Title: The Power of Local Self-Reductions
Authors: Richard Beigel and Howard Straubing
Abstract:
Identify a string x over {0,1} 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
{x - 1, ... , x - k}. 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 MOD6PH, 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
finite monoid, and then exploiting the algebraic structure of the
monoid.
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