On the logical definability of certain graph and poset languages

We show that it is equivalent, for certain sets of finite graphs, to be definable in CMS (counting monadic second-order logic, a natural extension of monadic second-order logic), and to be recognizable in an algebraic framework induced by the notion of modular decomposition of a finite graph. More precisely, we consider the set $F\_\infty$ of composition operations on graphs which occur in the modular decomposition of finite graphs. If $F$ is a subset of $F\_{\infty}$, we say that a graph is an \calF-graph if it can be decomposed using only operations in $F$. A set of $F$-graphs is recognizable if it is a union of classes in a finite-index equivalence relation which is preserved by the operations in $F$. We show that if $F$ is finite and its elements enjoy only a limited amount of commutativity -- a property which we call weak rigidity, then recognizability is equivalent to CMS-definability. This requirement is weak enough to be satisfied whenever all $F$-graphs are posets, that is, transitive dags. In particular, our result generalizes Kuske's recent result on series-parallel poset languages

Definability Equals Recognizability for kk-Outerplanar Graphs

One of the most famous algorithmic meta-theorems states that every graph property that can be defined by a sentence in counting monadic second order logic (CMSOL) can be checked in linear time for graphs of bounded treewidth, which is known as Courcelle's Theorem. 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. We prove this conjecture for $k$-outerplanar graphs, which are known to have treewidth at most $3k-1$.Comment: 40 pages, 8 figure

MSOL-Definability Equals Recognizability for Halin Graphs and Bounded Degree k-Outerplanar Graphs

One of the most famous algorithmic meta-theorems states that every graph property that can be defined by a sentence in counting monadic second order logic (CMSOL) can be checked in linear time for graphs of bounded treewidth, which is known as Courcelle's Theorem. 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. We prove this conjecture for a number of special cases in a stronger form. That is, we show that each recognizable property is definable in MSOL, i.e. the counting operation is not needed in our expressions. We give proofs for Halin graphs, bounded degree k-outerplanar graphs and some related graph classes. We furthermore show that the conjecture holds for any graph class that admits tree decompositions that can be defined in MSOL, thus providing a useful tool for future proofs

Definability equals recognizability for graphs of bounded treewidth

We prove a conjecture of Courcelle, which states that a graph property is definable in MSO with modular counting predicates on graphs of constant treewidth if, and only if it is recognizable in the following sense: constant-width tree decompositions of graphs satisfying the property can be recognized by tree automata. While the forward implication is a classic fact known as Courcelle's theorem, the converse direction remained openComment: 21 pages, an extended abstract will appear in the proceedings of LICS 201

Order Invariance on Decomposable Structures

Order-invariant formulas access an ordering on a structure's universe, but the model relation is independent of the used ordering. Order invariance is frequently used for logic-based approaches in computer science. Order-invariant formulas capture unordered problems of complexity classes and they model the independence of the answer to a database query from low-level aspects of databases. We study the expressive power of order-invariant monadic second-order (MSO) and first-order (FO) logic on restricted classes of structures that admit certain forms of tree decompositions (not necessarily of bounded width). While order-invariant MSO is more expressive than MSO and, even, CMSO (MSO with modulo-counting predicates), we show that order-invariant MSO and CMSO are equally expressive on graphs of bounded tree width and on planar graphs. This extends an earlier result for trees due to Courcelle. Moreover, we show that all properties definable in order-invariant FO are also definable in MSO on these classes. These results are applications of a theorem that shows how to lift up definability results for order-invariant logics from the bags of a graph's tree decomposition to the graph itself.Comment: Accepted for LICS 201

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

Definable decompositions for graphs of bounded linear cliquewidth

We prove that for every positive integer $k$, there exists an $\text{MSO}_1$-transduction that given a graph of linear cliquewidth at most $k$ outputs, nondeterministically, some cliquewidth decomposition of the graph of width bounded by a function of $k$. A direct corollary of this result is the equivalence of the notions of $\text{CMSO}_1$-definability and recognizability on graphs of bounded linear cliquewidth.Comment: 39 pages, 5 figures. The conference version of the manuscript appeared in the proceedings of LICS 201

Inverse monoids of higher-dimensional strings

International audienceHalfway between graph transformation theory and inverse semigroup theory, we define higher dimensional strings as bi-deterministic graphs with distinguished sets of input roots and output roots. We show that these generalized strings can be equipped with an associative product so that the resulting algebraic structure is an inverse semigroup. Its natural order is shown to capture existence of root preserving graph mor-phism. A simple set of generators is characterized. As a subsemigroup example, we show how all finite grids are finitely generated. Last, simple additional restrictions on products lead to the definition of subclasses with decidable Monadic Second Order (MSO) language theory

On the Monadic Second-Order Transduction Hierarchy

We compare classes of finite relational structures via monadic second-order transductions. More precisely, we study the preorder where we set C \subseteq K if, and only if, there exists a transduction {\tau} such that C\subseteq{\tau}(K). If we only consider classes of incidence structures we can completely describe the resulting hierarchy. It is linear of order type {\omega}+3. Each level can be characterised in terms of a suitable variant of tree-width. Canonical representatives of the various levels are: the class of all trees of height n, for each n \in N, of all paths, of all trees, and of all grids