1 Uncountable Sets continued...

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

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

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

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

6 Recursively Enumerable Languages and Recursive Languages

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

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

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

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

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

12 Recursive Recursively Enumerable Non Recursively Enumerable

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

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

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

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

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

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

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

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

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

22 1 Move 2 3

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

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

25 Proof: Input Tape Enumerator for Compare Machine that accepts

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

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

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

29 A Language which is not Recursively Enumerable

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

31 Consider alphabet Strings:

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

33 Example language accepted by Alternative representation

34

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

36

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

38

39 Theorem: Language is not recursively enumerable

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

41 Question:

42 Answer:

43 Question:

44 Answer:

45 Question:

46 Answer:

47 Similarly: for any Because either: or

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

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

50 A Language which is Recursively Enumerable and not Recursive

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

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

53 The language Theorem: is recursively enumerable

54 Proof: We will give a Turing Machine that accepts

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

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

57 Theorem: The language is not recursive

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

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

60 Therefore, is not recursive End of proof

61 Recursive Recursively Enumerable Non Recursively Enumerable