1. The Church-Turing Thesis http://cis.k.hosei.ac.jp/~yukita/

2. 2 Example M 1

3. 3 Definition 3.1

4. 4 Configuration 1011q 7 01111 q7q7 101101111___...

5. 5 Computation

6. 6 Recognizer and Decider

7. 7 Example 3.4 M 2

8. 8 Example 3.6 M 3

9. 9 Example 3.7 [Element distinctness problem] M 4

10. 10 Multitape Turing Machines

11. 11 Theorem 3.8 Every multitape Turing machine M has an equivalent single tape Turing machine S. S #01010#aaa_... #ba# M 01010_ aaa_ ba_

12. 12 Simulation

13. 13 Nondeterministic Turing Machines

14. 14 Theorem 3.10 Every nondeterministic Turing machine has an equivalent deterministic Turing machine. D 0010__... xx#0 12_ input tape 1x__ 332312113_ simulation tape address tape

15. 15 Proof

16. 16 Corollary 3.11 A language is Turing recognizable if and only if some nondeterministic Turing machine recognizes it.

17. 17 Enumerators

18. 18 Theorem 3.13 A language is Turing-recognizable if and only if some enumerator enumerates it.

19. 19 To avoid depth-first loops s1s1 s2s2 s3s3 s4s4 s5s5 s6s6

20. 20 Hilbert’s 10 th Problem

21. 21 Problem 3.18

22. 22 Example 3.14 An algorithm