Supportive Oracles for Parameterized Polynomial-Time Sub-Linear-Space Computations in Relation to L, NL, and P

Abstract

We focus our attention onto polynomial-time sub-linear-space computation for decision problems, which are parameterized by size parameters m(x), where the informal term “sub linear” means a function of the form $m(x)^{\varepsilon }\cdot polylog(|x|)$ on input instances x for a certain absolute constant $\varepsilon \in (0,1)$ and a certain polylogarithmic function polylog(n). The parameterized complexity class $\mathrm {PsubLIN}$ consists of all parameterized decision problems solvable simultaneously in polynomial time using sub-linear space. This complexity class is associated with the linear space hypothesis. There is no known inclusion relationships between $\mathrm {PsubLIN}$ and $\mathrm {para}\text {-}\,\!\mathrm {NL}$, where the prefix “para-” indicates the natural parameterization of a given complexity class. Toward circumstantial evidences for the inclusions and separations of the associated complexity classes, we seek their relativizations. However, the standard relativization of Turing machines is known to violate the relationships of $\mathrm {L}\subseteq \mathrm {NL}=\mathrm {co}\text {-}\,\!\mathrm {NL}\subseteq \mathrm {DSPACE}[O(\log ^2{n})]\cap \mathrm {P}$. We instead consider special oracles, called $\mathrm {NL}$-supportive oracles, which guarantee these relationships in the corresponding relativized worlds. This paper vigorously constructs such NL-supportive oracles that generate relativized worlds where, for example, $\mathrm {para}\text {-}\,\!\mathrm {L}\ne \mathrm {para}\text {-}\,\!\mathrm {NL}\nsubseteq \mathrm {PsubLIN}$ and $\mathrm {para}\text {-}\,\!\mathrm {L}\ne \mathrm {para}\text {-}\,\!\mathrm {NL}\subseteq \mathrm {PsubLIN}$.