How do you represent basic arithmetic using boolean function? Say you have an arithmetic problem involving two variables, how do you give a Boolean formula for that using standard techniques so that one gets a minimal formula for a given number of quantifiers?

Consider for example a function that decides if $(1)$ $x&gt;y$ or $(2)$ a function that decides $x|y$?
Hint: Use the standard reduction from CircuitSAT to SAT.  Equivalently: Use the Tseitin transform.

I assume you do know how to write a boolean circuit to test whether $x&gt;y$.  If not, consult a textbook.