We prove new upper and lower bounds on the number of iterations the \(k\)-dimensional Weisfeiler-Leman algorithm (\(k\)-WL) requires until stabilization. For \(k\geq 3\), we show that \(k\)-WL stabilizes after at most \(O(kn^{k-1}\log n)\) ...

We introduce a novel model-theoretic framework inspired from graph modification and based on the interplay between model theory and algorithmic graph minors. The core of our framework is a new compound logic operating with two types of sentences, ...

The width complexity measure plays a central role in resolution and other propositional proof systems like Polynomial Calculus (under the name of degree). The study of width lower bounds is the most used method for proving size lower bounds, and it is ...

We study functional dependencies together with two different probabilistic dependency notions: unary marginal identity and unary marginal distribution equivalence. A unary marginal identity states that two variables \(x\) and \(y\) are identically ...

We propose a novel SAT-based approach to graph generation. Our approach utilizes the interaction between a CDCL SAT solver and a special symmetry propagator where the SAT solver runs on an encoding of the desired graph property. The symmetry propagator ...

The k-spectrum is the set of all α > 0 such that G(n,n^−α) does not obey the 0-1 law for FO sentences with quantifier depth at most k. In this article, we prove that the minimum k such that the k-spectrum is infinite equals 5.

Satisfiability Modulo Theories (SMT) refers to the problem of deciding the satisfiability of a formula with respect to certain background first-order theories. In this article, we focus on Satisfiablity Modulo Integer Arithmetic, which is referred to as ...

First-order logic (FO) can express many algorithmic problems on graphs, such as the independent set and dominating set problem parameterized by solution size. However, FO cannot express the very simple algorithmic question whether two vertices are ...

Several forms of iterable belief change exist, differing in the kind of change and its strength: some operators introduce formulae, others remove them; some add formulae unconditionally, others only as additions to the previous beliefs; some only relative ...

We show that the number of variables and the quantifier depth needed to distinguish a pair of graphs by first-order logic sentences exactly match the complexity measures of clause width and depth needed to refute the corresponding graph isomorphism ...

We prove that the combinatorial Weisfeiler-Leman algorithm of dimension (3k+4) is a complete isomorphism test for the class of all graphs of rank width at most k. Rank width is a graph invariant that, similarly to tree width, measures the width of a ...

A recent strand of research in structural proof theory aims at exploring the notion of analytic calculi (i.e., those calculi that support general and modular proof-strategies for cut elimination) and at identifying classes of logics that can be captured ...

For any fixed positive integer k, let α_k denote the smallest α ∈ (0,1) such that the random graph sequence {G(n, n^-α)}_n does not satisfy the zero-one law for the set ε_k of all existential first-order sentences that are of quantifier depth at most k. This ...

Termination analysis of linear loops plays a key rôle in several areas of computer science, including program verification and abstract interpretation. Already for the simplest variants of linear loops the question of termination relates to deep open ...

Valued constraint satisfaction problems (VCSPs) are a large class of combinatorial optimisation problems. The computational complexity of VCSPs depends on the set of allowed cost functions in the input. Recently, the computational complexity of all VCSPs ...

We classify graphs and, more generally, finite relational structures that are identified by \(C^2\), that is, two-variable first-order logic with counting. Using this classification, we show that it can be decided in almost linear time whether a structure ...

We study the complexity of various fundamental counting problems that arise in the context of incomplete databases, i.e., relational databases that can contain unknown values in the form of labeled nulls. Specifically, we assume that the domains of these ...

We investigate the decidability of model checking logics of time, knowledge, and probability, with respect to two epistemic semantics: the clock and synchronous perfect recall semantics in partially observable discrete-time Markov chains. Decidability ...

We study the first-order (FO) model checking problem of dense graph classes, namely, those that have FO interpretations in (or are FO transductions of) some sparse graph classes. We give a structural characterization of the graph classes that are FO ...

The notion of bounded expansion captures uniform sparsity of graph classes and renders various algorithmic problems that are hard in general tractable. In particular, the model-checking problem for first-order logic is fixed-parameter tractable over ...