We establish a lower bound of 2n conditional branches 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 Post machine (Post's Formulation 1) that correctly decides the satisfiability of the conjunction of any two Boolean formulas from such a set by following an execution path that includes fewer than 2n
Nearby in the stack
conditional branches. By using multiple runs of this Post machine, with one run for each Boolean function of
n
variables, the proof derives a contradiction by showing that this Post 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 executes fewer than
2n
conditional branches. This lower bound of
2n
conditional branches 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 boxes in the symbol space and will hold even if a machine is capable of non-sequential access.