Alternating Hierarchies for Time-Space Tradeoffs

TL;DR

This paper introduces a new hierarchy, Eu-LinH, refining existing time-space complexity classes, and explores its structural properties and relationships to major complexity classes, aiming to shed light on fundamental open questions.

Contribution

It defines the Eu-LinH hierarchy as a sub-hierarchy of the linear time hierarchy and investigates its properties and connections to NL, SC, and NP, also introducing zeta-LinH to optimize space bounds.

Findings

Eu-LinH contains NL and SC.

Eu-LinH is contained in NP.

Structural properties of Eu-LinH are analyzed.

Abstract

Nepomnjascii's Theorem states that for all 0 <= \epsilon < 1 and k > 0 the class of languages recognized in nondeterministic time n^k and space n^\epsilon, NTISP[n^k, n^\epsilon ], is contained in the linear time hierarchy. By considering restrictions on the size of the universal quantifiers in the linear time hierarchy, this paper refines Nepomnjascii's result to give a sub- hierarchy, Eu-LinH, of the linear time hierarchy that is contained in NP and which contains NTISP[n^k, n^\epsilon ]. Hence, Eu-LinH contains NL and SC. This paper investigates basic structural properties of Eu-LinH. Then the relationships between Eu-LinH and the classes NL, SC, and NP are considered to see if they can shed light on the NL = NP or SC = NP questions. Finally, a new hierarchy, zeta -LinH, is defined to reduce the space requirements needed for the upper bound on Eu-LinH.