In this paper, we use resource-bounded dimension theory to investigate polynomial size circuits. We show that for every i ≥ 0, P/poly has ith order scaled p 3 -strong dimension 0. We also show that P/poly i.o. has p 3 -dimension 1/2, p 3 -strong dimension 1. Our results improve previous measure results of Lutz (1992) and dimension results of Hitchcock and Vinodchandran (2004).

The problem of counting unlabelled subtrees of a tree (that is, subtrees that are distinct up to isomorphism) is #P-complete, and hence equivalent in computational difficulty to evaluating the permanent of a 0,1-matrix.

Implications of a formal context (G,M, I) have a minimal implication basis, called Duquenne-Guigues basis or stem base. It is shown that the problem of deciding whether a set of attributes is a premise of the stem base is in coNP and determining the size of the stem base is polynomially Turing equivalent to a #P-complete problem.

We show that an essential part of the new (Jones-type) polynomial link invariants can be computed in subexponential time. This is in a sharp contrast to the result of Jaeger, Vertigan and Welsh that computing the whole polynomial and most of its evaluations is #P-hard.

We show that it is #P-complete to count the number of bases of matroids representable over a fixed infinite field or fields of fixed characteristic.