COMP-330 Theory of Computation. Fall Prof. Claude Crépeau. Lec. 16 : Turing Machines
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1 COMP-330 Theory of Computation Fall Prof. Claude Crépeau Lec. 16 : Turing Machines
2 COMP 330 Fall 2017: Lectures Schedule 1-2. Introduction 1.5. Some basic mathematics 2-3. Deterministic finite automata +Closure properties, 4. Nondeterministic finite automata 5. Minimization 6. Determinization+Kleene s theorem 7. Regular Expressions+GNFA 8. The pumping lemma 9. Myhill-Nerode theorem 10. Context-Free Grammar 11. PushDown automata 12.CFG - PDA equivalence 13. CFG Pumping lemma and applications 14. MIDTERM Deterministic CFLs Turing Machines and Church-Turing Thesis 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
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
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
5 All languages Computability Theory Decidable Languages Languages. we can describe Context-free Languages Regular Languages NON-decidable via Diagonalization NON-decidable via Reductions
6 Turing MACHINES Alan Turing
7 M1
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!
12 M 1 0 q
13 Turing Machines
14 TM Example
15
16 Definition of TM States q1 q2 q3 a,b,c Tape Alphabet a,b,c,a,b,c,_ Input Alphabet Transition function Start state Accept state Reject state b c,d q1 q2
17 Definition of TM 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 Reject state q2 q2
18 Definition of TM States q2 q1 q3 a,b,c Tape Alphabet a,b,c,a,b,c,_ Input Alphabet Transition function Start state q1 Accept state qacc Reject state qrej b c,d q1 q2
19 TM definition
20
21 TM Configuration
22 TM Computation
23 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. q i bv yields c qj v if δ(qi,b) = qj,c,r
24 TM Computation Start configuration: q0 w (w = input string) Accepting configuration: state = qaccept Rejecting configuration: state = qreject
25 TM Computation Turing Machine M accepts input w if there exists configurations C0, 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 recognized by M, denoted L(M).
26 TM Computation A TM decides a language if it recognizes it and halts (reaches an accepting or rejecting states) on all input strings.
27 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 literature. 2 Often named Recursive in the literature.
28 TM Examples
29 TM Examples
30 TM Computation
31
32 TM Examples
33 \ { #, _ } \ { x, _ }
34 TM Examples
35 TM Examples
36 TM Examples
37 More Turing MACHINES Multitape Turing Machines Non-Deterministic Turing Machines Enumerator Turing Machines Everything else...
38 Multitape TM
39 Multitape TM
40 Multitape TM
41 Multitape TM
42 Non-deterministic TM
43 Non-deterministic TM
44 Non-deterministic TM
45 Enumerator TM
46 Enumerator TM
47 Enumerator TM
48 Enumerator TM
49 Enumerator TM
50 Everything Else Lambda-calculus Alonzo Church Recursive Functions Programming languages: FORTRAN, PASCAL, C, JAVA,... Stephen Kleene LISP, SCHEME,... J. Barkley Rosser
51 Everything Else Lambda-calculus Alonzo Church Recursive Functions Programming languages: FORTRAN, PASCAL, C, JAVA,... Stephen Kleene LISP, SCHEME,... J. Barkley Rosser
52 Everything Else Lambda-calculus Alonzo Church Recursive Functions Programming languages: FORTRAN, PASCAL, C, JAVA,... Stephen Kleene LISP, SCHEME,... J. Barkley Rosser
53 Church-Turing Thesis Alonzo Church Alan Turing
54 Church-Turing Thesis
55 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.
56 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
57 Single variable Poly
58 COMP-330 Theory of Computation Fall Prof. Claude Crépeau Lec. 16 : Turing Machines & Church-Turing Thesis
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