Monadic second-order definable graph orderings

We study the question of whether, for a given class of finite graphs, one can define, for each graph of the class, a linear ordering in monadic second-order logic, possibly with the help of monadic parameters. We consider two variants of monadic second-order logic: one where we can only quantify over sets of vertices and one where we can also quantify over sets of edges. For several special cases, we present combinatorial characterisations of when such a linear ordering is definable. In some cases, for instance for graph classes that omit a fixed graph as a minor, the presented conditions are necessary and sufficient; in other cases, they are only necessary. Other graph classes we consider include complete bipartite graphs, split graphs, chordal graphs, and cographs. We prove that orderability is decidable for the so called HR-equational classes of graphs, which are described by equation systems and generalize the context-free languages

Weighted Automata and Monadic Second Order Logic

Let S be a commutative semiring. M. Droste and P. Gastin have introduced in 2005 weighted monadic second order logic WMSOL with weights in S. They use a syntactic fragment RMSOL of WMSOL to characterize word functions (power series) recognizable by weighted automata, where the semantics of quantifiers is used both as arithmetical operations and, in the boolean case, as quantification. Already in 2001, B. Courcelle, J.Makowsky and U. Rotics have introduced a formalism for graph parameters definable in Monadic Second order Logic, here called MSOLEVAL with values in a ring R. Their framework can be easily adapted to semirings S. This formalism clearly separates the logical part from the arithmetical part and also applies to word functions. In this paper we give two proofs that RMSOL and MSOLEVAL with values in S have the same expressive power over words. One proof shows directly that MSOLEVAL captures the functions recognizable by weighted automata. The other proof shows how to translate the formalisms from one into the other.Comment: In Proceedings GandALF 2013, arXiv:1307.416

Computabilities of Validity and Satisfiability in Probability Logics over Finite and Countable Models

The $\epsilon$-logic (which is called $\epsilon$E-logic in this paper) of Kuyper and Terwijn is a variant of first order logic with the same syntax, in which the models are equipped with probability measures and in which the $\forall x$ quantifier is interpreted as "there exists a set $A$ of measure $\ge 1 - \epsilon$ such that for each $x \in A$, ...." Previously, Kuyper and Terwijn proved that the general satisfiability and validity problems for this logic are, i) for rational $\epsilon \in (0, 1)$, respectively $\Sigma^1_1$-complete and $\Pi^1_1$-hard, and ii) for $\epsilon = 0$, respectively decidable and $\Sigma^0_1$-complete. The adjective "general" here means "uniformly over all languages." We extend these results in the scenario of finite models. In particular, we show that the problems of satisfiability by and validity over finite models in $\epsilon$E-logic are, i) for rational $\epsilon \in (0, 1)$, respectively $\Sigma^0_1$- and $\Pi^0_1$-complete, and ii) for $\epsilon = 0$, respectively decidable and $\Pi^0_1$-complete. Although partial results toward the countable case are also achieved, the computability of $\epsilon$E-logic over countable models still remains largely unsolved. In addition, most of the results, of this paper and of Kuyper and Terwijn, do not apply to individual languages with a finite number of unary predicates. Reducing this requirement continues to be a major point of research. On the positive side, we derive the decidability of the corresponding problems for monadic relational languages --- equality- and function-free languages with finitely many unary and zero other predicates. This result holds for all three of the unrestricted, the countable, and the finite model cases. Applications in computational learning theory, weighted graphs, and neural networks are discussed in the context of these decidability and undecidability results.Comment: 47 pages, 4 tables. Comments welcome. Fixed errors found by Rutger Kuype

