1. Reducibility http://cis.k.hosei.ac.jp/~yukita/

2. 2 Theorem 5.1 HALT TM is undecidable.

3. 3 Theorem 5.2 E TM is undecidable.

4. 4 Proof continued

5. 5 Theorem 5.3 REGULAR TM is undecidable.

6. 6 Continued Remark on Th. 5.3

7. 7 Theorem 5.4 EQ TM is undecidable.

8. 8 Definition 5.5 Computation History

9. 9 Definition 5.6 Linear bounded automata A restricted type of Turing machine whererin the tape head is not permitted to move off the portion of the tape containing the input. If the machine tries to move its head off either end of the input, the head stays where it is, in the same way that the head will not move off the left-hand end of an ordinary Turing machine. The idea of changing the alphabet suggests that the definition be modified in such a way that the amount of memory allowed is linear in n.

10. 10 Remark

11. 11 Lemma 5.7 Let M be an LBA with q states and g symbols in the tape alphabet. There are exactly qng n distinct configurations of M for a tape of length n.

12. 12 Theorem 5.8 A LBA is decidable.

13. 13 Theorem 5.9 E LBA is undecidable.

14. 14 Proof (continued)

15. 15 Remark on Th. 5.9

16. 16 Theorem 5.10 ALL CFG is undecidable.

17. 17 The Post Correspondence Problem

18. 18 Definitions of PCP and MPCP

19. 19 Theorem 5.11 PCP is undecidable.

20. 20 Proof continued

21. 21 Demonstration by example (1), (2)+(4) # # q 0 0 1 0 0 # # q 0 0 1 0 0 # 2 q 7 1 0 0 #

22. 22 Demonstration by example (2)+(4), (3)+(4) # 2 q 7 1 0 0 # # 2 q 7 1 0 0 # 2 0 q 5 0 0 # … # 2 0 q 5 0 0 # # 2 0 q 5 0 0 # 2 q 9 0 2 0 # …

23. 23 Demonstration by example (4)+(6) # # 2 1 q accept 0 2 # … # 2 1 q accept 0 2 #... # # 2 1 q accept 0 0 # 2 1 q accept 2 #... # q accept # …

24. 24 Demonstration by example (7) # q accept # #

25. 25 MPCP to PCP

26. 26 Definition 5.12 Computable Function

27. 27 Definition 5.15 Mapping Reducibility

28. 28 Theorem 5.16

29. 29 Example 5.18 Reduction from A TM to HALT TM

30. 30 Example 5.19 A TM  MPCP  PCP

31. 31 Example 5.20 E TM  EQ TM

32. 32 Example 5.21

33. 33 Theorem 5.22 Turing recognizability

34. 34 Remark

35. 35 Theorem 5.24 EQ TM is neither Turing-recognizable nor co- Turing-recognizable.

36. 36 Proof (continued)