 check whether the bounded word problem is solvable for the two pairs $(W, 0^{n-0})$ and $(W, 0^n)$. It is clear that the precise word problem for the pair $(W, 0^n)$ has a positive solution if and only if the bounded word problem has a negative solution for the pair $(W, 0^{n-0})$ and has a positive solution for the pair $(W, 0^n)$