50 research outputs found
The category of MSO transductions
MSO transductions are binary relations between structures which are defined
using monadic second-order logic. MSO transductions form a category, since they
are closed under composition. We show that many notions from language theory,
such as recognizability or tree decompositions, can be defined in an abstract
way that only refers to MSO transductions and their compositions
Fast Evaluation of Interlace Polynomials on Graphs of Bounded Treewidth
We consider the multivariate interlace polynomial introduced by Courcelle
(2008), which generalizes several interlace polynomials defined by Arratia,
Bollobas, and Sorkin (2004) and by Aigner and van der Holst (2004). We present
an algorithm to evaluate the multivariate interlace polynomial of a graph with
n vertices given a tree decomposition of the graph of width k. The best
previously known result (Courcelle 2008) employs a general logical framework
and leads to an algorithm with running time f(k)*n, where f(k) is doubly
exponential in k. Analyzing the GF(2)-rank of adjacency matrices in the context
of tree decompositions, we give a faster and more direct algorithm. Our
algorithm uses 2^{3k^2+O(k)}*n arithmetic operations and can be efficiently
implemented in parallel.Comment: v4: Minor error in Lemma 5.5 fixed, Section 6.6 added, minor
improvements. 44 pages, 14 figure
VC Density of Set Systems Definable in Tree-Like Graphs
We study set systems definable in graphs using variants of logic with different expressive power. Our focus is on the notion of Vapnik-Chervonenkis density: the smallest possible degree of a polynomial bounding the cardinalities of restrictions of such set systems. On one hand, we prove that if phi(x,y) is a fixed CMSO_1 formula and C is a class of graphs with uniformly bounded cliquewidth, then the set systems defined by phi in graphs from C have VC density at most |y|, which is the smallest bound that one could expect. We also show an analogous statement for the case when phi(x,y) is a CMSO_2 formula and C is a class of graphs with uniformly bounded treewidth. We complement these results by showing that if C has unbounded cliquewidth (respectively, treewidth), then, under some mild technical assumptions on C, the set systems definable by CMSO_1 (respectively, CMSO_2) formulas in graphs from C may have unbounded VC dimension, hence also unbounded VC density
Modulo-Counting First-Order Logic on Bounded Expansion Classes
We prove that, on bounded expansion classes, every first-order formula with
modulo counting is equivalent, in a linear-time computable monadic lift, to an
existential first-order formula. As a consequence, we derive, on bounded
expansion classes, that first-order transductions with modulo counting have the
same encoding power as existential first-order transductions. Also,
modulo-counting first-order model checking and computation of the size of sets
definable in modulo-counting first-order logic can be achieved in linear time
on bounded expansion classes. As an application, we prove that a class has
structurally bounded expansion if and only if is a class of bounded depth
vertex-minors of graphs in a bounded expansion class. We also show how our
results can be used to implement fast matrix calculus on bounded expansion
matrices over a finite field.Comment: submitted to CSGT2022 special issu
First-Order Interpretations of Bounded Expansion Classes
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 such graph classes. With the aim of generalizing such results to dense graphs, we introduce classes of graphs with structurally bounded expansion, defined as first-order interpretations of classes of bounded expansion. As a first step towards their algorithmic treatment, we provide their characterization analogous to the characterization of classes of bounded expansion via low treedepth decompositions, replacing treedepth by its dense analogue called shrubdepth
Cliquewidth and knowledge compilation
In this paper we study the role of cliquewidth in succinct representation of Boolean functions. Our main statement is the following: Let Z be a Boolean circuit having cliquewidth k. Then there is another circuit Z * computing the same function as Z having treewidth at most 18k + 2 and which has at most 4|Z| gates where |Z| is the number of gates of Z. In this sense, cliquewidth is not more ‘powerful’ than treewidth for the purpose of representation of Boolean functions. We believe this is quite a surprising fact because it contrasts the situation with graphs where an upper bound on the treewidth implies an upper bound on the cliquewidth but not vice versa.
We demonstrate the usefulness of the new theorem for knowledge compilation. In particular, we show that a circuit Z of cliquewidth k can be compiled into a Decomposable Negation Normal Form (dnnf) of size O(918k k 2|Z|) and the same runtime. To the best of our knowledge, this is the first result on efficient knowledge compilation parameterized by cliquewidth of a Boolean circuit