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

P-selectivity: Intersections and indices

AbstractThe P-selective sets (Selman, 1979) are those sets for which there is a polynomial-time algorithm that, given any two strings, determines which is “more likely” to belong to the set: if either of the strings is in the set, the algorithm chooses one that is in the set. We prove that, for each k, the k-ary Boolean connectives under which the P-selective sets are closed are exactly those that are either completely degenerate or almost-completely degenerate. We determine the complexity of the index set of the r.e. P-selective sets — ∑30-complete

Rotations of Periodic Strings and Short

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Rotation of Periodic Strings and Short Superstrings

This paper presents two simple approximation algorithms for the shortest superstring problem, with approximation ratios 2 2 3 ( 2:67) and 2 25 42 ( 2:596), improving the best previously published 2 3 4 approximation. The framework of our improved algorithms is similar to that of previous algorithms in the sense that they construct a superstring by computing some optimal cycle covers on the distance graph of the given strings, and then break and merge the cycles to finally obtain a Hamiltonian path, but we make use of new bounds on the overlap between two strings. We prove that for each periodic semi-infinite string ff = a1a2 \Delta \Delta \Delta of period q, there exists an integer k, such that for any (finite) string s of period p which is inequivalent to ff, the overlap between s and the rotation ff[k] = ak ak+1 \Delta \Delta \Delta is at most p+ 1 2 q. Moreover, if p q, then the overlap between s and ff[k] is not larger than 2 3 (p+q). In the previous shortes..

Logspace Reducibility: Models and Equivalences

We study the relative computational power of logspace reduction models. In particular, we study the relationships between one-way and two-way oracle tapes, resetting of the oracle head, and blanking of the oracle tape. We show that oracle models letting information persist between queries can be quite powerful, even if the information is not readable by the querying machine. We show that logspace f(n)-Turing reductions are stronger than polynomial-time f(n)-Turing reductions when f(n) = !(log n), and that this is optimal if P = L. 1 Introduction Efficient reductions are a central object of study in computational complexity theory. Polynomial-time reductions have received wide attention, and logspace-bounded reductions have also long been studied as a potentially finer-grained reducibility than polynomial-time reducibility. But the extent to which logspace reducibilities provably provide a finer-grained stratification has remained open. We resolve this with respect to relativizable t..

Logspace Reducibility: Models and Equivalences

We study the relative computational power of logspace reduction models. In particular, we study the relationships between one-way and two-way oracle tapes, resetting of the oracle head, and blanking of the oracle tape. We show that oracle models letting information persist between queries can be quite powerful, even if the information is not readable by the querying machine. We show that logspace f(n)-Turing reductions are stronger than polynomial-time f(n)-Turing reductions when f(n) = \omega (log n), and that this is optimal if P = L

Design and optimization of a novel throttling-inside-piston multi-stage hydraulic cylinder

In order to improve the performance of multi-stage cylinder to achieve rapid and steady extension, a novel throttling-inside-piston multi-stage hydraulic cylinder is proposed without changing physical size and piston working strokes of the traditional sequential multi-stage hydraulic cylinder. Based on internal structure optimal design and basic size parameter calculation, the mathematical model of the proposed throttling-inside-piston multi-stage hydraulic cylinder is studied using multi-rigid-body and impact-recovery dynamic theory. Then, the diameters of the orifices within the pistons are optimized using constraint optimization technique. Comparative simulation results show that the new type of throttling-inside-piston multi-stage hydraulic cylinder can significantly enhance the rapidity and steadiness for the erecting system

P-Selectivity: Intersections and Indices

The P-selective sets are those sets for which there is a polynomial-time algorithm that, given any two strings, determines which is ``more likely'' to belong to the set: if either of the strings is in the set, the algorithm chooses one that is in the set. We prove that, for each k, the k-ary Boolean connectives under which the P-selective sets are closed are exactly those that are either completely degenerate or almost-completely degenerate. We determine the complexity of the index set of the r.e. P-selective sets: \Sigma_3^0-complete

Polynomial-Time Multi-Selectivity

We introduce a generalization of Selman s P-selectivity that yields a more flexible notion of selectivity, called (polynomial-time) multi-selectivity, in which the selector is allowed to operate on multiple input strings. Since our introduction of this class, it has been used [HJRW] to prove the first known (and optimal) lower bounds for generalized selectivity-like classes in terms of EL2 , the second level of the extended low hierarchy. We study the resulting selectivity hierarchy, denoted by SH, which we prove does not collapse. In particular, we study the internal structure and the properties of SH and completely establish, in terms of incomparability and strict inclusion, the relations between our generalized selectivity classes and Ogihara s P-mc (polynomial-time membership-comparable) classes. Although SH is a strictly increasing infinite hierarchy, we show that the core results that hold for the P-selective sets and that prove them structurally simple also hold for SH. In particular, all sets in SH have small circuits, the NP sets in SH are in Low2 , the second level of the low hierarchy within NP, and SAT cannot be in SH unless P = NP. Finally, it is known that the P-selective sets are not closed under union or intersection. We provide an extended selectivity hierarchy that is based on SH and that is large enough to capture those closures of the P-selective sets, and yet, in contrast with the P-mc classes, is refined enough to distinguish them