Bruno Courcelle

The monadic second-order logic of graphs. I. recognizable sets of finite graphs

Abstract The notion of a recognizable set of finite graphs is introduced. Every set of finite graphs, that is definable in monadic second-order logic is recognizable, but not vice versa. The monadic second-order theory of a context-free set of graphs is decidable.

Upper bounds to the clique width of graphs

Abstract Hierarchical decompositions of graphs are interesting for algorithmic purposes. Many NP complete problems have linear complexity on graphs with tree-decompositions of bounded width. We investigate alternate hierarchical decompositions that apply to wider classes of graphs and still enjoy good algorithmic properties. These decompositions are motivated and inspired by the study of vertex-replacement context-free graph grammars. The complexity measure of graphs associated with these decompositions is called clique width . In this paper we bound the clique width of a graph in terms of its tree width on the one hand, and of the clique width of its edge complement on the other.

Fundamental properties of infinite trees

Abstrwt. Infinite trees naturally arise in the formaliration and the c~udy of fhc \cm;lntic*, 01 prog-amming languages. This paper investigates some of their i:omninatorial and :iIgcbri\ic propei ties that are especially relevant to semantics. This paper is concerned in particular with regtilar and algchraic itlfinitc trees, rlor ln.ith rzg\lI;lr or algebraic s4f.s of infinite trees. For this reason moss of the propertics s~atcd in rhi

Handle-rewriting hypergraph grammars

IVOIX become trivial when restricted either to tinite trees or to infinite words. It present:, a synthesis of various aspects of infinite trees, invcstigatcd bc diIlt*ic*nt ,tuthor\ III differenr contlbxts and hopes to he a unifying step towards a theor! of infinite trct.4 tlliit coultl take place near the theory of formal languages and the combina:,:r::c of tk* free monoi,.,

Graph expressions and graph rewritings

Abstract We introduce the handle-rewriting hypergraph grammars (HH grammars), based on the replacement of handles, i.e., of subhypergraphs consisting of one hyperedge together with its incident vertices. This extends hyperedge replacement, where only the hyperedge is replaced. A HH grammar is separated (an S-HH grammar) if nonterminal handles do not overlap. The S-HH grammars are context-free, and the sets they generate can be characterized as the least solutions of certain systems of equations. They generate the same sets of graphs as the NLC-like vertex-rewriting C-edNCE graph grammars that are also context-free.

Monadic second-order evaluations on tree-decomposable graphs

We define an algebraic structure for the set of finite graphs, a notion of graph expression for defining them, and a complete set of equational rules for manipulating graph expressions. (By agraph we mean an oriented hypergraph, the hyperedges of which are labeled with symbols from a fixed finite ranked alphabet and that is equipped with a finite sequence of distinguished vertices). The notion of a context-free graph grammar is introduced (based on the substitution of a graph for a hyperedge in a graph). The notion of an equational set of graphs follows in a standard way from the algebraic structure. As in the case of context-free languages, a set of graphs is contextfree iff it is equational. By working at the level of expressions, we derive from the algebraic formalism a notion of graph rewriting which is as powerful as the usual one (based on a categorical approach) introduced by Ehrig, Pfender, and Schneider.

On the fixed parameter complexity of graph enumeration problems definable in monadic second-order logic

Abstract Every graph generated by a hyperedge replacement graph-grammar can be represented by a tree, namely the derivation tree of the derivation sequence that produced it. Certain functions on graphs can be computed recursively on the derivation trees of these graphs. By using monadic second-order logic and semiring homomorphisms, we describe in a single formalism a large class of such functions. Polynomial and even linear algorithms can be constructed for some of these functions. We unify similar results obtained by Takamizawa (1982), Bern (1987), Arnborg (1991) and Habel (1989).

An algebraic theory of graph reduction

We discuss the parametrized complexity of counting and evaluation problems on graphs where the range of counting is denable in monadic second-order logic (MSOL). We show that for bounded tree-width these problems are solvable in polynomial time. The same holds for bounded clique width in the cases, where the decomposition, which establishes the bound on the clique-width, can be computed in polynomial time and for problems expressible by monadic second-order formulas without edge set quantication. Such quantications are allowed in the case of graphs with bounded tree-width. As applications we discuss in detail how this aects the parametrized complexity of the permanent and the hamiltonian of a matrix, and more generally, various generating functions of MSOL denable graph properties. Finally, our results are also applicable to SAT and ]SAT. ? 2001 Elsevier Science B.V. All rights reserved.

Monadic second-order definable graph transductions: a survey

We show how membership in classes of graphs definable in monadic second order logic and of bounded treewidth can be decided by finite sets of terminating reduction rules. The method is constructive in the sense that we describe an algorithm which will produce, from a formula in monadic second order logic and an integer k such that the class defined by the formula is of treewidth ≤ k, a set of rewrite rules that reduces any member of the class to one of finitely many graphs, in a number of steps bounded by the size of the graph. This reduction system corresponds to an algorithm that runs in time linear in the size of the graph.

An axiomatic definition of context-free rewriting and its application to NLC graph grammars☆☆☆

Abstract Formulas of monadic second-order logic can be used to specify graph transductions, i.e., multi-valued functions from graphs to graphs. We obtain in this way classes of graph transductions, called monadic second-order definable graph transductions (or, more simply, definable transductions ) that are closed under composition and preserve the two known classes of context-free sets of graphs, namely the class of hyperedge replacement (HR) and the class of vertex replacement (VR) sets. These two classes can be characterized in terms of definable transductions and recognizable sets of finite trees, independently of the rewriting mechanisms used to define the HR and VR grammars. When restricted to words, the definable transductions are strictly more powerful than the rational transductions such that the image of every finite word is finite; they do not preserve context-free languages. We also describe the sets of discrete (edgeless) labelled graphs that are the images of HR and VR sets under definable transductions: this gives a version of Parikhs theorem (i.e., the characterization of the commutative images of context-free languages) which extends the classical one and applies to HR and VR sets of graphs