On the Vapnik-Chervonenkis dimension of computer programs which use transcendental elementary operations

Abstract

We exhibit upper bounds for the Vapnik-Chervonenkis (VC) dimension of a wide family of concept classes that are defined by algorithms using analytic Pfaffian functions. We give upper bounds on the VC dimension of concept classes in which the membership test for whether an input belongs to a concept in the class can be performed either by a computation tree or by a circuit with sign gates containing Pfaffian functions as operators. These new bounds are polynomial both in the height of the tree and in the depth of the circuit. As consequence we obtain polynomial VC dimension not also for classes of concepts whose membership test can be defined by polynomial time algorithms but also for those defined by well-parallelizable sequential exponential time algorithms.