Probe Machine Based Computing Model for Solving Satisfiability Problem

Abstract

Probe Machine (PM) is a recently reported theoretical model with massive parallelism. Particularly, Turing Machine (TM) had been proven to be the special case of PM. The paper proposed a PM based computing model for satisfiability problem. PM is capable of searching pairs of data fiber that are complementary to pre-devised probes, leading to the connection of pairs of data in parallel. We encoded the assignment to variables in the given 3-SAT formula into data fibers and devised probes between pairs of data fiber, the unique satisfying assignment of a hard 3-SAT formula could be generated within just one step of probe operation. More generally, for an arbitrary 3-SAT formula with variables and clauses, we presented a method for deciding the satisfiability using the concept of potential probe. Complexity analysis shows the encoding complexity and time complexity of the proposed model are o(n) and o(1), respectively. The distinguishing characteristics of the proposed model lie in two aspects. On one hand, solution to NP-complete problem was generated in just one step of probe operation rather than found in vast solution space. On the other hand, the proposed model is highly parallel. Most important of all, the parallelism increases with problem size. This marks a giant step in computational theory. With the parallel search capability inherited in PM, the size of NP-complete search problems is expected to be increased further.