32 research outputs found
P-Selectivity, Immunity, and the Power of One Bit
We prove that P-sel, the class of all P-selective sets, is EXP-immune, but is
not EXP/1-immune. That is, we prove that some infinite P-selective set has no
infinite EXP-time subset, but we also prove that every infinite P-selective set
has some infinite subset in EXP/1. Informally put, the immunity of P-sel is so
fragile that it is pierced by a single bit of information.
The above claims follow from broader results that we obtain about the
immunity of the P-selective sets. In particular, we prove that for every
recursive function f, P-sel is DTIME(f)-immune. Yet we also prove that P-sel is
not \Pi_2^p/1-immune
Separating Cook Completeness from Karp-Levin Completeness Under a Worst-Case Hardness Hypothesis
We show that there is a language that is Turing complete for NP but not many-one complete for NP, under a worst-case hardness hypothesis. Our hypothesis asserts the existence of a non-deterministic, double-exponential time machine that runs in time O(2^2^n^c) (for some c > 1) accepting Sigma^* whose accepting computations cannot be computed by bounded-error, probabilistic machines running in time O(2^2^{beta * 2^n^c) (for some beta > 0). This is the first result that separates completeness notions for NP under a worst-case hardness hypothesis
Query complexity of membership comparable sets
AbstractThis paper investigates how many queries to k-membership comparable sets are needed in order to decide all (k+1)-membership comparable sets. For kā©¾2 this query complexity is at least linear and at most cubic. As a corollary, we obtain that more languages are O(logn)-membership comparable than truth-table reducible to P-selective sets
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 āmore likelyā 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 ā ā30-complete
Resource Bounded Immunity and Simplicity
Revisiting the thirty years-old notions of resource-bounded immunity and
simplicity, we investigate the structural characteristics of various immunity
notions: strong immunity, almost immunity, and hyperimmunity as well as their
corresponding simplicity notions. We also study limited immunity and
simplicity, called k-immunity and feasible k-immunity, and their simplicity
notions. Finally, we propose the k-immune hypothesis as a working hypothesis
that guarantees the existence of simple sets in NP.Comment: This is a complete version of the conference paper that appeared in
the Proceedings of the 3rd IFIP International Conference on Theoretical
Computer Science, Kluwer Academic Publishers, pp.81-95, Toulouse, France,
August 23-26, 200
On symmetric differences of NP-hard sets with weakly P-selective sets
AbstractThe symmetric differences of NP-hard sets with weakly-P-selective sets are investigated. We show that if there exist a weakly-P-selective set A and an NP-ā©½Pm-hard set H such that H - AĻµPbtt(sparse) and A ā HĻµPm(sparse) then P = NP. So no NP-ā©½Pm-hard set has sparse symmetric difference with any weakly-P-selective set unless P = NP. The proof of our main result is an interesting application of the tree prunning techniques (Fortune 1979; Mahaney 1982). In addition, we show that there exists a P-selective set which has exponentially dense symmetric difference with every set in Pbtt(sparse)