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Reducibility

Reducibility. http://cis.k.hosei.ac.jp/~yukita/. Theorem 5.1 HALT TM is undecidable. Theorem 5.2 E TM is undecidable. Proof continued. Theorem 5.3 REGULAR TM is undecidable. Continued Remark on Th. 5.3. Theorem 5.4 EQ TM is undecidable. Definition 5.5 Computation History.

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Reducibility

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

  2. Theorem 5.1 HALTTM is undecidable.

  3. Theorem 5.2 ETM is undecidable.

  4. Proof continued

  5. Theorem 5.3 REGULARTM is undecidable.

  6. Continued Remark on Th. 5.3

  7. Theorem 5.4 EQTM is undecidable.

  8. Definition 5.5 Computation History

  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. Remark

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

  12. Theorem 5.8 ALBA is decidable.

  13. Theorem 5.9 ELBA is undecidable.

  14. Proof (continued)

  15. Remark on Th. 5.9

  16. Theorem 5.10 ALLCFG is undecidable.

  17. The Post Correspondence Problem

  18. Definitions of PCP and MPCP

  19. Theorem 5.11 PCP is undecidable.

  20. Proof continued

  21. Demonstration by example(1), (2)+(4) # # q0 0 1 0 0 # # q0 0 1 0 0 # # q0 0 1 0 0 # 2 q7 1 0 0 #

  22. Demonstration by example(2)+(4), (3)+(4) # 2 q7 1 0 0 # # 2 q7 1 0 0 # 2 0 q5 0 0 # … # 2 0 q5 0 0 # # 2 0 q5 0 0 # 2 q9 0 2 0 # …

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

  24. Demonstration by example(7) # qaccept # # # qaccept # #

  25. MPCP to PCP

  26. Definition 5.12 Computable Function

  27. Definition 5.15 Mapping Reducibility

  28. Theorem 5.16

  29. Example 5.18 Reduction from ATM to HALTTM

  30. Example 5.19 ATMMPCP PCP

  31. Example 5.20 ETM EQTM

  32. Example 5.21

  33. Theorem 5.22 Turing recognizability

  34. Remark

  35. Theorem 5.24 EQTM is neither Turing-recognizable nor co-Turing-recognizable.

  36. Proof (continued)

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