Question

Which one is the correct way to show a RE union 0+1? I saw this two ways but I think both are correct. If both are correct why to complicate things?

Answer 1

They are both correct, as you stated.

The first one looks like it was generated using a set of standard rules — in this case, it’s overkill (and just looks silly), but in more complicated cases it’s easier to follow easy rules than to hold the whole thing in your head and write an equivalent NFA from scratch.

In general, an NFA can be rewritten such that it has a single final state (obviously there’s already only one start state).

Then, two NFAs in this form can be combined in such a way that the language they accept when combined is the union of the languages they accept individually — this corresponds to the or (+) in a regular expression. To combine the NFAs in this way, simply create a new node to act as the start state and connect it with ε-transitions to the start states of the two NFAs.

Then, in order to neatly end the NFA in a single final state (so that we can use this NFA recursively for other unions if we want), we create an extra node to serve as the unified final state and ε-connect the old final states (which lose their final status) to it.

Using the general rules above, it’s easy to arrive at the first diagram (two NFAs unioned together, the first matching 0, the other 1) — the second is easy to arrive at via common sense since it’s such a simple regex 😉