CIT 594 Turing Machine to Unrestricted Grammar
Spring 2012, David Matuszek
I think the trouble I had in class was simply that I got mixed up about which direction I was going: TM to unrestricted grammar, or the reverse.
In any case, here's an example of a Turing Machine that adds 1 to a binary number. The Turing Machine starts with the read/write head on the leftmost digit and ends with the read/write head on the rightmost digit.
Turing Machine  Grammar  Comments 

Input tape: 101 

The initial production (only) of the grammar specifies the desired input string. Abbreviations: 
start 0 0 R start 
s0 → 0s 
As with MPCP, going right is a single production; going left requires a separate production for each tape symbol. 
add1 1 0 L add1 
0a1 → a00 
For a move to the left,q_{x} a b L q_{y} becomes cq_{x}a → q_{y}cb for every c in the tape alphabet. For a move to the right, q_{x} a b R q_{y} becomes q_{x}a → bq_{y} 

f0 → 0f 
By just following the rules, we should have the production , but we don't really need that one. 
Here's an example of using both the Turing Machine and the Unrestricted Grammar to add 1 to the binary number 101
:
Turing Machine example using 101  Transition to use  Unrestricted Grammar example using 101  Production to use 

0 start 101 
start 1 1 R start 
S → #s101# 
s1 → 1s 
1 start 101 
start 0 0 R start 

s0 → 0s 
2 start 101 
start 1 1 R start 
→ #10s1# 
s1 → 1s 
3 start 101 
start # # L add1 
→ #101s# 
1s# → a1# 
4 add1 101 
add1 1 0 L add1 
→ #10a1# 
0a1 → a00 
5 add1 100 

→ #1a00# 
a0 → 1f 
6 finish 110 

→ #11f0# 
f0 → 0f 
7 finish 110 
finish # # L halt 
→ #110f# 
0f# → h0# 
8 halt 110 
→ #11h0# 