COMP-330 Theory of Computation. Fall Prof. Claude Crépeau. Lec. 14 : Turing Machines

Transcription

1 COMP-330 Theory of Computation Fall Prof. Claude Crépeau Lec. 14 : Turing Machines 1

2 COMP 330 Fall 2012: Lectures Schedule 1. Introduction 1.5. Some basic mathematics 2. Deterministic finite automata +Closure properties, 3. Nondeterministic finite automata 4. Determinization+Kleene s theorem 5-6. Regular Expressions, GNFA and Regular Languages 6-7. The pumping lemma 8. Minimization+Myhill-Nerode theorem 9. Context-free languages 10. Pushdown automata 11. Equivalence between CFG & PDA 12. The pumping lemma for CFLs 13. MIDTERM 14. Turing Machines 15. The Church-Turing Thesis 16. Introduction to computability 17. Models of computation Basic computability theory 18. Reducibility, undecidability and Rice s theorem 19. Undecidable problems about CFGs 20. Post Correspondence Problem 21. Validity of FOL is RE / Gödel s and Tarski s thms 22. Universality / The recursion theorem 23. Degrees of undecidability 24. Introduction to complexity 25. Introduction to Quantum Computing/Complexity 26. Review of course material 2

3 All languages Computability Theory Languages we can describe Decidable Languages Context-free Languages Regular NON-Regular Languages via Pumping Lemma Languages NON-Regular Languages via Reductions 3

4 All languages Computability Theory Languages we can describe Decidable Languages Context-free Languages Regular Languages NON-CFLs via Pumping Lemma NON-CFLs via Reductions 4

5 All languages Computability Theory Decidable Languages Languages. we can describe Context-free Languages Regular Languages NON-decidable via Diagonalization NON-decidable via Reductions 5

6 Turing MACHINES Alan Turing 6

7 M 1 7

8 M 1 q

9 M 1 1 q

10 M 1 0 q

11 M 1 1 q read and write! moves Right and Left! 11

12 M 1 0 q

13 Turing Machines 13

14 TM Example 14

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21 Definition of PDA States Input Alphabet Tape Alphabet Transition function Start state Accept state Reject state 16

22 Definition of PDA States q 1 q 2 q 3 Input Alphabet Tape Alphabet Transition function Start state Accept state Reject state 16

23 Definition of PDA States q 1 q 2 q 3 Input Alphabet a,b,c Tape Alphabet Transition function Start state Accept state Reject state 16

24 Definition of PDA States q 1 q 2 q 3 Input Alphabet Tape Alphabet a,b,c a,b,c,a,b,c,_ Transition function Start state Accept state Reject state 16

25 Definition of PDA States q1 q2 q3 a,b,c Tape Alphabet a,b,c,a,b,c,_ Input Alphabet Transition function b c,d q1 q2 Start state Accept state Reject state 16

26 Definition of PDA q2 q 3 q1 L or R States output head input symbol move Input Alphabet a,b,c symbol b c,d Tape Alphabet a,b,c,a,b,c,_ Transition function q1 b c,d Start state q1 Accept state q2 q2 Reject state 16

27 Definition of PDA States q1 q2 q3 a,b,c Tape Alphabet a,b,c,a,b,c,_ Input Alphabet Transition function b c,d q1 q2 Start state Accept state Reject state 16

28 Definition of PDA States q2 q1 q3 a,b,c Tape Alphabet a,b,c,a,b,c,_ Input Alphabet Transition function Start state q1 b c,d q1 q2 Accept state Reject state 16

29 Definition of PDA States q2 q1 q3 a,b,c Tape Alphabet a,b,c,a,b,c,_ Input Alphabet Transition function Start state q1 b c,d q1 q2 Accept state qacc Reject state 16

30 Definition of PDA States q2 q1 q3 a,b,c Tape Alphabet a,b,c,a,b,c,_ Input Alphabet Transition function Start state q1 b c,d q1 q2 Accept state qacc Reject state qrej 16

