We aim at investigating the solvability/insolvability of nondeterministic logarithmic-space (NL) decision, search, and optimization problems parameterized by natural size parameters using simultaneously polynomial time and sub-linear space. We are particularly focused on $\mathrm{2SAT}_3$ -- a restricted variant of the 2CNF Boolean (propositional) formula satisfiability problem in which each variable of a given 2CNF formula appears at most 3 times in the form of literals -- parameterized by the total number $m_{vbl}(\phi)$ of variables of each given Boolean formula $\phi$. We propose a new, practical working hypothesis, called the linear space hypothesis (LSH), which asserts that $(\mathrm{2SAT}_3,m_{vbl})$ cannot be solved in polynomial time using only ``sub-linear'' space (i.e., $m_{vbl}(x)^{\varepsilon}\, polylog(|x|)$ space for a constant $\varepsilon\in[0,1)$) on all instances $x$. Immediate consequences of LSH include $\mathrm{L}l\neq\mathrm{NL}$, $\mathrm{LOGDCFL}\neq\mathrm{LOGCFL}$, and $\mathrm{SC}\neq \mathrm{NSC}$. For our investigation, we fully utilize a key notion of ``short reductions'', under which the class $\mathrm{PsubLIN}$ of all parameterized polynomial-time sub-linear-space solvable problems is indeed closed.

翻译：我们旨在研究由自然规模参数参数化的非确定性对数空间（NL）决策、搜索和优化问题，在同时使用多项式时间和亚线性空间条件下的可解性/不可解性。我们特别关注$\mathrm{2SAT}_3$——2CNF布尔（命题）公式可满足性问题的一个受限变体，其中给定2CNF公式的每个变量以文字形式最多出现3次——并用每个给定布尔公式$\phi$的变量总数$m_{vbl}(\phi)$进行参数化。我们提出一个新的实用工作假设，称为线性空间假设（LSH），该假设断言$(\mathrm{2SAT}_3,m_{vbl})$无法在所有实例$x$上使用仅“亚线性”空间（即对于常数$\varepsilon\in[0,1)$，使用$m_{vbl}(x)^{\varepsilon}\, polylog(|x|)$空间）在多项式时间内求解。LSH的直接推论包括$\mathrm{L}\neq\mathrm{NL}$、$\mathrm{LOGDCFL}\neq\mathrm{LOGCFL}$以及$\mathrm{SC}\neq \mathrm{NSC}$。为进行我们的研究，我们充分利用了“短归约”这一关键概念，在该概念下，所有参数化多项式时间亚线性空间可解问题的类$\mathrm{PsubLIN}$确实是封闭的。