The Complexity of Very Simple Boolean Formulas with Applications

The concepts of $\textbf{SAT}$-hardness and $\textbf{SAT}$-completeness modulo npolylogn time and linear size reducibility, denoted by $\textbf{SAT}$-hard (npolylogn, n) and $\textbf{SAT}$-complete (npolylogn, n), respectively, are introduced. Regardless of whether $\textbf{P} = \textbf{NP}$ or $\textbf{P} \neq \textbf{NP}$, it is shown that intuitivelyEach $\textbf{SAT}$-hard (npolylogn, n) problem requires essentially at least as much deterministic time as, andEach $\textbf{SAT}$-complete (npolylogn, n) problem requires essentially the same deterministic time as the satisfiability problem for 3CNF formulas.It is proved that the $\leqq$, satisfiability, tautology, unique satisfiability, equivalence, and minimization problems are already $\textbf{SAT}$-complete (npolylogn, n), for very simple Boolean formulas and for very simple systems of Boolean equations. These completeness results are used to characterize the deterministic time complexities of a number of problems for lattices, propositional calculi, ...