As stangdon says in a comment:
You can't just say "by semicolon" - semicolon is a count noun, so it
has to be "a semicolon" or "semicolons".
Out of the remaining options, my preference is for the first ("Successive rules are separated by semicolons") but the second is also grammatical ("Successive rules are separated by a semicolon"). I don't think there's any overwhelming logical basis for one or the other.
Here is some evidence from Google Books that I think shows that people use plural nouns like this alongside "successive rules" (bolding added):
Composites, are rules that are created by merging at least two
successive rules of a parse tree of a solution. ("Online Modeling the Novice-Expert Shift in Programming Skills on a Rule-Schema-Case Partial Order," by Claus Möbus, Olaf Schröder and Heinz-Jürgen Thole, in Cognition and Computer Programming, edited by Karl F. Wender, Franz Schmalhofer, and Heinz-Dieter Böcker)
Note that this sentence starts with "Composites are rules..." and not "A composite is a rule...".
Error estimates for these rules are usually computed using
differences from successive rules... (Computation of Multivariate Normal and t Probabilities, by Alan Genz and Frank
Note that this says "differences from successive rules" and not "the difference from successive rules."