Relational Analysis; Sudoku
For a game of Sukoshu—which is a simpler version of Sudoku, consisting of one 4*4 board—the axiomatic expression of the rules is as follows:
1) ?x.?y.?z.?w.(cell(x,y,w) ? cell(x,z,w) ? same(y,z))
This sentence expresses the constraint whereby two cells in the same row cannot contain the same value.
2) ?x.?y.?z.?w.(cell(x,z,w) ? cell(y,z,w) ? same(x,y))
This sentence expresses the constraint whereby two cells in that same column cannot contain the same value.
And, 3) ?x.?y.?w.cell(x,y,w)
This sentence expresses the fact that every cell must contain at least one value.
Now, a game of Sudoku consists of a 9*9 board divided into nine 3*3 subboards. We want to axiomatically express the rules whereby, when filling empty cells, no numeral must be repeated in any row or column or 3*3 subboard.
How must we proceed? Do we preserve the sentences 1, 2, and 3 from the axiomatisation of the rules of Sukoshu and merely come up with a new sentence for the rule whereby no number is to be repeated in any 3*3 subboard?
1) ?x.?y.?z.?w.(cell(x,y,w) ? cell(x,z,w) ? same(y,z))
This sentence expresses the constraint whereby two cells in the same row cannot contain the same value.
2) ?x.?y.?z.?w.(cell(x,z,w) ? cell(y,z,w) ? same(x,y))
This sentence expresses the constraint whereby two cells in that same column cannot contain the same value.
And, 3) ?x.?y.?w.cell(x,y,w)
This sentence expresses the fact that every cell must contain at least one value.
Now, a game of Sudoku consists of a 9*9 board divided into nine 3*3 subboards. We want to axiomatically express the rules whereby, when filling empty cells, no numeral must be repeated in any row or column or 3*3 subboard.
How must we proceed? Do we preserve the sentences 1, 2, and 3 from the axiomatisation of the rules of Sukoshu and merely come up with a new sentence for the rule whereby no number is to be repeated in any 3*3 subboard?
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