Boolean Operations, Joins, and the Extended Low Hierarchy

We prove that the join of two sets may actually fall into a lower level of the extended low hierarchy than either of the sets. In particular, there exist sets that are not in the second level of the extended low hierarchy, EL_2, yet their join is in EL_2. That is, in terms of extended lowness, the join operator can lower complexity. Since in a strong intuitive sense the join does not lower complexity, our result suggests that the extended low hierarchy is unnatural as a complexity measure. We also study the closure properties of EL_ and prove that EL_2 is not closed under certain Boolean operations. To this end, we establish the first known (and optimal) EL_2 lower bounds for certain notions generalizing Selman's P-selectivity, which may be regarded as an interesting result in its own right.Comment: 12 page

Depth, Highness and DNR degrees

We study Bennett deep sequences in the context of recursion theory; in particular we investigate the notions of O(1)-deepK, O(1)-deepC , order-deep K and order-deep C sequences. Our main results are that Martin-Loef random sets are not order-deepC , that every many-one degree contains a set which is not O(1)-deepC , that O(1)-deepC sets and order-deepK sets have high or DNR Turing degree and that no K-trival set is O(1)-deepK.Comment: journal version, dmtc

Depth, Highness and DNR Degrees

A sequence is Bennett deep [5] if every recursive approximation of the Kolmogorov complexity of its initial segments from above satisfies that the difference between the approximation and the actual value of the Kolmogorov complexity of the initial segments dominates every constant function. We study for different lower bounds r of this difference between approximation and actual value of the initial segment complexity, which properties the corresponding r(n)-deep sets have. We prove that for r(n) = εn, depth coincides with highness on the Turing degrees. For smaller choices of r, i.e., r is any recursive order function, we show that depth implies either highness or diagonally-non-recursiveness (DNR). In particular, for left-r.e. sets, order depth already implies highness. As a corollary, we obtain that weakly-useful sets are either high or DNR. We prove that not all deep sets are high by constructing a low order-deep set. Bennett's depth is defined using prefix-free Kolmogorov complexity. We show that if one replaces prefix-free by plain Kolmogorov complexity in Bennett's depth definition, one obtains a notion which no longer satisfies the slow growth law (which stipulates that no shallow set truth-table computes a deep set); however, under this notion, random sets are not deep (at the unbounded recursive order magnitude). We improve Bennett's result that recursive sets are shallow by proving all K-trivial sets are shallow; our result is close to optimal. For Bennett's depth, the magnitude of compression improvement has to be achieved almost everywhere on the set. Bennett observed that relaxing to infinitely often is meaningless because every recursive set is infinitely often deep. We propose an alternative infinitely often depth notion that doesn't suffer this limitation (called i.o. depth).We show that every hyperimmune degree contains a i.o. deep set of magnitude εn, and construct a π01- class where every member is an i.o. deep set of magnitude εn. We prove that every non-recursive, non-DNR hyperimmune-free set is i.o. deep of constant magnitude, and that every nonrecursive many-one degree contains such a set

Depth, Highness and DNR Degrees

A sequence is Bennett deep [5] if every recursive approximation of the Kolmogorov complexity of its initial segments from above satisfies that the difference between the approximation and the actual value of the Kolmogorov complexity of the initial segments dominates every constant function. We study for different lower bounds r of this difference between approximation and actual value of the initial segment complexity, which properties the corresponding r(n)-deep sets have. We prove that for r(n) = εn, depth coincides with highness on the Turing degrees. For smaller choices of r, i.e., r is any recursive order function, we show that depth implies either highness or diagonally-non-recursiveness (DNR). In particular, for left-r.e. sets, order depth already implies highness. As a corollary, we obtain that weakly-useful sets are either high or DNR. We prove that not all deep sets are high by constructing a low order-deep set. Bennett's depth is defined using prefix-free Kolmogorov complexity. We show that if one replaces prefix-free by plain Kolmogorov complexity in Bennett's depth definition, one obtains a notion which no longer satisfies the slow growth law (which stipulates that no shallow set truth-table computes a deep set); however, under this notion, random sets are not deep (at the unbounded recursive order magnitude). We improve Bennett's result that recursive sets are shallow by proving all K-trivial sets are shallow; our result is close to optimal. For Bennett's depth, the magnitude of compression improvement has to be achieved almost everywhere on the set. Bennett observed that relaxing to infinitely often is meaningless because every recursive set is infinitely often deep. We propose an alternative infinitely often depth notion that doesn't suffer this limitation (called i.o. depth).We show that every hyperimmune degree contains a i.o. deep set of magnitude εn, and construct a π01- class where every member is an i.o. deep set of magnitude εn. We prove that every non-recursive, non-DNR hyperimmune-free set is i.o. deep of constant magnitude, and that every nonrecursive many-one degree contains such a set

Downward Collapse from a Weaker Hypothesis

Hemaspaandra et al. proved that, for $m > 0$ and $0 < i < k - 1$: if \Sigma_i^p \BoldfaceDelta DIFF_m(\Sigma_k^p) is closed under complementation, then $DIFF_m(\Sigma_k^p) = coDIFF_m(\Sigma_k^p)$. This sharply asymmetric result fails to apply to the case in which the hypothesis is weakened by allowing the $\Sigma_i^p$ to be replaced by any class in its difference hierarchy. We so extend the result by proving that, for $s,m > 0$ and $0 < i < k - 1$: if DIFF_s(\Sigma_i^p) \BoldfaceDelta DIFF_m(\Sigma_k^p) is closed under complementation, then $DIFF_m(\Sigma_k^p) = coDIFF_m(\Sigma_k^p)$

Query Order and the Polynomial Hierarchy

Hemaspaandra, Hempel, and Wechsung [cs.CC/9909020] initiated the field of query order, which studies the ways in which computational power is affected by the order in which information sources are accessed. The present paper studies, for the first time, query order as it applies to the levels of the polynomial hierarchy. We prove that the levels of the polynomial hierarchy are order-oblivious. Yet, we also show that these ordered query classes form new levels in the polynomial hierarchy unless the polynomial hierarchy collapses. We prove that all leaf language classes - and thus essentially all standard complexity classes - inherit all order-obliviousness results that hold for P.Comment: 14 page