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2. Turing Machines CSE2303 Formal Methods I Lecture 14

3. Overview Turing Machines Converting FA to TM Building Turing Machines

4. ... Finite String (input) All Blanks iiiiiiiv Tape Head Turing Machine Tape Head can: Move left and right. Read letters from the tape. Write letters onto the tape.

5. HALT 2 START 1 34 6 5 (a, A, R) (a, a, R) (B, B, R) (b, B, L) (A, A, R) (B, B, R) (B, B, L) (a, a, L) (A, A, R) ( , , R)

6. TM Components A Tape and Tape Head An Input Alphabet A Tape Alphabet A finite set of states –Each state is numbered by an integer  1. –Start State (1) –Halt State (2) A finite set of rules (letter, letter, direction) between states.

7. Definitions For a Turing Machine T Accept(T) –The set of strings leading to a HALT state. –Called the language accepted by T. Reject(T) –The set of strings that crash during execution. Loop(T) –The set of strings that loop forever.

8. Example HALT 2START 1 3 ( , , R) (b, b, R) (a, a, R) (b, b, R) Accept(T) = strings with a double aa Reject(T) = strings without a double aa that end in a Loop(T) =  or strings without a double aa that end in b

9. Regular Languages Every Regular Language can be accepted by a Turing Machine.

10. Convert FA to TM Change the - to START 1. Label all other states with a integer  3. Change the edge labels. –a to (a, a, R) –b to (b, b, R) Delete the + from all the Final states, and add an edge from each Final state to HALT 2, labeled with ( , , R).

11. EVEN-EVEN b b a ± b b aa a 3 45 START 1 (a, a, R) (b, b, R) (a, a, R) (b, b, R) (a, a, R) HALT 2 ( , , R)

12. Problem Build a Turing Machine that accepts the language {a n b n : n  0}.

13. If tape is blank, then Halt. Loop { If current letter is a, then change a to A & move right. Move right over any a’s and B’s. If current letter is b, then change b to B & move left. Move left over any B’s. If current letter is A, then move right & exit the loop. Else if current letter is a move left over any a’s. If current letter is A, then move right. } Move right over any B’s. If current letter is blank, then Halt.

14. HALT 2 START 1 34 6 5 (a, A, R) (a, a, R) (B, B, R) (b, B, L) (A, A, R) (B, B, R) (B, B, L) (a, a, L) (A, A, R) ( , , R)

15. Problem Build a Turing Machine that accepts the language {a n b n a n : n  0}.

16. Loop { If current letter is blank, then Halt. If current letter is a, then change a to A & move right. Move right over a*bb*. If current letter is a, then move left. If current letter is b, then change b to a & move right. Move right over any a’s. If current letter is blank, then delete 2 a’s on the left. Move left over any a’s and b’s. If current letter is A, then move right. }

17. TM for a n b n a n HALT 2START 1 345 987 6 ( , , L) (a, a, R) (b, a, R) (a, , L) (a, a, L) (b, b, L) (a, a, L) (b, b, R) (a, a, R) (a, A, R) (A, A, R) ( , , R)

18. Revision Know what a TM is, and how to use one. Be able to convert a FA into a TM. Be able to build a TM to define a language. Preparation Read –Chapter 25 in the Text book.