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

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

Reducing the Number of Solutions of NP Functions

AbstractWe study whether one can prune solutions from NP functions. Though it is known that, unless surprising complexity class collapses occur, one cannot reduce the number of accepting paths of NP machines, we nonetheless show that it often is possible to reduce the number of solutions of NP functions. For finite cardinality types, we give a sufficient condition for such solution reduction. We also give absolute and conditional necessary conditions for solution reduction, and in particular we show that in many cases solution reduction is impossible unless the polynomial hierarchy collapses

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 relation classes and solution relations

Die Dissertation On Relation Classes and Solution Relations ist in dem Gebiet der strukturellen Komplexitätstheorie einzuordnen. In einem ersten Teil wird die vollständige Inklusionsstruktur zwischen verschiedenen Relationenklassen aufgeklärt. Ein Großteil der Ergebnisse wird dabei mit Hilfe der Operatorenmethode erzielt. Im zweiten Teil werden Lösungsrelationen und easy-Sprachen betrachtet. Es wird der Fragestellung nachgegangen, welche Probleme durch eine vorgegebene Klasse von Relationen gelöst werden können