Deducibility in Sudoku

TL;DR

This paper introduces Sudoku logic, a formal system that rigorously defines and proves the conditions for solution uniqueness and symmetry in Sudoku puzzles.

Contribution

It formalizes Sudoku deduction, proves its soundness and completeness, and formalizes symmetry and solution uniqueness within a logical framework.

Findings

Sudoku puzzles have a formal logical characterization of solution uniqueness.

The paper proves the soundness and completeness of Sudoku logic.

A formal proof of Gurth's Symmetrical Placement Theorem is provided.

Abstract

In this paper we provide a formalism, Sudoku logic, in which a solution is logically deducible if for every cell of the grid we can provably exclude all but a single option. We prove that the deductive system of Sudoku logic is sound and complete, and as a consequence of that we prove that a Sudoku puzzle has a unique solution if and only if it has a deducible solution. Using the classification of fundamental Sudoku transformations by Adler and Adler we then formalize the notion of symmetry in Sudoku and provide a formal proof of Gurth's Symmetrical Placement Theorem. In the concluding section we present a Sudoku formula that captures the idea of a Sudoku grid having a unique solution. It turns out that this formula is an axiom of Sudoku logic, making it possible for us to offer to the Sudoku community a resolution of the Uniqueness Controversy: if we accept Sudoku logic as presented in this paper, there is no controversy! Uniqueness is an axiom and, as any other axiom, may freely be used in any Sudoku deduction.