Equality of NC and P

Determine whether the complexity class Nick’s Class (NC)—the set of decision problems solvable in polylogarithmic time on a parallel machine using a polynomial number of processors—equals the class P of problems solvable in polynomial time on a sequential machine. Establish whether every problem in P admits a polylogarithmic-time parallel algorithm under the PRAM model with polynomially many processors, thereby resolving the NC versus P question.

Background

Nick’s Class (NC) comprises problems that can be solved in polylogarithmic time with a polynomial number of parallel processors, typically formalized using the PRAM computational model. The class P consists of problems solvable in polynomial time on a sequential RAM model.

This question is foundational for parallel complexity theory and is highlighted in the context of the paper’s focus on parallel algorithms for DFA minimisation and related operations. While DFA minimisation is known to be in NC, broader relationships between parallel and sequential complexity classes remain unsettled, motivating exploration of practically efficient parallel algorithms.