On the Complexity of Recognizing Iterated Differences of Polyhedra

The iterated diierence of polyhedra V = P1n(P2n independently in 11] and 7] as a suucient condition for V to be exactly computable by a two-layered neural network. An algorithm checking whether V I R d is an iterated diierence of polyhedra is proposed in 11]. 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 diierence of polyhedra. This paper sheds some light on the nature of iterated diierence of polyhedra. The outcomes are : (i) an algorithm which always stops after a small number of iterations, (ii) suucient conditions for this algorithm to be polynomial and (iii) the proof that an iterated diierence of polyhedra can be exactly computed by a two-layered neural network using only essential hyperplanes.