31 TM definition 17

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33 TM Configuration 19

34 TM Computation 20

35 TM definition For all a,b,c Γ, u,v Γ*, qi,qi Q Config. ua qi bv yields config. u qj acv if δ(qi,b) = qj,c,l Config. ua qi bv yields config. uac qj v if δ(qi,b) = qj,c,r Special cases: Config. q i bv yields qj cv if δ(qi,b) = qj,c,l Config. qi bv yields c qj v if δ(qi,b) = qj,c,r 21

36 TM definition ua qi bv For all a,b,c Γ, u,v Γ*, qi,qi Q Config. ua qi bv yields config. u qj acv if δ(qi,b) = qj,c,l yields (L) Config. ua qi bv yields config. uac qj v if δ(qi,b) = qj,c,r u qj acv Special cases: Config. q i bv yields qj cv if δ(qi,b) = qj,c,l Config. qi bv yields c qj v if δ(qi,b) = qj,c,r 21

37 TM definition For all a,b,c Γ, u,v Γ*, qi,qi Q Config. ua qi bv yields config. u qj acv if δ(qi,b) = qj,c,l Config. ua qi bv yields config. uac qj v if δ(qi,b) = qj,c,r Special cases: Config. q i bv yields qj cv if δ(qi,b) = qj,c,l Config. qi bv yields c qj v if δ(qi,b) = qj,c,r 21

38 TM definition ua qi bv For all a,b,c Γ, u,v Γ*, qi,qi Q Config. ua qi bv yields config. u qj acv if δ(qi,b) = qj,c,l yields (R) Config. ua qi bv yields config. uac qj v if δ(qi,b) = qj,c,r uac qj v Special cases: Config. q i bv yields qj cv if δ(qi,b) = qj,c,l Config. qi bv yields c qj v if δ(qi,b) = qj,c,r 21

39 TM definition For all a,b,c Γ, u,v Γ*, qi,qi Q Config. ua qi bv yields config. u qj acv if δ(qi,b) = qj,c,l Config. ua qi bv yields config. uac qj v if δ(qi,b) = qj,c,r Special cases: Config. q i bv yields qj cv if δ(qi,b) = qj,c,l Config. qi bv yields c qj v if δ(qi,b) = qj,c,r 21

40 TM definition For all a,b,c Γ, u,v Γ*, qi,qi Q qi bv Config. ua qi bv yields config. u qj acv if δ(qi,b) = qj,c,l yields (L) Config. ua qi bv yields config. uac qj v if δ(qi,b) = qj,c,r qj cv Special cases: Config. q i bv yields qj cv if δ(qi,b) = qj,c,l Config. qi bv yields c qj v if δ(qi,b) = qj,c,r 21

41 TM definition For all a,b,c Γ, u,v Γ*, qi,qi Q Config. ua qi bv yields config. u qj acv if δ(qi,b) = qj,c,l Config. ua qi bv yields config. uac qj v if δ(qi,b) = qj,c,r Special cases: Config. q i bv yields qj cv if δ(qi,b) = qj,c,l Config. qi bv yields c qj v if δ(qi,b) = qj,c,r 21

42 TM definition For all a,b,c Γ, u,v Γ*, qi,qi Q qi bv Config. ua qi bv yields config. u qj acv if δ(qi,b) = qj,c,l yields (R) Config. ua qi bv yields config. uac qj v if δ(qi,b) = qj,c,r c qj v Special cases: Config. q i bv yields qj cv if δ(qi,b) = qj,c,l Config. qi bv yields c qj v if δ(qi,b) = qj,c,r 21

43 TM definition For all a,b,c Γ, u,v Γ*, qi,qi Q Config. ua qi bv yields config. u qj acv if δ(qi,b) = qj,c,l Config. ua qi bv yields config. uac qj v if δ(qi,b) = qj,c,r Special cases: Config. q i bv yields qj cv if δ(qi,b) = qj,c,l Config. qi bv yields c qj v if δ(qi,b) = qj,c,r 21

44 TM Computation Start configuration: q0 w (w = input string) Accepting configuration: state = qaccept Rejecting configuration: state = qreject 22

