We show that examinations of the expressive power of logical formulae
enriched by Lindstroem quantifiers over ordered finite structures
have a well-studied complexity-theoretic counterpart: the leaf
language approach to define complexity classes. Model classes of
formulae with Lindstroem quantifiers are nothing else than leaf
language definable sets. Along the ... more >>>

Building upon the known generalized-quantifier-based first-order
characterization of LOGCFL, we lay the groundwork for a deeper
investigation. Specifically, we examine subclasses of LOGCFL arising
from varying the arity and nesting of groupoidal quantifiers. Our
work extends the elaborate theory relating monoidal quantifiers to
NC^1 and its subclasses. In the ... more >>>

The reachability problem for graphs cannot be described, in the
sense of descriptive complexity theory, using a single first-order
formula. This is true both for directed and undirected graphs, both
in the finite and infinite. However, if we restrict ourselves to
graphs in which a certain graph parameter is fixed ... more >>>

It is well-known that every first-order property on words is expressible
using at most three variables. The subclass of properties expressible with
only two variables is also quite interesting and well-studied. We prove
precise structure theorems that characterize the exact expressive power of
first-order logic with two variables on words. ... more >>>

Algorithmic meta theorems are algorithmic results that apply to whole families of combinatorial problems, instead of just specific problems. These families are usually defined in terms of logic and graph theory. An archetypal algorithmic meta theorem is Courcelle's Theorem, which states that all graph properties definable in monadic second-order logic ... more >>>

We show that if $\mathcal C$ is a class of graphs which is
"nowhere dense" then first-order model-checking is
fixed-parameter tractable on $\mathcal C$. As all graph classes which exclude a fixed minor, or are of bounded local tree-width or locally exclude a minor are nowhere dense, this generalises algorithmic ... more >>>

Two graphs with adjacency matrices $\mathbf{A}$ and $\mathbf{B}$ are isomorphic if there exists a permutation matrix $\mathbf{P}$ for which the identity $\mathbf{P}^{\mathrm{T}} \mathbf{A} \mathbf{P} = \mathbf{B}$ holds. Multiplying through by $\mathbf{P}$ and relaxing the permutation matrix to a doubly stochastic matrix leads to the notion of fractional isomorphism. We show ... more >>>

We survey research that studies the connection between the computational complexity
of optimization problems on the one hand, and the duality gap between the primal and
dual optimization problems on the other. To our knowledge, this is the first survey that
connects the two very important areas. We further look ... more >>>

Kolaitis and Kopparty have shown that for any first-order formula with
parity quantifiers over the language of graphs there is a family of
multi-variate polynomials of constant-degree that agree with the
formula on all but a $2^{-\Omega(n)}$-fraction of the graphs with $n$
vertices. The proof yields a bound on the ... more >>>