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

1. 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 Produce all strings of length 2 ..........

24. Length 1 Produce strings: 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. Theorem: Let be an infinite countable set. The powerset of is uncountable. The power set of naturalnumbershasthe same cardinalityastheset of real numbers. (Usingthe Cantor–Bernstein–Schröder theorem, itis easy toprovethatthereexists a bijectionbetweentheset of reals and the power set of thenaturalnumbers).

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 Languagesand 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 Recursively Enumerable Recursive

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.