Take any operation that is done by any type of computer (e.g. a cpu on a modern laptop), which doesn't use any type of temporary memory storage.
I.e. this computer operation computes a function $f(x)=y$, without using any form of memory storage.
Q1: Is it possible for any arbitrary such $f$ to be represented by a formula $\phi(x,y)$ in propositional logic, such that $\phi(x,y) =True$ iff $f(x)=y$?
What if we relax the requirement that no temporary memory can be used?
EDIT: after reading the responses so far, perhaps it makes sense to phrase the question in the opposite way:
Q1b: What restrictions on $f(x)$ are necessary and sufficient for it to be representable in propositional logic?