Fractal Parallelism: Solving SAT in Bounded Space and Time

Abstract geometrical computation can solve NP-complete problems e ciently: any boolean constraint satisfaction problem, instance of SAT, can be solved in bounded space and time with simple geometrical constructions involving only drawing parallel lines on a Euclidean space-time plane. Complexity as the maximal length of a sequence of consecutive segments is quadratic. The geometrical algorithm achieves massive parallelism: an exponential number of cases are explored simultaneously. The construction relies on a fractal pattern and requires the same amount of space and time independently of the SAT formula. Key-words. Abstract geometrical computation; Signal machine; Fractal; SAT; Massive parallelism; Model of computation.