Modeling Sudoku as a CNF Formula Spring 2017

Sudoku Rules: • Each cell filled with a number 1… 9 • Each •

Defining the Variables • asserts than the cell in row and column is assigned

Assigning Numbers (1) • One number in 1… 9 in cell in row i,

Assigning Numbers (2) • The cell in row i column j cannot take two

Assigning Numbers (3) • Each cell must contain a number: • Every cell contains

Restricting Rows, Columns • Every row contains every number: • Every column contains every

Restricting 3 x 3 Boxes • Every 3 x 3 box contains every number:

Redundant Clauses • Sudoku problem can be modeled in many ways • May involve

Defining Initial Setup • Initial setup is defined by including unary clauses containing the

Sudoku CNF Formula • • 729 variables 324 clauses with 9 literals 2916 clauses

Solving Sudoku • Total number of possible assignments: • Testing one billion assignments a

Slides: 13

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Modeling Sudoku as a CNF Formula Spring 2017 CSCE 235 H Introduction to Discrete Structures URL: cse. unl. edu/~cse 235 h All questions: Piazza

Sudoku Rules: • Each cell filled with a number 1… 9 • Each • row, • column, and • 3 x 3 box contains all nine numbers (i. e. , no duplicates) CSCE 235 H Modeling Sudoku 2

Sudoku Rules: • Each cell filled with a number 1… 9 • Each • row, • column, and • 3 x 3 box contains all nine numbers (i. e. , no duplicates) CSCE 235 H Modeling Sudoku 3

Defining the Variables • asserts than the cell in row and column is assigned value. CSCE 235 H Modeling Sudoku 4

Assigning Numbers (1) • One number in 1… 9 in cell in row i, column j • Every cell contains at least one number: CSCE 235 H Modeling Sudoku 5

Assigning Numbers (2) • The cell in row i column j cannot take two numbers CSCE 235 H Modeling Sudoku 6

Assigning Numbers (3) • Each cell must contain a number: • Every cell contains at most one number: CSCE 235 H Modeling Sudoku 7

Restricting Rows, Columns • Every row contains every number: • Every column contains every number: CSCE 235 H Modeling Sudoku 8

Restricting 3 x 3 Boxes • Every 3 x 3 box contains every number: CSCE 235 H Modeling Sudoku 9

Redundant Clauses • Sudoku problem can be modeled in many ways • May involve redundant clauses that can be removed to obtain an equivalent formula • Can be generated using inference rules on other clauses in the problem • Redundant clauses may be useful and speed the solving process CSCE 235 H Modeling Sudoku 10

Defining Initial Setup • Initial setup is defined by including unary clauses containing the variables corresponding to the values in the filled cells CSCE 235 H Modeling Sudoku 11

Sudoku CNF Formula • • 729 variables 324 clauses with 9 literals 2916 clauses with 2 literals 3240 total clauses (+ clauses for initial setup) CSCE 235 H Modeling Sudoku 12

Solving Sudoku • Total number of possible assignments: • Testing one billion assignments a second: • Modern SAT solvers can solve Sudoku in milliseconds by aggressively pruning the search tree CSCE 235 H Modeling Sudoku 13