On the Complexity of Recognizing Iterated Differences of Polyhedra on the Complexity of Recognizing Iterated Differences of Polyhedra

The iterated di erence of polyhedra V P n P n Pk has been proposed independently in and as a su cient condition for V to be exactly computable by a two layered neural network An algorithm checking whether V IR is an iterated di erence of polyhedra is proposed in However this algorithm is not practically usable because it has a high computational complexity and it was only conjectured to stop with a negative answer when applied to a region which is not an iterated di erence of polyhedra This paper sheds some light on the nature of iterated di erence of polyhedra The outcomes are i an algorithm which always stops after a small number of iterations ii su cient conditions for this algorithm to be polynomial and iii the proof that an iterated di erence of polyhedra can be exactly computed by a two layered neural network using only essential hyperplanes