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CS 405/505 ASSIGNMENT
Due Wednesday, March 28, 2007
- A Sudoku (http://www.sudoko.com) puzzle is solved on a 9 × 9 grid, which may be subdi- vided as a 3 × 3 grid of 3 × 3 grids. The object of the puzzle is to fill in the grid so that every row, every column, and every 3 × 3 box contains the digits 1 through 9. Usually some elements of the grid are already filled in when the puzzle begins. An example of such a puzzle is:
5 2 7 6 9 8 1 4 3 6 9 1 7 4 3 5 2 8 8 4 3 2 1 5 6 7 9 2 6 8 5 3 9 7 1 4 9 7 5 4 6 1 8 3 2 1 3 4 8 2 7 9 6 5 3 5 6 1 8 2 4 9 7 4 8 9 3 7 6 2 5 1 7 1 2 9 5 4 3 8 6
Note that the digits in bold face are the ones already given. This problem is to write a set of Prolog predicates which solve a Sudoku puzzle.
(a) Write a Prolog predicate puzzle (P) to instantiate P to the puzzle above as a list of 9 rows of entries, each row being a list of 9 entries. Use [] to indicate the entries which are not set initially (i.e., those that are not bold). (b) Write a Prolog predicate row (P, N, R) which takes a puzzle P and extracts the N-th row, with N = 1 being the first row, into R. If P does not have an N-th row, then the predicate fails. Test your predicate on the following, assuming P is the example puzzle given above: row(P, 2, R) ⇒ R = [[], [], 1, 7, [], 3, 5, [], []] row(P, 9, R) ⇒ R = [[], 1, [], [], [], [], [], 8, []] row(P, 0, R) ⇒ no (c) Write a Prolog predicate column (P, N, C) which takes a puzzle P and extracts the N- th column, with N = 1 being the first column, into C. If P does not have an N-th column, then the predicate fails. Test your predicate on the following, assuming P is the example puzzle given above: column(P, 2, C) ⇒ C = [2, [], [], [], 7, [], [], [], 1] column(P, 9, C) ⇒ C = [[], [], 9, [], [], [], 7, [], []] column(P, 0, C) ⇒ no (d) Write a Prolog predicate box (P, I, J, B) which takes a puzzle P and extracts the box B at coordinates I and J, with (I, J) = (1, 1) being the first box. If P does not have the indicated box, then the predicate fails. Test your predicate on the following, assuming P is the example puzzle given above: box(P, 1, 2, B) ⇒ B = [[], [], [], 7, [], 3, [], 1, []] box(P, 3, 3, B) ⇒ B = [4, [], 7, 2, [], [], [], 8, []] box(P, 2, 4, B) ⇒ no
(e) Write a Prolog predicate elements (L, E) which takes a row, column or box L and returns the values of the fixed (i.e., a specific integer rather than a list of possibilities, including []) row, column or box elements as a linear list. Test your predicate on the following: elements([7, 1, 2, 9, 5, 4, 3, 8, 6], E) ⇒ E = [7, 1, 2, 9, 5, 4, 3, 8, 6] elements([2, [], [], [], 7, [], [], [], 1], E) ⇒ E = [2, 7, 1] elements([[2, 4, 6, 9], [2, 4, 6, 9], 1, 7, [2, 4, 6, 9], 3, 5, [2, 4, 6, 9], [2, 4, 6, 9]], E) ⇒ E = [1, 7, 3, 5] (f) Write a Prolog predicate initialize (L1, L2) which takes a row, column or box L and returns L1, with all null values set to the initial set of candidates (i.e., those digits not yet fixed), as L2. Test your predicate on the following: initialize([7, 1, 2, 9, 5, 4, 3, 8, 6], L) ⇒ L = [7, 1, 2, 9, 5, 4, 3, 8, 6] initialize([2, [], [], [], 7, [], [], [], 1], L) ⇒ L = [2, [3, 4, 5, 6, 8, 9], [3, 4, 5, 6, 8, 9], [3, 4, 5, 6, 8, 9], 7, [3, 4, 5, 6, 8, 9], [3, 4, 5, 6, 8, 9], [3, 4, 5, 6, 8, 9], 1] initialize([[], [], 1, 7, [], 3, 5, [], []]) ⇒ L = [[2, 4, 6, 8, 9], [2, 4, 6, 8, 9], 1, 7, [2, 4, 6, 8, 9], 3, 5, [2, 4, 6, 8, 9], [2, 4, 6, 8, 9]] (g) Write a Prolog predicate update (L1, L2) which takes a row, column or box L1 and returns L1 with all fixed digits removed from the candidate sets, as L2. Test your predicate on the following: update([7, 1, 2, 9, 5, 4, 3, 8, 6], L) ⇒ L = [7, 1, 2, 9, 5, 4, 3, 8, 6] update([2, [3, 4, 5, 6, 8, 9], 4, [3, 4, 5, 6, 8, 9], 7, [3, 4, 5, 6, 8, 9], 5, [3, 4, 5, 6, 8, 9], 1], L) ⇒ L = [2, [3, 6, 8, 9], 4, [3, 6, 8, 9], 7, [3, 6, 8, 9], 5, [3, 6, 8, 9], 1] update([[2, 4, 6, 8, 9], 9, 1, 7, 4, 3, 5, 2, [2, 4, 6, 8, 9]], L) ⇒ L = [[6, 8], 9, 1, 7, 4, 3, 5, 2, [6, 8]] (h) You will be provided with a predicate solve (P, S) which takes a puzzle P and returns the solution S. This predicate will call the predicates you have written above. Run solve to test the system.
- Consider the Core program segment S below:
a := x; b := y; while (a != b) loop if (a < b) then b := b - a; else a := a - b; end if; end loop; z := a;
Using the axiomatic semantics of Core, show that the assertion {x > 0 ∧ y > 0 } S {z = gcd (x, y)} is correct, assuming the following mathematical properties of gcd.
