Polynomials of Bounded Tree-Width

Basic algorithms in real algebraic geometry and their complexity: from Sturm's theorem to the existential theory of reals. polar varieties, real equation solving, and data structures: The hypersurface case. A partial k-arboretum of graphs with bounded tree-width (tutorial). a theory of computation and complexity over the real numbers: NP-completeness, recursive functions and universal machines. Bull. The Feferman-Vaught theorem holds also true in a corresponding way for the other parsing operations introduced above. It is true as well for other classes of parameters like the clique-width of a graph and its corresponding graph operations, see CMR00b]. It is then used for computational purposes as follows. Given a MS 2 () formula 2 F n;r we precompute a table of all Hintikka formulas in H n;r and how three of them t together with respect to the Feferman-Vaught theorem for all the parsing operations. The formula is written in its equivalent form as disjunction of Hintikka formulas, say = 1 _: : :_ s : Now, given a structure of bounded tree-width we rst construct its tree decomposition and the corresponding parse tree according to theorems 6 and theorem:DowneyFellows99. From the precomputed table of Hintikka formulas we obtain a corresponding decomposition of the i 's. Then we evaluate bottom-up from the leaves of the parse tree to its root the Hintikka formulas in the decomposition (using again the precomputed table) and check whether is true on the input. Since the parse tree has linear size in the size of the structure the running time is linear (though with large constants). Some technicalities have to be taken care of. We demonstrate them again for the join operator. Theorem 17 depends on the shape (x; y) an assignment z induces on the free rst-order variables. Therefore, if the above algorithm is performed bottom up, we have to take into account all possible assignments for z, that is all possible patterns z induces on the free rst-order variables. Suppose there are s many of them (where s only depends on the given formula). If we consider all decomposition patterns of these s many variables along the parse tree of a given structure of bounded tree-width we see that their number only depends on a function f(s) in s. For every xed pattern we perform the above algorithm in linear time, giving an O(n) algorithm in total. The interested reader might try to perform a proof …