By Professor Bruno Courcelle, Dr Joost Engelfriet

The learn of graph constitution has complicated lately with nice strides: finite graphs should be defined algebraically, allowing them to be developed out of extra uncomplicated components. individually the homes of graphs might be studied in a logical language referred to as monadic second-order common sense. during this e-book, those positive factors of graph constitution are introduced jointly for the 1st time in a presentation that unifies and synthesizes examine over the past 25 years. the writer not just offers a radical description of the idea, but additionally info its functions, at the one hand to the development of graph algorithms, and, at the different to the extension of formal language idea to finite graphs. as a result the ebook can be of curiosity to graduate scholars and researchers in graph conception, finite version idea, formal language concept, and complexity idea.

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**Additional info for Graph Structure and Monadic Second-Order Logic: A Language-Theoretic Approach**

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We are grateful to the participants of the European workshops and projects on graph grammars and graph transformations, initiated by H. Ehrig and G. Rozenberg, for the many years of discussion and collaboration. Without being a member of the Institut Universitaire de France (IUF), B. Courcelle could not have worked on this book. He thanks M. Nivat and W. Thomas who presented his application to IUF, and all those who supported it by writing recommendation letters. He dedicates his work to the memory of Ph.

Furthermore, it fits very well with the notion of an equational set. 5. Following Mezei and Wright [MezWri], we say that a subset L of an F-algebra M (where F is finite) is recognizable if L = h−1 (N ) for some homomorphism of F-algebras h : M → Q, where Q is a finite F-algebra and N ⊆ Q. We will denote by Rec(M) the family of recognizable subsets of M. The above definition of recognizability of L is equivalent to saying that the property L P of the elements of M , such that P L (x) is True if and only if x ∈ L, belongs to a finite F-inductive set of properties.

Then P is universally valid on L if and only if L¬P = ∅, which is decidable. 5 defined by the equation L = f LL ∪ {x, y}. 9 “discovers” this fact. 2 Inductive sets of properties and recognizability 35 and odd length respectively. These languages are defined by the two equations: K0 = f K0 K1 ∪ f K1 K0 , K1 = f K0 K0 ∪ f K1 K1 ∪ {x, y}. It is easy to see that K0 is empty (just look at the corresponding context-free grammar). Hence K1 = L and every word of L has odd length. It is useful to have a proof by fixed-point induction that a property is universally valid on an equational set although an algorithm can also give the answer, because a proof is more informative than the yes or no answer of an algorithm: it shows the properties of all components of the solution of the equation system that “contribute” to the validity of the proved property.