Understanding Linear Bounded Automata and Turing Machines in Computer Science

1. Linear Bounded AutomataLBAs

2. Linear Bounded Automata are like Turing Machines with a restriction: The working space of the tape is the space of the input string

3. Input string Working space of tape Right-end marker Left-end marker All computation is done between end markers

4. We define LBA’s to be NonDeterministic Open Problem: NonDeterministic LBA’s have same power with Deterministic LBA’s ?

5. Example languages accepted by LBAs: LBA’s have more power than NPDA’s

6. Later in class we will prove: LBA’s have less power than Turing Machines

7. A Universal Turing Machine

8. Limitation of Turing Machines: Turing Machines are “hardwired” They execute only one program Real Computers are reprogrammable

9. Solution: Universal Turing Machine • is a reprogrammable machine • simulates any other Turing Machine

10. Universal Turing machine simulates any Turing machine Input of Universal Machine: Description of transitions of Initial tape contents of

11. Description of Universal Turing Machine Tape Contents of State of

12. Alphabet Encoding Symbols: Encoding:

13. State Encoding States: Encoding: Head Move Encoding Move: Encoding:

14. Transition Encoding Transition: Encoding: separator

15. Machine Encoding Transitions: Encoding: separator

16. Input of Universal Turing Machine: encoding of the simulated machine

17. A Turing Machine is described with a string of 0’s and 1’s The set of Turing machines form a language: each string of the language is the encoding of a Turing Machine

18. Countable Sets

19. Infinite sets are either: • Countable • Uncountable

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

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

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

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

24. Better Approach

30. Rational Numbers: Correspondence: Positive Integers:

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

32. 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

33. strings Enumeration Machine for output Finite time:

34. Enumeration Machine Configuration Time 0 Time

35. Time Time

36. A set is countable if there is an enumeration procedure for it

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

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

39. Proper Order Better procedure: Produce all strings of length 1 Produce all strings of length 2 Produce all strings of length 2 ..........

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

41. Theorem: The set of all Turing Machines is countable

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

43. 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

44. Uncountable Sets

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

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

47. Proof: Since is countable, we can write Element of

48. Elements of the powerset have the form:

49. We encode each element of the power set with a string of 0’s and 1’s Encoding Powerset element

50. Let’s assume for contradiction that the powerset is countable. We can enumerate the elements of the powerset