The 2CNF Boolean formula satisfiability problem and the linear space hypothesis

Abstract

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 $m_{vbl}\left(\varphi \right)$ 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., $m_{vbl}{\left(x\right)}^{\epsilon }polylog\left(\right|x\left|\right)$ space for a constant $\epsilon \in [0,1)$) on all instances x. Immediate consequences of LSH include $L\ne \mathrm{NL}$, $\mathrm{LOGDCFL}\ne \mathrm{LOGCFL}$, and $\mathrm{SC}\ne \mathrm{NSC}$. For our investigation, we fully utilize a key notion of “short reductions”, under which the class PsubLIN of all parameterized polynomial-time sub-linear-space solvable problems is indeed closed.