Polynomial-Time Multi-Selectivity

TL;DR

This paper introduces polynomial-time multi-selectivity, a generalized notion of selectivity, and studies its hierarchy, revealing structural properties, complexity implications, and relationships with existing classes, advancing understanding of selectivity in computational complexity.

Contribution

It defines and analyzes the multi-selectivity hierarchy, SH, proving it does not collapse and establishing its structural properties and relations to other complexity classes.

Findings

SH does not collapse and is an infinite hierarchy.

Sets in SH have small circuits and NP sets in SH are in Low_2.

SAT is not in SH unless P=NP.

Abstract

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 to prove the first known (and optimal) lower bounds for generalized selectivity-like classes in terms of EL_2, 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 Low_2, the second level of the low hierarchy within NP; and SAT cannot be in SH unless P = NP. Finally, it is known that P-Sel, the class of P-selective sets, is 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.