45 TM Computation Turing Machine M accepts input w if there exists configurations C 0, C1,..., Cm such that C0 is a start configuration Ci yields Ci+1 for 0 i<m Cm is an accepting configuration. The collection of strings that M accepts is the language of M or the language recongnized by M, denoted L(M). 23

46 TM Computation 24

47 TM Computation A TM decides a language if it recognizes it and halts (reaches an accepting or rejecting states) on all input strings. 24

48 TM Computation A TM decides a language if it recognizes it and halts (reaches an accepting or rejecting states) on all input strings. 24

49 TM Computation A TM decides a language if it recognizes it and halts (reaches an accepting or rejecting states) on all input strings. 1 Often named Recursively-Enumerable in the litterature. 2 Often named Recursive in the litterature. 24

50 TM Examples 25

51 TM Examples 26

52 TM Examples 27

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80 TM Examples 29

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82 TM Examples 31

83 TM Examples 32

84 TM Examples 33

85 More Turing MACHINES Multitape Turing Machines Non-Deterministic Turing Machines Enumerator Turing Machines Everything else... 34

86 Multitape TM 35

87 Multitape TM 36

88 Multitape TM 36

89 Multitape TM 37

90 Multitape TM 38

91 Non-deterministic TM 39

92 Non-deterministic TM 40

93 Non-deterministic TM 41

94 Non-deterministic TM 41

95 Non-deterministic TM 41

96 Enumerator TM 42

97 Enumerator TM 43

98 Enumerator TM 44

99 Enumerator TM 45

100 Enumerator TM 45

101 Enumerator TM 45

102 Everything Else Alonzo Church Stephen Kleene J. Barkley Rosser 46

103 Everything Else Lambda-calculus Alonzo Church Stephen Kleene J. Barkley Rosser 46

104 Everything Else Lambda-calculus Alonzo Church Recursive Functions Stephen Kleene J. Barkley Rosser 46

105 Everything Else Lambda-calculus Alonzo Church Recursive Functions Programming languages: Stephen Kleene J. Barkley Rosser 46

106 Everything Else Lambda-calculus Alonzo Church Recursive Functions Programming languages: FORTRAN, PASCAL, C, JAVA,... Stephen Kleene J. Barkley Rosser 46

107 Everything Else Lambda-calculus Alonzo Church Recursive Functions Programming languages: FORTRAN, PASCAL, C, JAVA,... Stephen Kleene LISP, SCHEME,... J. Barkley Rosser 46

108 Church-Turing Thesis Alonzo Church Alan Turing 47

109 Church-Turing Thesis 48

110 Paris, 1900 David Hilbert 49

111 Paris, 1900 David Hilbert Speaking on 8 August 1900, at the Paris 2 nd International Congress of Mathematicians, at La Sorbonne, German mathematician David Hilbert presented ten problems in mathematics. 49

112 Paris, 1900 David Hilbert Speaking on 8 August 1900, at the Paris 2 nd International Congress of Mathematicians, at La Sorbonne, German mathematician David Hilbert presented ten problems in mathematics. The problems were all unsolved at the time, and several of them turned out to be very influential for 20 th century mathematics. 49

113 Hilbert s 10 th problem Let P be a polynomial in several variables: P(x,y,z)=24x 2 y 3 +17x+5y+25 Is there a set of integers for x,y,z such that P(x,y,z)=0? This problem is undecidable... but is Turing-Recognizable... Needed a formal model of computing to prove impossibility. Yuri Matiyasevich 50

114 Single variable Poly 51

115 Single variable Poly 51

116 Format & Notations Represent objects as strings O 1, O 2,..., O k is the string representing objects O 1, O 2,..., O k Many encodings are possible. Implicitly, at begining of an algorithm, check that input is in the correct format, otherwise reject. 52

117 Format & Notations 53

118 Format & Notations 54

119 COMP-330 Theory of Computation Fall Prof. Claude Crépeau Lec. 14 : Turing Machines & Church-Turing Thesis 55