Cell-like P-systems using deterministic update rules to solve Sudoku

Abstract

Sudoku, of order n, is a logical puzzle with an objective to fill a partially completed $n^{2} \times n^{2}$ grid, such that it’s each row, column and n × n sub-grid, also called box, contains the digits ranging from 1 to n ², exactly once. It is known to be a NP-complete combinatorial problem. In this paper, a parallel and distributed framework of cell-like P-systems is presented to solve Sudoku puzzles. For this, the number of membranes including skin membrane is equal to the size of puzzle, i.e., n ² for Sudoku of order n. This P-system model has total of 6 rules to solve the puzzle, out of which five are evolution or update rules while one is communication rule. The model solves “easy”, “medium”, and “hard” puzzles of the studied database with 100% success rate. However, in the “evil” category some of the problems could not be solved, the reason of which is also explained in this paper. From the numerical results, it is concluded that the majority of the Sudoku puzzles could be solved in a very small computational time.