Monadic Second-Order Logic for Graphs: Algorithmic and Language Theoretical Applications

This tutorial will present an overview of the use of Monadic Second-Order Logic to describe sets of finite graphs and graph transformations, in relation with the notions of tree-width and clique-width. It will review applications to the construction of algorithms, to Graph Theory and to the extension to graphs of Formal Language Theory concepts. We first explain the role of Logic. A graph, either finite or infinite, can be considered as a logical structure whose domain (the ground set of the logical structure) consists of the set of vertices ; a binary relation on this set represents adjacency. Graph properties can be expressed by logical formulas of different languages and classified accordingly. First-order formulas are rather weak in this respect because they can only express local properties, like having degree or diameter at most k for fixed k. Most properties of interest in Graph Theory can be expressed in second-order logic (this language allows quantifications on relations of arbitrary but fixed arity), but unfortunately, little can be obtained from such expressions. Monadic second-order formulas are second-order formulas that only use quantifications on unary relations, i.e., on sets. They can express many basic and useful graph properties like connectivity, k-colorability, planarity and minor inclusion, just to take a few examples. These properties are said to bemonadic secondorder expressible and the corresponding sets of graphs are monadic second-order definable. Many algorithmic properties follow from such logical descriptions. In particular, every monadic second-order definable set of finite graphs of bounded tree-width has a linear time recognition algorithm ([1], [2], [4], [5], [6]). Monadic second-order formulas are also used in Formal Language Theory to describe languages, i.e., sets of words or terms. A fundamental result in this field is that monadic second-order formulas and finite automata have the same expressive power. It is fundamental for the theory and practice of model-checking, Supported by the GRAAL project of "Agence Nationale pour la Recherche".