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

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2 Theorem 5.1 HALT TM is undecidable.

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3 Theorem 5.2 E TM is undecidable.

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4 Proof continued

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5 Theorem 5.3 REGULAR TM is undecidable.

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6 Continued Remark on Th. 5.3

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7 Theorem 5.4 EQ TM is undecidable.

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8 Definition 5.5 Computation History

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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.

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10 Remark

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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.

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12 Theorem 5.8 A LBA is decidable.

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13 Theorem 5.9 E LBA is undecidable.

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14 Proof (continued)

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15 Remark on Th. 5.9

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16 Theorem 5.10 ALL CFG is undecidable.

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17 The Post Correspondence Problem

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18 Definitions of PCP and MPCP

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19 Theorem 5.11 PCP is undecidable.

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20 Proof continued

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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 #

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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 # …

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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 # …

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

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25 MPCP to PCP

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26 Definition 5.12 Computable Function

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27 Definition 5.15 Mapping Reducibility

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28 Theorem 5.16

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29 Example 5.18 Reduction from A TM to HALT TM

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30 Example 5.19 A TM MPCP PCP

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31 Example 5.20 E TM EQ TM

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32 Example 5.21

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33 Theorem 5.22 Turing recognizability

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34 Remark

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

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36 Proof (continued)

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