Algebraic Properties for Selector Functions

TL;DR

This paper explores algebraic properties of P-selective sets, showing that associativity implies commutativity, and proposes algebraic conditions under which their deterministic advice complexity is linear, potentially matching known lower bounds.

Contribution

It proves that associativity implies commutativity for P-selective sets and introduces an algebraic condition that could ensure linear advice complexity, improving understanding of their computational complexity.

Findings

Associativity implies commutativity for P-selective sets.

An algebraic condition is identified that could guarantee linear advice complexity.

If all P-selective sets are associatively P-selective, then their advice complexity is linear.

Abstract

The nondeterministic advice complexity of the P-selective sets is known to be exactly linear. Regarding the deterministic advice complexity of the P-selective sets--i.e., the amount of Karp--Lipton advice needed for polynomial-time machines to recognize them in general--the best current upper bound is quadratic [Ko, 1983] and the best current lower bound is linear [Hemaspaandra and Torenvliet, 1996]. We prove that every associatively P-selective set is commutatively, associatively P-selective. Using this, we establish an algebraic sufficient condition for the P-selective sets to have a linear upper bound (which thus would match the existing lower bound) on their deterministic advice complexity: If all P-selective sets are associatively P-selective then the deterministic advice complexity of the P-selective sets is linear. The weakest previously known sufficient condition was P=NP. We also establish related results for algebraic properties of, and advice complexity of, the nondeterministically selective sets.