Hausdorff dimension assigns a dimension value to each subset of an arbitrary metric space. In Euclidean space, this concept coincides with our intuition that

Abstract Resource-bounded dimension is a notion of computational information density of in-finite sequences based on computationally bounded gamblers. This paper develops the theory of pushdown dimension and explores its relationship with finite-state dimension.The pushdown dimension of any sequence is trivially bounded above by its finite-state dimension, since a pushdown gambler can simulate any finite-state gambler. We show thatfor every rational 0 < d < 1, there exists a sequence with finite-state dimension d whosepushdown dimension is at most d/2. This provides a stronger quantitative analogue of thewell-known fact that pushdown automata decide strictly more languages than finite-state

We show that hard sets S for NP must have exponential density, for some ɛ> 0 and infinitely many n, unless coNP ⊆ i.e. |S=n | ≥ 2nɛ NP/poly and the polynomial-time hierarchy collapses. This result holds for Turing reductions that make n1−ɛ queries. In addition we study the instance complexity of NP-hard problems and show that hard sets also have an exponential amount of instances that have instance complexity nδ for some δ> 0. This result also holds for Turing reductions that make n1−ɛ queries. 1

We use the connection between resource-bounded dimension and the online mistake-boundmodel of learning to show that the following classes have polynomial-time dimension zero. 1. The class of problems which reduce to nondense sets via a majority reduction. 2. The class of problems which reduce to nondense sets via an iterated reduction that com-poses a bounded-query truth-table reduction with a conjunctive reduction. As corollary,