs of bounded twin-width

We prove that every class of graphs $\mathscr C$ that is monadically stable and has bounded twin-width can be transduced from some class with bounded sparse twin-width. This generalizes analogous results for classes of bounded linear cliquewidth and of bounded cliquewidth. It also implies that monadically stable classes of bounded twin-widthare linearly $\chi$-bounded.Comment: 44 pages, 2 figure

LOIS: an application of SMT solvers *

Abstract We present an implemented programming language called LOIS, which allows iterating through certain infinite sets, in finite time. We argue that this language offers a new application of SMT solvers to verification of infinite-state systems, by showing that many known algorithms can easily be implemented using LOIS, which in turn invokes SMT solvers for various theories. In many applications, ω-categorical theories with quantifier elimination are of particular interest. Our tests indicate that state-of-the art solvers perform poorly on such theories, as they are outperformed by orders of magnitude by a simple quantifier-elimination procedure

Register Automata with Extrema Constraints, and an Application to Two-Variable Logic

We introduce a model of register automata over infinite trees with extrema constraints. Such an automaton can store elements of a linearly ordered domain in its registers, and can compare those values to the suprema and infima of register values in subtrees. We show that the emptiness problem for these automata is decidable. As an application, we prove decidability of the countable satisfiability problem for two-variable logic in the presence of a tree order, a linear order, and arbitrary atoms that are MSO definable from the tree order. As a consequence, the satisfiability problem for two-variable logic with arbitrary predicates, two of them interpreted by linear orders, is decidable

Indiscernibles and Flatness in Monadically Stable and Monadically NIP Classes

Monadically stable and monadically NIP classes of structures were initially studied in the context of model theory and defined in logical terms. They have recently attracted attention in the area of structural graph theory, as they generalize notions such as nowhere denseness, bounded cliquewidth, and bounded twinwidth. Our main result is the - to the best of our knowledge first - purely combinatorial characterization of monadically stable classes of graphs, in terms of a property dubbed flip-flatness. A class $\mathcal{C}$ of graphs is flip-flat if for every fixed radius $r$, every sufficiently large set of vertices of a graph $G \in \mathcal{C}$ contains a large subset of vertices with mutual distance larger than $r$, where the distance is measured in some graph $G'$ that can be obtained from $G$ by performing a bounded number of flips that swap edges and non-edges within a subset of vertices. Flip-flatness generalizes the notion of uniform quasi-wideness, which characterizes nowhere dense classes and had a key impact on the combinatorial and algorithmic treatment of nowhere dense classes. To obtain this result, we develop tools that also apply to the more general monadically NIP classes, based on the notion of indiscernible sequences from model theory. We show that in monadically stable and monadically NIP classes indiscernible sequences impose a strong combinatorial structure on their definable neighborhoods. All our proofs are constructive and yield efficient algorithms.Comment: v2: revised presentation; renamed flip-wideness to flip-flatness; changed the title from "Indiscernibles and Wideness [...]" to "Indiscernibles and Flatness [...]