# Equivalent CSPs

Consider two constraint satisfaction problems $LaTeX: P_1$ and $LaTeX: P_2$ and a sequence $LaTeX: X$ of their common variables. We say that $LaTeX: P_1$ and $LaTeX: P_2$ are equivalent w.r.t $LaTeX: X$ if
• for every solution $LaTeX: d$ to $LaTeX: P_1$ a solution to $LaTeX: P_2$ exists that coincides with $LaTeX: d$ on the variables in X, and
• for every solution $LaTeX: e$ to $LaTeX: P_2$ a solution to $LaTeX: P_1$ exists that coincides with $LaTeX: e$ on the variables in X.