Computations by fly-automata beyond monadic second-order logic

Courcelle, Bruno; Durand, Irène

doi:10.1016/j.tcs.2015.12.026

Cited by 13 publications

(26 citation statements)

References 37 publications

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“…These results indicate that the tools of [7,8] can be applied to the verification of MSO 2 properties of graphs of bounded tree-width given by their treedecompositions. The software AUTOGRAPH 12 can be used basically as it is (up to minor syntactic adaptations) although the algebras of terms describing tree-decompositions and of terms defining clique-width are fairly different (as discussed in [4]).…”

Section: Resultsmentioning

confidence: 88%

“…For every MSO formula ϕ(X 1 , ..., X p ), one can define a finite set B and an integer i such that, for each effectively given set C of labels, one can construct a deterministic FA A ϕ(X1,...,Xp),C over F (p) C that recognizes the language L ϕ(X1,...,Xp),C and whose set of states Q(C) satisfies the following properties: 8 We use this observation in Section 4.2. 9 This definition is used in [9] to prove that the (Strong) Recognizability Theorem, Theorem 5.68(1) of [10], can be proved via the construction of automata.…”

Section: Automatamentioning

confidence: 99%

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Fly-automata for checking MSO2 graph properties

Courcelle¹

2018

Discrete Applied Mathematics

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A more descriptive but too long title would be : Constructing flyautomata to check properties of graphs of bounded tree-width expressed by monadic second-order formulas written with edge quantifications. Such properties are called MSO2 in short. Fly-automata (FA) run bottom-up on terms denoting graphs and compute "on the fly" the necessary states and transitions instead of looking into huge, actually unimplementable tables. In previous works, we have constructed FA that process terms denoting graphs of bounded clique-width, in order to check their monadic second-order (MSO) properties (expressed by formulas without edge quantifications). Here, we adapt these FA to incidence graphs, so that they can check MSO2 properties of graphs of bounded tree-width. This is possible because: (1) an MSO2 property of a graph is nothing but an MSO property of its incidence graph and (2) the clique-width of the incidence graph of a graph is linearly bounded in terms of its tree-width. Our constructions are actually implementable and usable. We detail concrete constructions of automata in this perspective.keywords: monadic second-order logic, edge quantification, tree-width, clique-width, algorithmic meta-theorem, fly-automaton.2 By a result of Bodlaender (see [1,12,13]), a tree-decomposition of G of width k can be computed in linear time if there exists one. Hence the variant of (b) where a tree-decomposition is not given but must be computed also holds, but this variant is not a consequence of (a). Furthermore, the linear time decomposition algorithm is not practically implementable.3 Automata A ϕ,k can also be defined for formulas ϕ with free variables. See Theorem 5.4 FA can have infinitely many states: a state can record the (unbounded) number of occurrences of a particular symbol. We can thus construct fly-automata that check properties that are not MSO expressible (for example that a graph is regular or can be partitioned into p disjoint regular graphs). These automata yield FPT or XP algorithms [12,13] for clique-width as parameter. By equipping fly-automata with output functions, we can make them compute values attached to graphs: for example, assuming that the input graph is s-colorable, the minimum size of X 1 in an s-coloring (X 1 , . . . , Xs). We can compute the number of p-vertex colorings, and also the number of so-called of acyclic 4-colorings of Petersen's graph: 10800. The number of acyclic 3-colorings of McGee's graph is 57024. See [8].

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Section: Resultsmentioning

confidence: 88%

Section: Automatamentioning

confidence: 99%

Fly-automata for checking MSO2 graph properties

Courcelle¹

2018

Discrete Applied Mathematics

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“…(We will not manipulate tree-decompositions.) We denote by twd(G) the tree-width of a graph G. Similarly, for clique-width 9 , denoted by cwd(G), we refer the reader to [12,15,16,17,18,19].…”

Section: Tree-width and Clique-widthmentioning

confidence: 99%

“…Remark about the claim : The survey article [30] states that if one adds or deletes an edge to a graph, one can increase or decrease its clique-width by at most 17 2 (Theorem 9). Hence, if one adds m edges to a graph, one can increase its clique-width by at most 2m.…”

Section: Planarity and Does Not Modify ω(X Y )mentioning

confidence: 99%

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On quasi-planar graphs: Clique-width and logical description

Courcelle

2020

Discrete Applied Mathematics

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Motivated by the construction of FPT graph algorithms parameterized by clique-width or tree-width, we study graph classes for which treewidth and clique-width are linearly related. This is the case for all graph classes of bounded expansion, but in view of concrete applications, we want to have "small" constants in the comparisons between these width parameters.We focus our attention on graphs that can be drawn in the plane with limited edge crossings, for an example, at most p crossings for each edge. These graphs are called p-planar. We consider a more general situation where the graph of edge crossings must belong to a fixed class of graphs D. For p-planar graphs, D is the class of graphs of degree at most p. We prove that tree-width and clique-width are linearly related for graphs drawable with a graph of crossings of bounded average degree.We prove that the class of 1-planar graphs, although conceptually close to that of planar graphs, is not characterized by a monadic second-order sentence. We identify two subclasses that are.

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Model Checking with Fly-Automata

Courcelle

Durand

2016

Encyclopedia of Algorithms

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Problem definition The verification of monadic second-order (MSO) graph properties, equivalently, the model-checking problem for MSO logic over finite binary relational structures, is fixed-parameter tractable (FPT) for the parameter consisting of the formula that expresses the property and the tree-width or the clique-width of the input graph or structure. How to build usable algorithms for this problem ? The proof of the general theorem (an algorithmic meta-theorem, cf. [12]) is based on the description of the inputs by algebraic terms and the construction of finite automata that accepts the terms describing the satisfying inputs. But these automata are in practice much too large to be constructed [11, 14]. A typical number of states is 2 2 10 and lower-bounds match this number. Can one use automata and overcome this difficulty ? Key results We propose to use fly-automata (FA) [3]. They are automata whose states are described and not listed, and whose transitions are computed on the fly and not tabulated. When running on a term of size 1000, a fly-automaton with 2 2 10 states computes only 1000 transitions if it is deterministic. FA can have infinitely many states. For example, a state can record, among other things, the (unbounded)

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Computations by fly-automata beyond monadic second-order logic

Cited by 13 publications

References 37 publications

Fly-automata for checking MSO2 graph properties

Fly-automata for checking MSO2 graph properties

On quasi-planar graphs: Clique-width and logical description

Model Checking with Fly-Automata

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