| Description |
This study explores methods of applying Non-Euclidean Geometry to the Boolean Satisfiability Problem (SAT). When presented with a knowledge base in Conjunctive Normal Form (CNF) with n atoms, it can be represented as an n-dimensional hypercube, where each corner corresponds to a unique combination of the logical truth assignments to the atoms. A geometric approach to solving SAT, CHOP-SAT [10] performs cuts on the hypercube's corners, with each chop arising from a conjunct in the CNF sentence. This process eliminates non-solution points and preserves only those corners that satisfy the CNF sentence within the feasible region in Euclidean Geometry. The SAT problem is solvable if corners within the feasible region of the hypercube are detected following the cuts. These corners signify the existence of a solution within the given constraints. The Poincar´e disk is a model within Non-Euclidean Geometry that is represented as a unit disk in Euclidean geometry, but which has an associated metric which makes the boundary of the disk infinitely distant from the center. The corners of a hypercube superscribed about the unit disk can be projected onto the unit disk's boundary. Since these solution points are the only points at infinite distance from the center of the unit disk, the hope is that there will be a low-cost computational method to find them. The specific goal of this study is to investigate whether performing CHOP-SAT in this Non- Euclidean Geometry representation can yield an efficient algorithm for solving the Boolean Satisfiability Problem (SAT). |
