Given binary numbers $b_1, b_2, ... , b_m < 2^n$ as well as their complements $b'_1, b'_2, ..., b'_m$ (with leading $1$'s if necessary, such that $b_i + b'_i = 2^n - 1$), is it possible to quickly determine whether a number $k < 2^n$ can be constructed by repeatedly applying bitwise AND and OR?
For example, given 8, 10, 12 and 7, 5, 3 number 9 can be constructed as follows: (3 & 5) | 8 = 9. Note that the order of operations is always from left to right, like (((a * b) * c) * d) * e ...
If it's not possible, maybe there is a weaker test which tells whether in order to construct $k$ it would be necessary to use an XOR or XNOR, since they can't be simulated by using AND and OR sequentially?