Universality of $(+, \oplus)$ over $\mathbb{Z}_2^n$
Nope.  For instance, you can't express the function $f(x)=x&gt;&gt;1$.  In general, the least-significant bit of $f(x)$ depends only on the least-significant bit of $x$.  You can prove that by structural induction, using the following two properties: $(a \oplus b) \bmod 2 = (a \bmod 2) \oplus (b \bmod 2)$ and $(a + b) \bmod 2 = (a \bmod 2) + (b \bmod 2) \bmod 2$.