1. Constructing Complex Turing Machines
Idea:
Use a modular approach similar to the one use to construct complex programs
Example (programming):
Given two nn matrixes A and B to compute ABT we:
Construct a procedure to compute BT, transverse(B,C)
Construct a procedure to multiply two matrices, multiply(B,C,D)
Call transverse(B,C) and then multiply(A,C,D). D contains the answer

2. Graphical Conventions
We will use a convention similar to a flow chart to indicate how to put together Turing machines (TMs).
Convention. Let A and B be 2 TMs, and a be a character in  then: A a B A;
if a then B
Means that:
If A halts in a configuration of the form (h,w1aw2), where w1or w2 are any words
Then B starts with the configuration (s,w1aw2), where s is the initial state in B

3. Graphical Conventions (2)M a a M;
If a then A
else B A B
Means that:
If M halts in a configuration of the form (h,w1aw2), then A starts with the configuration (s,w1aw2), where s is the initial state of A
If M halts in a configuration of the form (h,w1w2) with   a, then B starts with the configuration (s’,w1 w2), where s’ is the initial state in B

4. Graphical Conventions (3)
while (not a) do A a A
Means that:
If the initial configuration has the form (s,w1w2), then A starts with this configuration if   a, where s is the initial state in A
If A halts in a configuration of the form (h,w1w2) with   a, then A starts with the configuration (s,w1 w2)

5. Basic Turing Machines a
Denotes “write a on the current cell” (a is a character in ) 
Denotes “move to the right of the current cell” (the book uses “R” instead of “”) 
Denotes “move to the left of the current cell” (the book uses “L” instead of “”)

6. Other Simple Turing Machines
What it does a
“if current cell has an a, then replace it with a b” b a 
“if current cell has a character a different than , then move once to the right and write that a”

7. The  Turing Machine –  –   What it does?
Move to the right until if finds 

8. The  Turing Machine
Draw diagram for    –

9.  and  – –  –
Move to the left until the first non-blank space cell is found   –
Move to the right until the first non-blank space cell is found 

10. The Conventions 2 simple rules: Rule # 1: Rule # 2: R
“move to the right until you find “ R
“move to the right until you find a symbol other than “
Note: first check. Then move (think of a “while”)
“Never forget Rule # 1”
(Mr Miogi, The Karate Kid)

11. Example: The (ab)* Turing Machine1  ?  a b a b  a b a a