We present logic based methods for constructing XP and FPT graph algorithms, parameterized by tree-width or clique-width. We will use fly-automata introduced in a previous article. They make it possible to check properties that are not monadic second-order expressible because their states may include counters, so that their set of states may be infinite. We equip these automata with output functions, so that they can compute values associated with terms or graphs. We present tools for constructing easily algorithms by combining predefined automata for basic functions and properties.
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