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

## TL;DR

This paper investigates the relationships between certain parameterized complexity classes related to polynomial-time sub-linear-space computations, using specially constructed oracles to explore potential class separations and inclusions.

## Contribution

It introduces NL-supportive oracles that preserve key class relationships and demonstrates their use in analyzing the inclusion and separation of PsubLIN with para-NL and para-L.

## Key findings

- Constructed NL-supportive oracles for relativized worlds
- Showed para-L can be unequal to para-NL in these worlds
- Demonstrated possible inclusion of para-NL in PsubLIN under certain oracles

## 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 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 PsubLIN and para-NL (nondeterministic log-space class), 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 L$\subseteq$NL=co-NL$\subseteq$DSPACE[O($\log^2{n}$)]$\cap$P. We instead consider special oracles, called 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, para-L$\neq$para-NL$\nsubseteq$PsubLIN and para-L$\neq$para-NL$\subseteq$PsubLIN.