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

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

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

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)

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 “”)

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”

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

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

 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 

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)

Example: The (ab)* Turing Machine 1   ?  a b a b  a b a a