Partiality Insights

 No, I don't mean political nepotism nor do I mean favoritism within families. 

By partiality, I just mean the prosaic fact that, sometimes, functions are not defined everywhere. 

From humdrum 'step functions' 

to the beautiful poetry of '1/x':

Anyone, mathematician or not, can see the beautiful poetry of missing a zero, but gaining two infinities!

However, Category Theory has only total functions and we need to deal, with grace, as point out Cockett and Garner, with partial functions. What can we do?

It turns out that several partial solutions are available and here are some of the ones I know about:

1. We can use some 3-valued logic, where the third truth-value is some sort of undefined (and there are a few extra choices to be made here);

2. We can use the exceptions monad T(A)= A+1, where A stand for the normal values and 1 is the error of type A;

3. We can talk, like Fourman and Scott do, of "existentials that are uniquely defined";

4. We can try to choose between de Paola and Heller's 'dominical categories', Rosolini's 'P-categories' or Cockett's restriction categories.

Now, if we were to do dialectica constructions, paying attention to partiality, which of these alternatives would be easiest for us? Are there other constructions that are better?