24 research outputs found
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 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 -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
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 of graphs is
flip-flat if for every fixed radius , every sufficiently large set of
vertices of a graph contains a large subset of vertices
with mutual distance larger than , where the distance is measured in some
graph that can be obtained from 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 [...]
First-Order Model Checking on Monadically Stable Graph Classes
A graph class is called monadically stable if one cannot
interpret, in first-order logic, arbitrary large linear orders in colored
graphs from . We prove that the model checking problem for
first-order logic is fixed-parameter tractable on every monadically stable
graph class. This extends the results of [Grohe, Kreutzer, and Siebertz; J. ACM
'17] for nowhere dense classes and of [Dreier, M\"ahlmann, and Siebertz; STOC
'23] for structurally nowhere dense classes to all monadically stable classes.
As a complementary hardness result, we prove that for every hereditary graph
class that is edge-stable (excludes some half-graph as a
semi-induced subgraph) but not monadically stable, first-order model checking
is -hard on , and -hard when
restricted to existential sentences. This confirms, in the special case of
edge-stable classes, an on-going conjecture that the notion of monadic NIP
delimits the tractability of first-order model checking on hereditary classes
of graphs.
For our tractability result, we first prove that monadically stable graph
classes have almost linear neighborhood complexity. Using this, we construct
sparse neighborhood covers for monadically stable classes, which provides the
missing ingredient for the algorithm of [Dreier, M\"ahlmann, and Siebertz; STOC
'23]. The key component of this construction is the usage of orders with low
crossing number [Welzl; SoCG '88], a tool from the area of range queries.
For our hardness result, we prove a new characterization of monadically
stable graph classes in terms of forbidden induced subgraphs. We then use this
characterization to show that in hereditary classes that are edge-stable but
not monadically stable, one can effectively interpret the class of all graphs
using only existential formulas.Comment: 55 pages, 13 figure