This is a very simple question, but I couldn't find a reference and I just wanted to check my facts. I was looking for a state machine similar to pushdown automata but where the stack is restricted to one element (i.e. a register).
It occurred to me that if the register has an infinite alphabet (of possible values) it will basically be equivalent to a PDA. Every configuration of the stack could be represented by a single element of the register's alphabet. (E.g. let's say that the register's alphabet is the set of natural numbers). Is this correct?
Furthermore, a DFSM is equivalent in strength to a NFSM. However, I assume that (DFSM + 1 register) would not be equivalent to a (NFSM + 1 register) since DPDA and NPDA are not equivalent?