The Power of Local Self-Reductions

Identify a string x over f0; 1g with the positive integer whose binary representation is 1x. We say that a self-reduction is k-local if on input x all queries belong to fx ? 1; : : :; x ? kg. We show that all k-locally self-reducible sets belong to PSPACE. However, the power of k-local self-reductions changes drastically between k = 2 and k = 3. Although all 2-locally self-reducible sets belong to MOD 6 PH, some 3-locally self-reducible sets are PSPACE-complete. Furthermore, there exists a 6-locally self-reducible PSPACE-complete set whose self-reduction is an m-reduction (in fact, a permutation). We prove all these results by showing that such languages are equivalent in complexity to the problem of multiplying an exponentially long sequence of uniformly generated elements in a nite monoid, and then exploiting the algebraic structure of the monoid.