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1 Uncountable Sets continued...

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2 Theorem: Let be an infinite countable set. The powerset of is uncountable

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3 Application: Languages Example Alphabet : Set of Strings: infinite and countable Powerset: all languages uncountable

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4 Languages: Uncountable Turing machines: Countable There are infinitely many more languages than Turing Machines

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5 There are some languages not accepted by Turing Machines These languages cannot be described by algorithms

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6 Recursively Enumerable Languages and Recursive Languages

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7 Definition: A language is recursively enumerable if some Turing machine accepts it

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8 For string : Let be a recursively enumerable language and be the Turing Machine that accepts it ifthen halts in a final state ifthen halts in some state or loops forever

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9 Definition: A language is recursive if some Turing machine accepts it and halts on any input string In other words: A language is recursive if there is a membership algorithm for it

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10 For string : Let be a recursive language and be the Turing Machine that accepts it ifthen halts in a final state ifthen halts in a non-final state

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11 We will prove: 1. There is a specific language which is not recursively enumerable 2. There is a specific language which is recursively enumerable but not recursive

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12 Recursive Recursively Enumerable Non Recursively Enumerable

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13 First we prove: If a language is recursive then there is an enumeration procedure for it A language is recursively enumerable if and only if there is an enumeration procedure for it

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14 Theorem: if a language is recursive then there is an enumeration procedure for it

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15 Proof: Enumeration Machine Accepts Enumerates all strings of input alphabet

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16 Enumeration procedure Repeat: generates a string checks if YES: print to output NO: ignore End of proof

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17 Theorem: if language is recursively enumerable then there is an enumeration procedure for it

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18 Proof: Enumeration Machine Accepts Enumerates all strings of input alphabet

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19 Enumeration procedure Repeat:generates a string checks if YES: print to output NO: ignore NAIVE APPROACH Problem: If machine may loop forever

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20 executes first step on BETTER APPROACH Generates second string executes first step on second step on Generates first string

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21 Generates third string executes first step on second step on third step on And so on............

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22 1 Move 2 3

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23 If for string machine halts in a final state then it prints on the output End of proof

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24 Theorem: If for language there is an enumeration procedure then is recursively enumerable

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25 Proof: Input Tape Enumerator for Compare Machine that accepts

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26 Turing machine that accepts Repeat: Using the enumerator, generate the next string of For input string Compare generated string with If same, accept and exit loop End of proof

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27 This is not a membership algorithm. Why? Question: Answer: The enumeration procedure may not produce strings in proper order

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28 We have proven: A language is recursively enumerable if and only if there is an enumeration procedure for it

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29 A Language which is not Recursively Enumerable

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30 We want to find a language that is not Recursively Enumerable This language is not accepted by any Turing Machine

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31 Consider alphabet Strings:

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32 Consider Turing Machines that accept languages over alphabet They are countable:

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33 Example language accepted by Alternative representation

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35 Consider the language consists from the 1’s in the diagonal

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37 Consider the language consists from the 0’s in the diagonal

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39 Theorem: Language is not recursively enumerable

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40 Proof: is recursively enumerable Assume for contradiction that There must exist some machine that accepts

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41 Question:

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42 Answer:

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43 Question:

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44 Answer:

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45 Question:

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46 Answer:

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47 Similarly: for any Because either: or

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48 Therefore the machine cannot exist CONTRADICTION!!! Therefore the language is not recursively enumerable End of proof

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49 Observation: There is no algorithm that describes (otherwise it would be accepted by a Turing Machine)

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50 A Language which is Recursively Enumerable and not Recursive

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51 We want to find a language which There is a Turing Machine that accepts the language The machine doesn’t necessarily halt on any input Is recursively enumerable But not recursive

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52 We will prove that the language Is recursively enumerable but not recursive

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53 The language Theorem: is recursively enumerable

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54 Proof: We will give a Turing Machine that accepts

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55 Turing Machine that accepts For any input string Write Find Turing machine (using the enumeration procedure for Turing Machines) Simulate on input If accepts, then accept End of proof

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56 Observation: Recursively enumerable Not recursively enumerable (Thus, not recursive)

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57 Theorem: The language is not recursive

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58 Proof: Assume for contradiction that is recursive Then is recursive: Take the Turing Machine that accepts halts on any input If accepts then reject If rejects then accept

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59 Therefore: recursive But we know: not recursively enumerable thus, not recursive CONTRADICTION!!!!

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60 Therefore, is not recursive End of proof

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61 Recursive Recursively Enumerable Non Recursively Enumerable

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