We want to show that a language L = L(M) for some TM M iff L = L(G) for some unrestricted grammar G. The following two theorems completes the proof.
Theorem : Let G = ( N, 
, P, S ) be an unrestricted grammar. Then the language L(G) generated by G is recursively enumerable.
Proof : To prove the theorem, we construct a 3-type nondeterministic TM M that accepts L(G). Tape 1 always holds any given input string w 
. A production 
of G is represented as 
where # is a special tape symbol of M such that #
. All the production of G with this representation are written on tape-2 of M. Two productions are separated by the string ##. The idea is that M's computation simulates derivations of G. Tape 3 is used to simulate the derivative of G. On many input string w, the computation of TM = M consists of the following steps:
- w is written on tape 1.
- S is written on the first cell of tape 3.
- A production is chosen from tape 2 (we assume that all the productions of G are written on tape 2)