CD5560 FABER Formal Languages, Automata and Models of Computation Lecture 13

1. CD5560 FABER Formal Languages, Automata and Models of Computation Lecture 13 Mälardalen University 2010

2. ContentAlan Turing and Hilbert Program UniversalTuring Machine Chomsky Hierarchy DecidabilityReducibilityUncomputable FunctionsRice’s TheoremInteractive Computing, Persistent TM’s (Dina Goldin)

3. http://www.turing.org.uk/turing/ Who was Alan Turing? Founder of computer science, mathematician, philosopher, codebreaker, visionary man before his time. http://www.cs.usfca.edu/www.AlanTuring.net/turing_archive/index.html-Jack Copeland and Diane Proudfoot http://www.turing.org.uk/turing/ The Alan Turing Home PageAndrew Hodges

4. 1912 (23 June): Birth, London1926-31: Sherborne School1930: Death of friend Christopher Morcom1931-34: Undergraduate at King's College, Cambridge University1932-35: Quantum mechanics, probability, logic1935: Elected fellow of King's College, Cambridge1936: The Turing machine, computability, universal machine1936-38: Princeton University. Ph.D. Logic, algebra, number theory1938-39: Return to Cambridge. Introduced to German Enigma cipher machine1939-40: The Bombe, machine for Enigma decryption1939-42: Breaking of U-boat Enigma, saving battle of the Atlantic Alan Turing

5. 1943-45: Chief Anglo-American crypto consultant. Electronic work.1945: National Physical Laboratory, London1946: Computer and software design leading the world.1947-48: Programming, neural nets, and artificial intelligence1948: Manchester University1949: First serious mathematical use of a computer1950: The Turing Test for machine intelligence1951: Elected FRS. Non-linear theory of biological growth1952: Arrested as a homosexual, loss of security clearance1953-54: Unfinished work in biology and physics1954 (7 June): Death (suicide) by cyanide poisoning, Wilmslow, Cheshire. Alan Turing

6. Hilbert’s Program, 1900 Hilbert’s hope was that mathematics would be reducible to finding proofs (manipulating the strings of symbols) from a fixed system of axioms, axioms that everyone could agree were true. Can all of mathematics be made algorithmic, or will there always be new problems that outstrip any given algorithm, and so require creative acts of mind to solve?

7. TURING MACHINES

8. Turing’s "Machines". These machines are humans who calculate. (Wittgenstein) A man provided with paper, pencil, and rubber, and subject to strict discipline, is in effect a universal machine. (Turing)

9. Standard Turing Machine Tape ...... ...... Read-Write head Control Unit

10. The Tape No boundaries -- infinite length ...... ...... Read-Write head The head moves Left or Right

11. ...... ...... Read-Write head The head at each time step: 1. Reads a symbol 2. Writes a symbol 3. Moves Left or Right

12. Example Time 0 ...... ...... Time 1 ...... ...... 1. Reads 2. Writes 3. Moves Left

13. The Input String Input string Blank symbol ...... ...... head Head starts at the leftmost position of the input string

14. States & Transitions Write Read Move Left Move Right

15. Time 1 ...... ...... Time 2 ...... ......

16. Determinism Turing Machines are deterministic Not Allowed Allowed No lambda transitions allowed in standard TM!

17. Formal Definitions for Turing Machines

18. Transition Function

19. Turing Machine Input alphabet Tape alphabet States Final states Transition function Initial state blank

20. Universal Turing Machine

21. Turing Machines are “hardwired” they execute only one program A limitation of Turing Machines: Better are reprogrammable machines.

22. Solution: Universal Turing Machine Characteristics: • Reprogrammable machine • Simulates any other Turing Machine

23. Universal Turing Machine simulates any other Turing Machine Input of Universal Turing Machine • Description of transitions of • Initial tape contents of

24. Tape 1 Three tapes Description of Tape 2 Universal Turing Machine Tape Contents of Tape 3 State of

25. Tape 1 Description of We describe Turing machine as a string of symbols: We encode as a string of symbols

26. Alphabet Encoding Symbols: Encoding:

27. States: Encoding: Move: Encoding: State Encoding Head Move Encoding

28. Transition Encoding Transition: Encoding: separator

29. Machine Encoding Transitions: Encoding: separator

30. Tape 1 contents of Universal Turing Machine: encoding of the simulated machine as a binary string of 0’s and 1’s

31. A Turing Machine is described with a binary string of 0’s and 1’s. Therefore: The set of Turing machines forms a language: Each string of the language is the binary encoding of a Turing Machine.

32. Language of Turing Machines (Turing Machine 1) L = {010100101, 00100100101111, 111010011110010101, ……} (Turing Machine 2) ……

33. The Chomsky Hierarchy

34. The Chomsky Language Hierarchy Recursively-enumerable Recursive Context-sensitive Context-free Regular Non-recursively enumerable

35. Unrestricted Grammars Productions String of variables and terminals String of variables and terminals

36. Example of unrestricted grammar

37. Theorem A language is recursively enumerable if and only if it is generated by an unrestricted grammar.

38. Context-Sensitive Grammars Productions String of variables and terminals String of variables and terminals and

39. The language is context-sensitive:

40. Theorem A language is context sensitive if and only if it is accepted by a Linear-Bounded automaton.

41. Linear Bounded Automata (LBAs) are the same as Turing Machines with one difference: The input string tape space is the only tape space allowed to use.

42. Linear Bounded Automaton (LBA) Input string Working space in tape Left-end marker Right-end marker All computation is done between end markers.

43. Observation There is a language which is context-sensitive but not recursive.

44. Decidability

45. Does Machine have three states ? • Is string a binary number? • Does DFA accept any input? Consider problems with answer YES or NO. Examples

46. Does Machine have three states ? • Is string a binary number? • Does DFA accept any input? A problem is decidable if some Turing machine solves (decides) the problem. Decidable problems:

47. The Turing machine that solves a problem answers YES or NO for each instance. YES Input problem instance Turing Machine NO

48. The machine that decides a problem: • If the answer is YES • then halts in a yes state • If the answer is NO • then halts in a no state These states may not be the final states.

49. Turing Machine that decides a problem YES NO YES and NO states are halting states

50. Difference between Recursive Languages (“Acceptera”) and Decidable problems (“Avgöra”) For decidable problems: The YES states may not be final states.