Counting Monadic Second Order Logic Research Articles

Generation and Polynomial Parsing of Graph Languages with Non-Structural Reentrancies

Abstract Graph-based semantic representations are popular in natural language processing, where it is often convenient to model linguistic concepts as nodes and relations as edges between them. Several attempts have been made to find a generative device that is sufficiently powerful to describe languages of semantic graphs, while at the same allowing efficient parsing. We contribute to this line of work by introducing graph extension grammar, a variant of the contextual hyperedge replacement grammars proposed by Hoffmann et al. Contextual hyperedge replacement can generate graphs with non-structural reentrancies, a type of node-sharing that is very common in formalisms such as abstract meaning representation, but that context-free types of graph grammars cannot model. To provide our formalism with a way to place reentrancies in a linguistically meaningful way, we endow rules with logical formulas in counting monadic second-order logic. We then present a parsing algorithm and show as our main result that this algorithm runs in polynomial time on graph languages generated by a subclass of our grammars, the so-called local graph extension grammars.

Tree Automata and Pigeonhole Classes of Matroids: II

Let $\psi$ be a sentence in the counting monadic second-order logic of matroids and let F be a finite field. Hlineny's Theorem says that we can test whether F-representable matroids satisfy $\psi$ using an algorithm that is fixed-parameter tractable with respect to branch-width. In a previous paper we proved there is a similar fixed-parameter tractable algorithm that can test the members of any efficiently pigeonhole class. In this sequel we apply results from the first paper and thereby extend Hlineny's Theorem to the classes of fundamental transversal matroids, lattice path matroids, bicircular matroids, and H-gain-graphic matroids, when H is a finite group. As a consequence, we can obtain a new proof of Courcelle's Theorem.

Beyond Classes of Graphs with “Few” Minimal Separators: FPT Results Through Potential Maximal Cliques

Let $$\mathcal {P}(G,X)$$ be a property associating a boolean value to each pair (G, X) where G is a graph and X is a vertex subset. Assume that $$\mathcal {P}$$ is expressible in counting monadic second order logic (CMSO) and let t be an integer constant. We consider the following optimization problem: given an input graph $$G=(V,E)$$ , find subsets $$X\subseteq F \subseteq V$$ such that the treewidth of G[F] is at most t, property $$\mathcal {P}(G[F],X)$$ is true and X is of maximum size under these conditions. The problem generalizes many classical algorithmic questions, e.g., Longest Induced Path, Maximum Induced Forest, Independent $$\mathcal {H}$$ -Packing, etc. Fomin et al. (SIAM J Comput 44(1):54–87, 2015) proved that the problem is polynomial on the class of graph $${\mathcal {G}}_{{\text {poly}}}$$ , i.e. the graphs having at most $${\text {poly}}(n)$$ minimal separators for some polynomial $${\text {poly}}$$ . Here we consider the class $${\mathcal {G}}_{{\text {poly}}}+ kv$$ , formed by graphs of $${\mathcal {G}}_{{\text {poly}}}$$ to which we add a set of at most k vertices with arbitrary adjacencies, called modulator. We prove that the generic optimization problem is fixed parameter tractable on $${\mathcal {G}}_{{\text {poly}}}+ kv$$ , with parameter k, if the modulator is also part of the input.

Definability equals recognizability for [formula omitted]-outerplanar graphs and [formula omitted]-chordal partial [formula omitted]-trees

One of the most famous algorithmic meta-theorems states that every graph property which can be defined in counting monadic second order logic (CMSOL) can be checked in linear time on graphs of bounded treewidth, which is known as Courcelle’s Theorem (Courcelle, 1990). These algorithms are constructed as finite state tree automata and hence every CMSOL-definable graph property is recognizable. Courcelle also conjectured that the converse holds, i.e. every recognizable graph property is definable in CMSOL for graphs of bounded treewidth. In this paper we prove two special cases of this conjecture, first for the class of k-outerplanar graphs, which are known to have treewidth at most 3k−1 (Bodlaender, 1998) and for graphs of bounded treewidth without chordless cycles of length at least some constant ℓ.We furthermore show that for a proof of Courcelle’s Conjecture it is sufficient to show that all members of a graph class admit constant width tree decompositions whose bags and edges can be identified with MSOL-predicates. For graph classes that admit MSOL-definable constant width tree decompositions that have bounded degree or allow for a linear ordering of all nodes with the same parent we even give a stronger result: In that case, the counting predicates of CMSOL are not needed.

(Meta) Kernelization

In a parameterized problem, every instance I comes with a positive integer k . The problem is said to admit a polynomial kernel if, in polynomial time, one can reduce the size of the instance I to a polynomial in k while preserving the answer. In this work, we give two meta-theorems on kernelization. The first theorem says that all problems expressible in counting monadic second-order logic and satisfying a coverability property admit a polynomial kernel on graphs of bounded genus. Our second result is that all problems that have finite integer index and satisfy a weaker coverability property admit a linear kernel on graphs of bounded genus. These theorems unify and extend all previously known kernelization results for planar graph problems.

Recognizability Equals Definability for Graphs of Bounded Treewidth and Bounded Chordality

We prove that every recognizable family of graphs of bounded treewidth and bounded chordality is definable in counting monadic second-order logic.

Large Induced Subgraphs via Triangulations and CMSO

We obtain an algorithmic metatheorem for the following optimization problem. Let $\varphi$ be a counting monadic second order logic (CMSO) formula and $t\geq 0$ be an integer. For a given graph $G=(V,E)$, the task is to maximize $|X|$ subject to the following: there is a set $ F\subseteq V$ such that $X\subseteq F $, the subgraph $G[F]$ induced by $F$ is of treewidth at most $t$, and the structure $(G[F],X)$ models $\varphi$, i.e., $(G[F],X)\models\varphi$. We give an algorithm solving this optimization problem on any $n$-vertex graph $G$ in time ${\cal O}(|\Pi_G| \cdot n^{t+4}\cdot f(t,\varphi))$, where $\Pi_G$ is the set of all potential maximal cliques in $G$ and $f$ is a function of $t$ and $\varphi$ only. Pipelined with the known bounds on the number of potential maximal cliques in different graph classes, there are a plethora of algorithmic consequences extending and subsuming many known results on polynomial-time algorithms for graph classes. We also show that all potential maximal cliques of $G$ can be enumerated in time ${\cal O}(1.7347^n)$. This implies the existence of an exact exponential algorithm of running time ${\cal O}(1.7347^n)$ for many NP-hard problems related to finding maximum induced subgraphs with different properties.

On spectra of sentences of monadic second order logic with counting

Abstract.We show that the spectrum of a sentence ϕ in Counting Monadic Second Order Logic (CMSOL) using one binary relation symbol and finitely many unary relation symbols, is ultimately periodic, provided all the models of ϕ are of clique width at most k, for some fixed k. We prove a similar statement for arbitrary finite relational vocabularies τ and a variant of clique width for τ-structures. This includes the cases where the models of ϕ are of tree width at most k. For the case of bounded tree-width, the ultimate periodicity is even proved for Guarded Second Order Logic GSOL. We also generalize this result to many-sorted spectra, which can be viewed as an analogue of Parikh's Theorem on context-free languages, and its analogues for context-free graph grammars due to Habel and Courcelle.Our work was inspired by Gurevich and Shelah (2003), who showed ultimate periodicity of the spectrum for sentences of Monadic Second Order Logic where only finitely many unary predicates and one unary function are allowed. This restriction implies that the models are all of tree width at most 2, and hence it follows from our result.