Recursively Enumerable and Recursive Languages Costas Busch RPI

Recursively Enumerable and Recursive Languages Costas Busch - RPI 1

Definition: A language is recursively enumerable if some Turing machine accepts it Costas Busch - RPI 2

Let be a recursively enumerable language and the Turing Machine that accepts it For string : if then halts in a final state if then halts in a non-final state or loops forever Costas Busch - RPI 3

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 Costas Busch - RPI 4

Let be a recursive language and the Turing Machine that accepts it For string : if then halts in a final state if then halts in a non-final state Costas Busch - RPI 5

We will prove: 1. There is a specific language which is not recursively enumerable (not accepted by any Turing Machine) 2. There is a specific language which is recursively enumerable but not recursive Costas Busch - RPI 6

Non Recursively Enumerable Recursive Costas Busch - RPI 7

A Language which is not Recursively Enumerable Costas Busch - RPI 8

We want to find a language that is not Recursively Enumerable This language is not accepted by any Turing Machine Costas Busch - RPI 9

Consider alphabet Strings: Costas Busch - RPI 10

Consider Turing Machines that accept languages over alphabet They are countable: Costas Busch - RPI 11

Example language accepted by Alternative representation Costas Busch - RPI 12

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Consider the language consists from the 1’s in the diagonal Costas Busch - RPI 14

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Consider the language consists of the 0’s in the diagonal Costas Busch - RPI 16

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Theorem: Language is not recursively enumerable Costas Busch - RPI 18

Proof: Assume for contradiction that is recursively enumerable There must exist some machine that accepts Costas Busch - RPI 19

Question: Costas Busch - RPI 20

Answer: Costas Busch - RPI 21

Question: Costas Busch - RPI 22

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Question: Costas Busch - RPI 24

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for any Similarly: Because either: or Costas Busch - RPI 26

Therefore, the machine cannot exist Therefore, the language is not recursively enumerable End of Proof Costas Busch - RPI 27

Observation: There is no algorithm that describes (otherwise would be accepted by some Turing Machine) Costas Busch - RPI 28

Non Recursively Enumerable Recursive Costas Busch - RPI 29

A Language which is Recursively Enumerable and not Recursive Costas Busch - RPI 30

We want to find a language which Is recursively enumerable But not recursive There is a Turing Machine that accepts the language The machine doesn’t halt on some input Costas Busch - RPI 31

We will prove that the language Is recursively enumerable but not recursive Costas Busch - RPI 32

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Theorem: The language is recursively enumerable Costas Busch - RPI 34

Proof: We will give a Turing Machine that accepts Costas Busch - RPI 35

Turing Machine that accepts For any input string • Compute , for which • Find Turing machine (using an enumeration procedure for Turing Machines) • Simulate • If on input accepts, then accept End of Proof Costas Busch - RPI 36

Observation: Recursively enumerable Not recursively enumerable (Thus, also not recursive) Costas Busch - RPI 37

Theorem: The language is not recursive Costas Busch - RPI 38

Proof: Assume for contradiction that Then is recursive: Take the Turing Machine that accepts halts on any input: If If accepts then rejects then accept Costas Busch - RPI 39

Therefore: is recursive But we know: is not recursively enumerable thus, not recursive CONTRADICTION!!!! Costas Busch - RPI 40

Therefore, is not recursive End of Proof Costas Busch - RPI 41

Non Recursively Enumerable Recursive Costas Busch - RPI 42

Turing acceptable languages and Enumeration Procedures Costas Busch - RPI 43

We will prove: (weak result) • If a language is recursive then there is an enumeration procedure for it (strong result) • A language is recursively enumerable if and only if there is an enumeration procedure for it Costas Busch - RPI 44

Theorem: if a language is recursive then there is an enumeration procedure for it Costas Busch - RPI 45

Proof: Enumeration Machine Enumerates all strings of input alphabet Costas Busch - RPI Accepts 46

If the alphabet is then can enumerate strings as follows: Costas Busch - RPI 47

Enumeration procedure Repeat: generates a string checks if YES: print NO: to output ignore End of Proof Costas Busch - RPI 48

Example: Enumeration Output Costas Busch - RPI 49

Theorem: if language is recursively enumerable then there is an enumeration procedure for it Costas Busch - RPI 50

Proof: Enumeration Machine Enumerates all strings of input alphabet Costas Busch - RPI Accepts 51

If the alphabet is then can enumerate strings as follows: Costas Busch - RPI 52

NAIVE APPROACH Enumeration procedure Repeat: generates a string checks if YES: print NO: Problem: If machine to output ignore may loop forever Costas Busch - RPI 53

BETTER APPROACH Generates first string executes first step on Generates second string executes first step on second step on Costas Busch - RPI 54

Generates third string executes first step on second step on third step on And so on. . . Costas Busch - RPI 55

Step in string 1 1 2 2 3 3 Costas Busch - RPI 56

If for any string machine halts in a final state then it prints on the output End of Proof Costas Busch - RPI 57

Theorem: If for language there is an enumeration procedure then is recursively enumerable Costas Busch - RPI 58

Proof: Input Tape Machine that accepts Enumerator for Compare Costas Busch - RPI 59

Turing machine that accepts For input string Repeat: • Using the enumerator, generate the next string of • Compare generated string with If same, accept and exit loop End of Proof Costas Busch - RPI 60

We have proven: A language is recursively enumerable if and only if there is an enumeration procedure for it Costas Busch - RPI 61