L is regular$$L is recognized by a robust NFA

You can do it by doing the following:

To prove the left$-$implies$-$right $($

$$

$$

$)$ part: assume you have a DFA$/$NFA$/$RE for a language $($whichever you prefer$),$ then you need to show you can build a robust NFA for that language. Starting from a DFA is easier for this, in my opinion.

To prove the other direction $($right$-$implies$-$left,

$$

$$

$)$: assume you have a robust NFA, you need to show one can build an NFA$/$DFA$/$RE for that language. Building an NFA is easier for this, again in my opinion. A robust NFA is an NFA that accepts a string w only if there is a way for it to accept w using only half of the transitions in its state diagram. Show that robust NFAs recognize the class of regular languages.