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

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

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

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

8. Better Approach

14. Rational Numbers:
Correspondence:
Positive Integers:

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

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

17. strings
Enumeration
Machine for
output
Finite time:

18. Enumeration Machine
Configuration
Time 0
Time

19. Time
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 2

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

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

26. Enumeration Procedure:
Repeat
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

27. Uncountable Sets

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

29. Let be an infinite countable set. The powerset of is uncountable.
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 *
Encoding
Powerset
element
*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. Let be a recursively enumerable language
and be the Turing Machine that accepts it.
For a string : if
then halts in a final state if
then 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. Let be a recursive language
and be the Turing Machine that accepts it.
For a string : if
then halts in a final state. if
then 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. 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
Enumerates all
strings of input alphabet
Accepts

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
Enumerates all
strings of input alphabet
Accepts

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

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

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

59. Move 1 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
Machine that
accepts
Enumerator
for
Compare

63. 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

64. Question:
This is not a membership algorithm.
Why?
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

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

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

76. Theorem:
Language is not recursively enumerable.

77. Proof:
Assume on the contrary that
is recursively enumerable
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
Is recursively
enumerable
But not
recursive
There is a
Turing Machine
that accepts
the language
The machine
doesn’t
necessarily halt
on any input

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

90. Theorem:
The language
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. Non Recursively Enumerable