  We propose a new approach to SAT solving which solves SAT problems in vector
spaces as a cost minimization problem of a non-negative differentiable cost
function J^sat. In our approach, a solution, i.e., satisfying assignment, for a
SAT problem in n variables is represented by a binary vector u in {0,1}^n that
makes J^sat(u) zero. We search for such u in a vector space R^n by cost
minimization, i.e., starting from an initial u_0 and minimizing J to zero while
iteratively updating u by Newton's method. We implemented our approach as a
matrix-based differential SAT solver MatSat. Although existing main-stream SAT
solvers decide each bit of a solution assignment one by one, be they of
conflict driven clause learning (CDCL) type or of stochastic local search (SLS)
type, MatSat fundamentally differs from them in that it continuously approach a
solution in a vector space. We conducted an experiment to measure the
scalability of MatSat with random 3-SAT problems in which MatSat could find a
solution up to n=10^5 variables. We also compared MatSat with four
state-of-the-art SAT solvers including winners of SAT competition 2018 and SAT
Race 2019 in terms of time for finding a solution, using a random benchmark set
from SAT 2018 competition and an artificial random 3-SAT instance set. The
result shows that MatSat comes in second in both test sets and outperforms all
the CDCL type solvers.
