New Complexity Bounds for Cylindrical Decompositions of Sub-Pfaffian Sets

Abstract

Tarski-Seidenberg principle plays a key role in real algebraic geometry and its applications. It is also constructive and some efficient quantifier elimination algorithms appeared recently. However, the principle is wrong for first-order theories involving certain real analytic functions (e.g., an exponential function). In this case a weaker statement is sometimes true, a possibility to eliminate one sort ofq uantifiers (either ∀ or ∃). We construct an algorithm for a cylindrical cell decomposition ofa closed cube In ⊂ Rn compatible with a semianalytic subset S ⊂ In, defined by analytic functions from a certain broad finitely defined class (Pfaffian functions), modulo an oracle for deciding emptiness of such sets. In particular the algorithm is able to eliminate one sort ofq uantifiers from a first-order formula. The complexity of the algorithm and the bounds on the output are doubly exponential in O(n2).KeywordsInductive HypothesisOpen DomainReal Analytic FunctionCell DecompositionCylindrical CellThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.