Complexity of cylindrical decompositions of sub-Pfaffian sets

We construct an algorithm for a cylindrical cell decomposition of a closed cube image compatible with a “restricted” sub-Pfaffian subset Ysubset ofIn, provided an oracle deciding consistency of a system of Pfaffian equations and inequalities is given. In particular, the algorithm produces the complement image. The complexity bound of the algorithm, the number and formats of cells are doubly exponential in n3.