On the reducibility of sets inside NP to sets with low information content

We study whether sets inside NP can be reduced to sets with low information content but possibly still high computational complexity. Examples of sets with low information content are tally sets, sparse sets, P-selective sets and membership comparable sets. For the graph automorphism and isomorphism problems GA and GI, for the directed graph reachability problem GAP, for the determinant function det, and for logspace self-reducible languages we establish the following results: 1. If GA is ≤ptt-reducible to a P-selective set, then GA ∈ P. 2. If GI is O(log n)-membership comparable, then GI ∈ RP. 3. If GAP is logspace O(1)-membership comparable, then GAP ∈ L. 4. If det is ≤ T -reducible to an L-selective set, then det ∈ FL. 5. If A is logspace self-reducible and ≤ T -reducible to an L-selective set, then A ∈ L. The last result is a strong logspace version of the characterisation of P as the class of self-reducible P-selective languages. As P and NL have logspace self-reducible complete sets, it also establishes a logspace analogue of the conjecture that if SAT is ≤pT-reducible to a P-selective set, then SAT ∈ P.