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\begin{itemize} \item \emph{Row-constraints:} \emph{If} a cell in a row has a certain value, \emph{then} no other cell in that row can have that value. For each $i$, and each $k$ we have: $$x_{i1k} \imp \lnot x_{i2k} \land \lnot x_{i3k} \land ... \land \lnot x_{i9k}$$ $$x_{i2k} \imp \lnot x_{i1k} \land \lnot x_{i2k} \land ... \land \lnot x_{i9k}$$ $$\vdots$$ $$x_{i9k} \imp \lnot x_{i1k} \land \lnot x_{i2k} \land ... \land \lnot x_{i8k}$$
\item \emph{Column-constraints:} \emph{If} a cell in a column has a certain value, \emph{then} no other cell in that column can have that value. For each $j$, and each $k$ we have: $$x_{1jk} \imp \lnot x_{2jk} \land \lnot x_{3jk} \land ... \land \lnot x_{9jk}$$ $$x_{2jk} \imp \lnot x_{1jk} \land \lnot x_{2jk} \land ... \land \lnot x_{9jk}$$ $$\vdots$$ $$x_{9jk} \imp \lnot x_{1jk} \land \lnot x_{2jk} \land ... \land \lnot x_{8jk}$$
\item \emph{Square-constraints:} \emph{If} a cell in a 3x3 square has a certain value, \emph{then} no other cell in that square can have that value. For the first square, we have for each $k$: $$x_{11k} \imp \lnot x_{12k} \land \lnot x_{13k} \land \lnot x_{21k} \land \lnot x_{22k} \land \lnot x_{23k} \land \lnot x_{31k} \land \lnot x_{32k} \land \lnot x_{33k}$$ $$\vdots$$ $$x_{33k} \imp \lnot x_{11k} \land \lnot x_{12k} \land \lnot x_{13k} \land \lnot x_{21k} \land \lnot x_{22k} \land \lnot x_{23k} \land \lnot x_{31k} \land \lnot x_{32k}$$ The constraints for the remaining squares are similar.
\item \emph{Predefined-number-constraints:} If a cell has a predefined value, we need to set the corresponding variable to true, e.g., the cell in the fifth row and the fifth column has the value 9. Therefore we have $$x_{559}.$$
\item \emph{Cell-constraints:} Each cell must contain a number ranging from one to nine. For each $i$, and each $j$ we have $$x_{ij1} \lor x_{ij2} \lor ... \lor x_{ij9}.$$ On its own, this constraint would allow for a cell to have more than one value. However, this is not possible due to the other constraints.\end{itemize}
To construct the final propositional formula, all constraints need to be connected via conjunctions.A satisfying assignment for the final formula represents one possible solution for the Sudoku puzzle.In case that there does not exists a solution, the SAT sovler would return UNSAT.
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