I'm sure you all are familiar with Theorem 5.3 from Sipser's TOC book:

S = "On input (M,w) where M is a TM and w is a string: 1. Construct the code of TM M2 as follows: M2 = "On input x: (a) If x = 0n1n for some n ≥ 0, accept. (b) If x = 0n1n, run M on w and if M accepts w, then accept." 2. Run R on (M2). 3. If R accepts, accept; if R rejects, reject."

I'm hoping to find an explanation as to how accepting (a) helps here. If x is of the form 0n1n then M2 accepts, R accepts, and S accepts. But we have accepted a nonregular language and M is not even considered. So R will accept this particular nonregular language (or)? epsilon star if M accepts w. Since R is obviously outside of M2 how does it know what caused M2 to accpet?

Does anyone have a different perspective on this? All of my searches seem to simply regurgitate Theorem 5.3 and don't offer much else.

notof the form $0^n 1^n$..." $\endgroup$ – Ricardo Andrade Sep 18 '13 at 12:46