arXiv subjects

Edith Hemaspaandra

Publications and source records attributed to Edith Hemaspaandra.

63 records · Page 4Linked to original sources

arXiv1999

Translating Equality Downwards

Downward translation of equality refers to cases where a collapse of some pair of complexity classes would induce a collapse of some other pair of complexity classes that (a priori) one expects are smaller. Recently, the first downward translation of equality was obtained that applied to the polynomial hierarchy-in particular, to bounded access to its levels [cs.CC/9910007]. In this paper, we provide a much broader downward translation that extends not only that downward translation but also that translation's elegant enhancement by Buhrman and Fortnow. Our work also sheds light on previous research on the structure of refined polynomial hierarchies, and strengthens the connection between the collapse of bounded query hierarchies and the collapse of the polynomial hierarchy.

cs.CC↗

arXiv1999

What's Up with Downward Collapse: Using the Easy-Hard Technique to Link Boolean and Polynomial Hierarchy Collapses

During the past decade, nine papers have obtained increasingly strong consequences from the assumption that boolean or bounded-query hierarchies collapse. The final four papers of this nine-paper progression actually achieve downward collapse---that is, they show that high-level collapses induce collapses at (what beforehand were thought to be) lower complexity levels. For example, for each $k\geq 2$ it is now known that if $\psigkone=\psigktwo$ then $\ph=\sigmak$. This article surveys the history, the results, and the technique---the so-called easy-hard method---of these nine papers.

cs.CC↗

arXiv1999

R_{1-tt}^{SN}(NP) Distinguishes Robust Many-One and Turing Completeness

Do complexity classes have many-one complete sets if and only if they have Turing-complete sets? We prove that there is a relativized world in which a relatively natural complexity class-namely a downward closure of NP, \rsnnp - has Turing-complete sets but has no many-one complete sets. In fact, we show that in the same relativized world this class has 2-truth-table complete sets but lacks 1-truth-table complete sets. As part of the groundwork for our result, we prove that \rsnnp has many equivalent forms having to do with ordered and parallel access to $\np$ and $\npinterconp$.

cs.CC↗

arXiv1999

An Introduction to Query Order

Hemaspaandra, Hempel, and Wechsung [cs.CC/9909020] raised the following questions: If one is allowed one question to each of two different information sources, does the order in which one asks the questions affect the class of problems that one can solve with the given access? If so, which order yields the greater computational power? The answers to these questions have been learned-inasfar as they can be learned without resolving whether or not the polynomial hierarchy collapses-for both the polynomial hierarchy and the boolean hierarchy. In the polynomial hierarchy, query order never matters. In the boolean hierarchy, query order sometimes does not matter and, unless the polynomial hierarchy collapses, sometimes does matter. Furthermore, the study of query order has yielded dividends in seemingly unrelated areas, such as bottleneck computations and downward translation of equality. In this article, we present some of the central results on query order. The article is written in such a way as to encourage the reader to try his or her own hand at proving some of these results. We also give literature pointers to the quickly growing set of related results and applications.

cs.CC↗

arXiv1999

A Downward Collapse within the Polynomial Hierarchy

Downward collapse (a.k.a. upward separation) refers to cases where the equality of two larger classes implies the equality of two smaller classes. We provide an unqualified downward collapse result completely within the polynomial hierarchy. In particular, we prove that, for k > 2, if $\psigkone = \psigktwo$ then $\sigmak = \pik = \ph$. We extend this to obtain a more general downward collapse result.

cs.CC↗

arXiv1999

Exact Analysis of Dodgson Elections: Lewis Carroll's 1876 Voting System is Complete for Parallel Access to NP

In 1876, Lewis Carroll proposed a voting system in which the winner is the candidate who with the fewest changes in voters' preferences becomes a Condorcet winner---a candidate who beats all other candidates in pairwise majority-rule elections. Bartholdi, Tovey, and Trick provided a lower bound---NP-hardness---on the computational complexity of determining the election winner in Carroll's system. We provide a stronger lower bound and an upper bound that matches our lower bound. In particular, determining the winner in Carroll's system is complete for parallel access to NP, i.e., it is complete for $\thetatwo$, for which it becomes the most natural complete problem known. It follows that determining the winner in Carroll's elections is not NP-complete unless the polynomial hierarchy collapses.

cs.CC↗

arXiv1999

Raising NP Lower Bounds to Parallel NP Lower Bounds

A decade ago, a beautiful paper by Wagner developed a ``toolkit'' that in certain cases allows one to prove problems hard for parallel access to NP. However, the problems his toolkit applies to most directly are not overly natural. During the past year, problems that previously were known only to be NP-hard or coNP-hard have been shown to be hard even for the class of sets solvable via parallel access to NP. Many of these problems are longstanding and extremely natural, such as the Minimum Equivalent Expression problem (which was the original motivation for creating the polynomial hierarchy), the problem of determining the winner in the election system introduced by Lewis Carroll in 1876, and the problem of determining on which inputs heuristic algorithms perform well. In the present article, we survey this recent progress in raising lower bounds.

cs.CC↗

arXiv1999

On the Power of Positive Turing Reductions

In the early 1980s, Selman's seminal work on positive Turing reductions showed that positive Turing reduction to NP yields no greater computational power than NP itself. Thus, positive Turing and Turing reducibility to NP differ sharply unless the polynomial hierarchy collapses. We show that the situation is quite different for DP, the next level of the boolean hierarchy. In particular, positive Turing reduction to DP already yields all (and only) sets Turing reducibility to NP. Thus, positive Turing and Turing reducibility to DP yield the same class. Additionally, we show that an even weaker class, P(NP[1]), can be substituted for DP in this context.

cs.CC↗

arXiv1998

Downward Collapse from a Weaker Hypothesis

Hemaspaandra et al. proved that, for $m > 0$ and $0 < i < k - 1$: if $Σ_i^p \BoldfaceDelta DIFF_m(Σ_k^p)$ is closed under complementation, then $DIFF_m(Σ_k^p) = coDIFF_m(Σ_k^p)$. This sharply asymmetric result fails to apply to the case in which the hypothesis is weakened by allowing the $Σ_i^p$ 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 $DIFF_s(Σ_i^p) \BoldfaceDelta DIFF_m(Σ_k^p)$ is closed under complementation, then $DIFF_m(Σ_k^p) = coDIFF_m(Σ_k^p)$.

cs.CC↗