The monadic second-order logic of graphs. II: Infinite graphs of bounded width.

*(English)*Zbl 0694.68043From the author’s introduction: “Finite graphs can be denoted by algebraic expressions M. Bauderon and B. Courcelle [Math. Syst. Theory 20, No.2/3, 83-127 (1987; Zbl 0641.68115)] that are constructed over an infinite signature \({\mathbb{H}}\). Infinite graph expressions can be defined, and they denote countable graphs. With finitely many operators, we cannot denote all countable graphs, but only those of finite width. These infinite graph expressions are actually infinite trees. On the other hand, Courcelle has defined... a monadic second order logical language appropriate for expressing properties of graphs. Our main theorem states that given a finite subset \({\mathbb{K}}\) of \({\mathbb{H}}\), and a monadic second-order formula \(\phi\), we can construct a monadic second-order formula that defines the set \({\mathbb{E}}({\mathbb{K}},\phi)\) of finite or infinite graph expressions constructed over \({\mathbb{K}}\), the corresponding graph of which satisfies \(\phi\). From Rabin’s theorem, saying that the monadic second-order theory of the complete binary tree is decidable, it follows that the emptiness of the set \({\mathbb{E}}({\mathbb{K}},\phi)\) is decidable. Hence the monadic second-order theory of all countable graphs, of width less than some fixed integer is decidable.” The paper is carefully and clearly written, with enough examples to make reading relatively easy.

Reviewer: S.Bloom

##### MSC:

68R10 | Graph theory (including graph drawing) in computer science |

03B15 | Higher-order logic; type theory (MSC2010) |

05C99 | Graph theory |

68Q60 | Specification and verification (program logics, model checking, etc.) |

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\textit{B. Courcelle}, Math. Syst. Theory 21, No. 4, 187--221 (1989; Zbl 0694.68043)

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##### References:

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