Question

So, it’s finals time and I came across this problem in an old exam:

Give a regular expression that denotes diff(x) where:

- diff(x) is the number of 1's in x minus the number of 0's in x
- 1 <= diff(x) <= 3

e.g.

 diff(10110100111) = 7-4 = 3
 diff(11100011) = 5-3 = 2
 diff(10011) = 3-2 = 1

Answer 1

it should not be possible to build a regex as desired. if it was you’d have a finite state automaton necessarily implementing an unbounded counter in order to distinguish between inputs 0^n1^n111 and 0^n1^n1111. obviously this cannot be achieved, at least in terms of theory (it can be achieved, however, if the difference between the number of 1s and 0s in any prefix of x is bounded by a constant).

that might be irrelevant in practice as virtually every common regex engine is more powerful than a regex recognizer but it might be relevant in the context of an exam question.