Brief Overview of Attempts

In my initial attempts to develop a polynomial SAT solver, I explored the use of matrices as a foundational structure. This approach was driven by the desire to leverage the power of CUDA technology for rapid computation. I represented logical statements as 2-clauses, focusing on implications, and converted every statement to its equivalent implication form. For more complex statements, such as those in the form ((x and y) implies z), I employed a distinct data structure, recognizing that these were not reducible to simple 2-clauses.

As my journey progressed, I shifted my focus to the properties of logical operators. My current approach is centered on identifying the most complex form that can exhaustively express all states satisfying a statement. By adjusting operators to this comprehensive form, based on their interdefinability, I aim to manage complexity more effectively. Rather than allowing complexity to escalate exponentially with the intricacy of the formula, I strive to cap complexity while maintaining exhaustive representation. This approach envisions transforming the escalation of complexity into a function of a fixed upper limit, multiplied by a constant factor—the number of operators involved.

This evolution in my strategy reflects a deepening understanding of the challenges and potential solutions in developing an efficient polynomial SAT solver. It's a journey of balancing complexity with comprehensiveness, continually seeking innovative ways to streamline logical reasoning.