Abstract: Polynomial-time many-one reductions provide the standard notion of completeness for complexity classes. However, as first explicated by Berman and Hartmanis in their work on the isomorphism conjecture, all natural complete problems are actually complete under reductions with stronger properties. We study the length-increasing property and show under various computational hardness assumptions that all PSPACE-complete problems are complete via length-increasing reductions that are computable with a small amount of nonuniform advice.
If there is a problem in PSPACE that requires exponential time, then polynomial size advice suffices to give length-increasing reductions to all PSPACE-complete sets. Under the stronger assumption that linear space requires exponential-size NP-oracle circuits, we reduce the advice to logarithmic size. Our proofs make use of pseudorandom generators, hardness versus randomness tradeoffs, and worst-case to average-case hardness reductions.