A Lower Bound of $2^n$ Conditional Branches for Boolean Satisfiability on Post Machines

TL;DR

This paper proves a fundamental lower bound of 2^n conditional branches needed by Post machines to decide Boolean satisfiability for full representations of n-variable Boolean functions, highlighting inherent computational complexity.

Contribution

It establishes a universal lower bound of 2^n conditional branches for any full representation of Boolean functions, regardless of formula minimization or access method.

Findings

Lower bound applies to all full representations of Boolean functions.

The bound does not hold for certain restricted formulas like 2CNF, XOR-SAT, HORN-SAT.

The lower bound is independent of sequential or non-sequential access.

Abstract

We establish a lower bound of $2^n$ 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 $2^n$ 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 $2^n$ conditional branches. This lower bound of $2^n$ 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.