Distinguishing Complexity and Symmetry of Information

We describe how to use polynomial-time Kolmogorov distinguishing complexity to give an approximate measure of the size of sets. For any set S, we show that relative to S the polynomial time distinguishing complexity of every element of length n of S is bounded by 2 log jjS =n jj + O(log n). This lemma enables us to give a characterization of sparse sets using distinguishing complexity. We use this new lemma as a catalyst to study symmetry of information for polynomial-time distinguishing complexity. Longpr e and Mocas and Longpr e and Watanabe showed that if certain one-way functions exist then symmetry of information fails for the standard polynomial-time Kolmogorov complexity. We try to recover symmetry of information by studying Kolmogorov distinguishing complexity using our new approximating measure idea to avoid the problems with one-way functions and indexing of strings in (small) sets. We show problems with even formalizing the symmetry of information question for polynomial-time deterministic distinguishing complexity. Nondeterministic distinguishing complexity gives us more hope but we show that symmetry of information still seems unlikely due to the apparent inability of nondeterminism to approximately count.