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Communication Dans Un Congrès Année : 2010

Tractable constructions of finite automata from monadic second-order formula

Résumé

It is well-know from [DF, FG, CMR] that the model-checking problem for MSO logic on graphs is fixed-parameter tractable (FPT) with respect to tree-width and clique-width. The proof uses tree-decompositions (for tree-width as parameter) and \ $k$-expressions (for clique-width as parameter; see below), and the construction of a finite tree-automaton from an MSO sentence, expressing the property to check. These two points are difficult : although tree-width $\leq k$ can be checked in linear time, the corresponding algorithm (by Bodlaender, see [DF]) is not practically usable. The situation is similar for bounded clique-width (see [HliOum]). Graphs can be given \emph{with} their decompositions witnessing tree-width or clique-width $\leq k$, but another difficulty arises : the automata to be constructed are much too large and computations abort by lack of memory space. This is actually unavoidable if one wants an algorithm taking as input any MSO sentence (see, e.g., [FriGro]). One possibility is to forget the idea of implementing the general theorem, and to work directly on particular problems: see\ [G1,G2] or [GH]. Another one, explored here (also in [KL] in a different way) consists in finding fragments of MSO logic having an interesting expressive power, and for which automata constructions (or other constructions) are tractable. What we propose is based on the following ideas : (1) Do not alternate quantifiers and do not determinize automata. (2) Write MSO\ formulas with Boolean set terms (see below definitions) (3) Precompile basic graph properties into "small" finite automata. We do not capture all MSO\ graph properties, but we can formalize in this way coloring and partitionning problems, and also some domination problems to take a few examples. In this abstract, we only discuss graphs of bounded clique-width, but the ideas extend to graphs of bounded tree-width and MSO formulas with edge set quantifications.
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hal-00522591 , version 1 (01-10-2010)

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  • HAL Id : hal-00522591 , version 1

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Bruno Courcelle, Irène A. Durand. Tractable constructions of finite automata from monadic second-order formula. Workshop on Logical Approaches to Barriers in Computing and Complexity, Feb 2010, Greifswald, Germany. ⟨hal-00522591⟩

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