[Paper Review] Query Order and the Polynomial Hierarchy

Edith Hemaspaandra, Lane A. Hemaspaandra+1 more|ArXiv.org|Oct 1, 1999

Advanced Algebra and Logic3 citations

TL;DR

This paper investigates how the order of queries to NP oracles affects computational power within the polynomial hierarchy. It proves that while the levels of the polynomial hierarchy are order-oblivious (i.e., query order does not change their power), ordered query classes form new, distinct levels unless the hierarchy collapses, thus revealing a new structure within PH that is sensitive to query sequence.

ABSTRACT

Hemaspaandra, Hempel, and Wechsung [cs.CC/9909020] initiated the field of query order, which studies the ways in which computational power is affected by the order in which information sources are accessed. The present paper studies, for the first time, query order as it applies to the levels of the polynomial hierarchy. We prove that the levels of the polynomial hierarchy are order-oblivious. Yet, we also show that these ordered query classes form new levels in the polynomial hierarchy unless the polynomial hierarchy collapses. We prove that all leaf language classes - and thus essentially all standard complexity classes - inherit all order-obliviousness results that hold for P.

Motivation & Objective

• To investigate the impact of query order on computational power within the polynomial hierarchy.
• To determine whether the levels of the polynomial hierarchy are sensitive to the sequence in which queries are made.
• To explore whether ordered query classes yield new complexity classes beyond the standard hierarchy.
• To examine the inheritance of order-obliviousness properties from P to broader complexity classes.

Proposed method

• The paper analyzes classes defined by ordered sequences of queries to NP oracles, focusing on query order in the context of the polynomial hierarchy.
• It employs reductions and oracle machine constructions to compare the power of different query orderings.
• It proves that all leaf language classes inherit the order-obliviousness properties of P, extending known results.
• It establishes that unless the polynomial hierarchy collapses, ordered query classes form strictly new levels in the hierarchy.
• It uses the concept of leaf language definability to unify results across standard complexity classes.
• It leverages earlier results from Hemaspaandra et al. (cs.CC/9909020) on query order to build a framework for hierarchical analysis.

Experimental results

Research questions

• RQ1 Does the order in which queries are made to NP oracles affect the computational power of classes within the polynomial hierarchy?
• RQ2 Are the levels of the polynomial hierarchy invariant under query order, or does ordering induce new complexity classes?
• RQ3 Can ordered query classes be shown to form new levels in the polynomial hierarchy, and under what conditions?
• RQ4 Do standard complexity classes, such as those defined via leaf languages, inherit the order-obliviousness properties of P?
• RQ5 What is the relationship between query order and the collapse of the polynomial hierarchy?

Key findings

• The levels of the polynomial hierarchy are order-oblivious: the power of a class does not depend on the sequence of queries to NP oracles.
• Despite order-obliviousness at the level of hierarchy levels, ordered query classes form new, distinct classes unless the polynomial hierarchy collapses.
• All leaf language classes inherit the order-obliviousness properties of P, indicating broad applicability of these results.
• The existence of new ordered query classes implies that query order introduces structural complexity not captured by the standard hierarchy.
• The results suggest that query order is a meaningful parameter in complexity theory, even when the overall hierarchy remains unchanged.
• The paper establishes that the polynomial hierarchy does not collapse if and only if there exist distinct ordered query classes at each level.