Deciding the Vapnik-Cervonenkis dimension is Σ3 p-complete

Linial et al. (1988) raised the question of how difficult the computation of the Vapnik-Cervonenkis dimension of a concept class over a finite universe is. Papadimitriou and Yannakakis (1993) obtained a first answer using matrix representations of concept classes. However, this approach does not capture classes having exponential size, like monomials, which are encountered in learning theory. We choose a more natural representation, which leads us to redefine the VC DIMENSION problem. We establish that VC DIMENSION is Σ₃^p -complete, thereby giving a rare natural example of a Σ₃^p-complete problem