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 $m_{vbl}(ϕ)$ of variables of each given Boolean formula $ϕ$. 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.

(A4, 10pt, 28 pages) This current article corrects and extends its preliminary report in the Proc. of the 42nd International Symposium on Mathematical Foundations of Computer Science (MFCS 2017), August 21-25, 2017, Aalborg, Denmark, Leibniz International Proceedings in Informatics (LIPIcs), Schloss Dagstuhl - Leibniz-Zentrum fuer Informatik 2017, vol. 83, pp. 62:1-62:14, 2017