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 2SAT3—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 mvbl(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 (2SAT3,mvbl) cannot be solved in polynomial time using only “sub-linear” space (i.e., mvbl(x)εpolylog(|x|) space for a constant ε∈[0,1)) on all instances x. Immediate consequences of LSH include L≠NL, LOGDCFL≠LOGCFL, and SC≠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.