A PDA is a well-defined formal object. It is the 6-tuple $(Q,\Sigma,\Gamma,\delta,q_0,F)$ and it computes according to the formal definition given in class. So there **is** a definition, exactly the formal one given in class.

We are trying to understand what can and can't be done with a PDA given this formal definition. "Guessing" is a way to give an intuition for the formal properties of a PDA. It hints to the fact that the transition function of a PDA allow you at each step of the computation to proceed in more than one (but still finite) way. Each possible computation is a "guess", and if one of those succeeds then you accept the word. So the only guessing allowed is one that can be described using a formal PDA according to the formal definition given. So you can not add power to the PDA - it does what it does, but if you are able to show that the type of guessing that you suggested can be achieved using the formal definition, then yes.

About the specific guessing you suggested, **if** I understand what you mean, then it seems that it's possible - remembering one bit can be done with a state, and you need to "guess" sometime later that the next bit is different and compare the bit in the input with the one stored in the state.