1 1 CDT314 FABER Formal Languages, Automata and Models of Computation Lecture 15-1 Mälardalen University 2012

2 Countable Sets Based on course materials by Prof. Costas Busch

3 Infinite sets are either: Countable Uncountable

4 Countable set: There is a one to one correspondence between elements of the set and positive integers

5 Example: Even integers: The set of even integers is countable Positive integers: Correspondence: corresponds to

6 Example:The set of rational numbers is countable Rational numbers:

7 Naive Approach Rational numbers: Positive integers: Correspondence: Doesn’t work: we will never count numbers with nominator 2

8 Better Approach

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14 Rational Numbers: Correspondence: Positive Integers:

15 We proved: the set of rational numbers is countable by giving an enumeration procedure

16 Definition An enumeration procedure for is a Turing machine that generates any string of in finite number of steps. Let be a set of strings

17 Enumeration Machine for Finite time: strings output

18 Enumeration Machine Configuration Time 0 Time

19 Time

20 A set is countable if there is an enumeration procedure for it

21 Example: The set of all strings is countable. We will describe the enumeration procedure.

22 Naive procedure: Produce the strings in lexicographic order: Doesn’t work! Strings starting with will never be produced.

23 Better procedure: Proper Order Produce all strings of length 1 Produce all strings of length

24 Produce strings: Length 2 Length 3 Length 1 Proper Order

25 Theorem: The set of all Turing Machines is countable. Proof: Find an enumeration procedure for the set of Turing Machine strings. Any Turing machine is a finite string Encoded with a sequence of 0’s and 1’s.

26 1. Generate the next string of 0’s and 1’s in proper order 2. Check if the string defines a Turing Machine if YES: print string on output if NO: ignore string Enumeration Procedure: Repeat

27 Uncountable Sets

28 A set is uncountable if it is not countable Definition:

29 Theorem: Let be an infinite countable set. The powerset of is uncountable. The power set of natural numbers has the same cardinality as the set of real numbers. (Using the Cantor–Bernstein–Schröder theorem, it is easy to prove that there exists a bijection between the set of reals and the power set of the natural numbers).

30 Proof: Since is countable, we can write Element of

31 Elements of the powerset have the form:

32 We encode each element of the power set with a string of 0’s and 1’s * Powerset element Encoding *Cantor’s diagonal argument

33 Let’s assume the contrary, that the powerset is countable. We can enumerate the elements of the powerset.

34 Powerset element Encoding

35 Take the powerset element whose bits are the complements of the diagonal.

36 New element: (Diagonal complement)

37 The new element must be some This is impossible: The i-th bit must be the complement of itself.

38 We have contradiction! Therefore the powerset is uncountable.

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

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

41 Languages: uncountable Turing machines: countable There are infinitely many more languages than Turing machines!

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

43 Recursively Enumerable Languages and Recursive Languages

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

45 For a 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

46 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

47 For a 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.

48 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.

49 Recursive Recursively Enumerable Non Recursively Enumerable

50 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.

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

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

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

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

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

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

57 executes first step on BETTER APPROACH generates second string executes first step on second step on generates first string

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

59 1 Move 2 3

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

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

62 Proof: Input Tape Enumerator for Compare Machine that accepts

63 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

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

65 We have shown: A language is recursively enumerable if and only if there is an enumeration procedure for it.

66 A Language which is not Recursively Enumerable

67 We search for a language that is not Recursively Enumerable. This language is not accepted by any Turing Machine.

68 Consider alphabet Strings:

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

70 Example language accepted by Alternative representation

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72 Consider the language consists of the 1’s on the diagonal

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74 Consider the language consists from of 0’s on the diagonal

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76 Theorem: Language is not recursively enumerable.

77 Proof: is recursively enumerable Assume on the contrary that There must exist some machine that accepts

78 Question:

79 Answer:

80 Question:

81 Answer:

82 Question:

83 Answer:

84 Similarly: for any Because either: or

85 Therefore the machine cannot exist CONTRADICTION!!! The language is not recursively enumerable. End of proof

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

87 A Language which is Recursively Enumerable and not Recursive

88 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

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

90 The language Theorem: is recursively enumerable

91 Proof: We will give a Turing Machine that accepts

92 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

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

94 Theorem: The language is not recursive.

95 Proof: Assume on the contrary that is recursive. Then is recursive: Take the Turing Machine that accepts halts on any input If accepts then reject If rejects then accept

96 Therefore: recursive But we know: not recursively enumerable thus, not recursive CONTRADICTION! Therefore, is not recursive End of proof

97 Recursive Recursively Enumerable Non Recursively Enumerable