Finiteness conditions for graph algebras over tropical semirings

Connection matrices for graph parameters with values in a field have been introduced by M. Freedman, L. Lov{\'a}sz and A. Schrijver (2007). Graph parameters with connection matrices of finite rank can be computed in polynomial time on graph classes of bounded tree-width. We introduce join matrices, a generalization of connection matrices, and allow graph parameters to take values in the tropical rings (max-plus algebras) over the real numbers. We show that rank-finiteness of join matrices implies that these graph parameters can be computed in polynomial time on graph classes of bounded clique-width. In the case of graph parameters with values in arbitrary commutative semirings, this remains true for graph classes of bounded linear clique-width. B. Godlin, T. Kotek and J.A. Makowsky (2008) showed that definability of a graph parameter in Monadic Second Order Logic implies rank finiteness. We also show that there are uncountably many integer valued graph parameters with connection matrices or join matrices of fixed finite rank. This shows that rank finiteness is a much weaker assumption than any definability assumption.Comment: 12 pages, accepted for presentation at FPSAC 2014 (Chicago, June 29 -July 3, 2014), to appear in Discrete Mathematics and Theoretical Computer Scienc

Deciding first-order properties of nowhere dense graphs

Nowhere dense graph classes, introduced by Nesetril and Ossona de Mendez, form a large variety of classes of "sparse graphs" including the class of planar graphs, actually all classes with excluded minors, and also bounded degree graphs and graph classes of bounded expansion. We show that deciding properties of graphs definable in first-order logic is fixed-parameter tractable on nowhere dense graph classes. At least for graph classes closed under taking subgraphs, this result is optimal: it was known before that for all classes C of graphs closed under taking subgraphs, if deciding first-order properties of graphs in C is fixed-parameter tractable, then C must be nowhere dense (under a reasonable complexity theoretic assumption). As a by-product, we give an algorithmic construction of sparse neighbourhood covers for nowhere dense graphs. This extends and improves previous constructions of neighbourhood covers for graph classes with excluded minors. At the same time, our construction is considerably simpler than those. Our proofs are based on a new game-theoretic characterisation of nowhere dense graphs that allows for a recursive version of locality-based algorithms on these classes. On the logical side, we prove a "rank-preserving" version of Gaifman's locality theorem.Comment: 30 page

Monadic second-order model-checking on decomposable matroids

A notion of branch-width, which generalizes the one known for graphs, can be defined for matroids. We first give a proof of the polynomial time model-checking of monadic second-order formulas on representable matroids of bounded branch-width, by reduction to monadic second-order formulas on trees. This proof is much simpler than the one previously known. We also provide a link between our logical approach and a grammar that allows to build matroids of bounded branch-width. Finally, we introduce a new class of non-necessarily representable matroids, described by a grammar and on which monadic second-order formulas can be checked in linear time.Comment: 32 pages, journal paper. Revision: the last part has been removed and the writing improve

Data-Compression for Parametrized Counting Problems on Sparse Graphs

We study the concept of compactor, which may be seen as a counting-analogue of kernelization in counting parameterized complexity. For a function F:Sigma^* -> N and a parameterization kappa: Sigma^* -> N, a compactor (P,M) consists of a polynomial-time computable function P, called condenser, and a computable function M, called extractor, such that F=M o P, and the condensing P(x) of x has length at most s(kappa(x)), for any input x in Sigma^*. If s is a polynomial function, then the compactor is said to be of polynomial-size. Although the study on counting-analogue of kernelization is not unprecedented, it has received little attention so far. We study a family of vertex-certified counting problems on graphs that are MSOL-expressible; that is, for an MSOL-formula phi with one free set variable to be interpreted as a vertex subset, we want to count all A subseteq V(G) where |A|=k and (G,A) models phi. In this paper, we prove that every vertex-certified counting problems on graphs that is MSOL-expressible and treewidth modulable, when parameterized by k, admits a polynomial-size compactor on H-topological-minor-free graphs with condensing time O(k^2n^2) and decoding time 2^{O(k)}. This implies the existence of an FPT-algorithm of running time O(n^2 k^2)+2^{O(k)}. All aforementioned complexities are under the Uniform Cost Measure (UCM) model where numbers can be stored in constant space and arithmetic operations can be done in constant time