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

Flip-Breakability: A Combinatorial Dichotomy for Monadically Dependent Graph Classes

A conjecture in algorithmic model theory predicts that the model-checking problem for first-order logic is fixed-parameter tractable on a hereditary graph class if and only if the class is monadically dependent. Originating in model theory, this notion is defined in terms of logic, and encompasses nowhere dense classes, monadically stable classes, and classes of bounded twin-width. Working towards this conjecture, we provide the first two combinatorial characterizations of monadically dependent graph classes. This yields the following dichotomy. On the structure side, we characterize monadic dependence by a Ramsey-theoretic property called flip-breakability. This notion generalizes the notions of uniform quasi-wideness, flip-flatness, and bounded grid rank, which characterize nowhere denseness, monadic stability, and bounded twin-width, respectively, and played a key role in their respective model checking algorithms. Natural restrictions of flip-breakability additionally characterize bounded treewidth and cliquewidth and bounded treedepth and shrubdepth. On the non-structure side, we characterize monadic dependence by explicitly listing few families of forbidden induced subgraphs. This result is analogous to the characterization of nowhere denseness via forbidden subdivided cliques, and allows us to resolve one half of the motivating conjecture: First-order model checking is AW[$*$]-hard on every hereditary graph class that is monadically independent. The result moreover implies that hereditary graph classes which are small, have almost bounded twin-width, or have almost bounded flip-width, are monadically dependent. Lastly, we lift our result to also obtain a combinatorial dichotomy in the more general setting of monadically dependent classes of binary structures.Comment: v2: added section "Conclusions and Future Work

Definable isomorphism problem

We investigate the isomorphism problem in the setting of definable sets (equivalent to sets with atoms): given two definable relational structures, are they related by a definable isomorphism? Under mild assumptions on the underlying structure of atoms, we prove decidability of the problem. The core result is parameter-elimination: existence of an isomorphism definable with parameters implies existence of an isomorphism definable without parameters

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