Theorem : Let L be a recursively enumerable language. Then L = L(G) for some unrestricted grammar G.

Proof : Since L is r.e, it is ampled by a deterministic TM we want to construct an unrestricted grammar whose derivations simulates the computations of M, such that . That is, for any string iff symbol for some and . For this is to happen we need to represent IDs of TM M by strings of terminals & nonterminals in G and must have productions in such that

S q₀w w.

That is ,

1. The initial ID q₀w must be derivable from S.
2. Induction production in G to simulate every move of M.
3. If M eventually enters a final state, then transform the string to w.