A Lower Bound for Boolean Satisfiability on Turing Machines

We establish a lower bound for deciding the satisfiability of the conjunction of any two Boolean formulas from a set called a full representation of Boolean functions of n variables a set containing a Boolean formula to represent each Boolean function of n variables. The contradiction proof first assumes that there exists a Turing machine with k symbols in its tape alphabet that correctly decides the satisfiability of the conjunction of any two Boolean formulas from such a set by making fewer than 2logk2 moves. By using multiple runs of this Turing machine, with one run for each Boolean function of n variables, the proof derives a contradiction by showing that this Turing machine is unable to correctly decide the satisfiability of the conjunction of at least one pair of Boolean formulas from a full representation of n-variable Boolean functions if the machine makes fewer than 2logk2 moves. This lower bound holds for any full representation of Boolean functions of n variables, even if a full representation consists solely of minimized Boolean formulas derived by a Boolean minimization method. We discuss why the lower bound fails to hold for satisfiability of certain restricted formulas, such as 2CNF satisfiability, XOR-SAT, and HORN-SAT. We also relate the lower bound to 3CNF satisfiability. The lower bound does not depend on sequentiality of access to the tape squares and will hold even if a machine is capable of non-sequential access.