How to compute "A1 x. .. x An subset B" fast if B is fixed?

Let B be any fixed subset of {1,...,d}^n.
On a RAM, is there a representation of B of size polynomial in d and at
most exponential in n so that the following query can be answered in
time polynomial in d and n:
Let A1, ..., An be any subsets of {1,...,d}. Is A1 x ... x An subset B?

Can anyone help? Ideas, references are welcome.
Sasha.
.

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