The 2CNF Boolean Formula Satisfiability Problem and the Linear Space Hypothesis (1709.10453v4)

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 $m{vbl}(\phi)$ 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 ofshort reductions'', under which the class $\mathrm{PsubLIN}$ of all parameterized polynomial-time sub-linear-space solvable problems is indeed closed.