Complexity Lower Bounds for Computation Trees with ElementaryTranscendental Function

We consider computation trees which admit as gate functions along with the usual arithmetic operations also algebraic or transcendental functions like exp; log; sin; square root (deened in the relevant domains) or much more general, Pfaaan functions. A new method for proving lower bounds on the depth of these trees is developed which allows to prove a lower bound (p logN) for testing membership to a convex polyhedron with N facets of all dimensions, provided that N is large enough. This method diiers essentially from the approaches adopted for algebraic computation 1 Pfaaan computation trees We consider the following computation model, a generalization of the algebraic computation trees (see, e.g., 1], 26]). Deenition 1. Pfaaan computation tree T is a tree at every node v of which a Pfaaan function f v in variables X 1 ; : : :; X n is attached, which satisses the following properties. Let f v0 ; : : :; f vl ; f vl+1 = f v be the functions attached to all the nodes along the branch T v of T leading from the root v 0 to v l+1 = v. We assume that the Pfaaan function f v satisses the following diierential equation (see 20]): df v = where g v;j are polynomials with real coeecients. The tree T branches at v to its three sons according to the sign of f v (cf. 1]). Thereby, to each node v one can assign (by induction on the depth l + 1 of v) a set U v R n consisting of all the points for which the sign conditions for the functions f v0 ; : : :; f vl along the branch T v are valid. Thus, at the induction step, one assigns to three sons of v the sets U v \ ff v > 0g; U v \ ff v = 0g; U v \ ff v < 0g; respectively. We assume also that the function f v is real analytic in U v. To each leaf w of T an output \yes" or \no" is assigned, we call the set U w accepting set if to w \yes" is assigned. We say that T tests the membership problem to the union of all accepting sets (sf. 1]). Taking polynomials as the gate functions f v in T , we come to the algebraic computation trees. The …