CAPES MI. Turing Machines and decidable problems. Laure Gonnord

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1 CAPES MI Turing Machines and decidable problems Laure Gonnord Université Claude Bernard Lyon1 2017

2 Motivation 1 Motivation 2 Turing Machines 3 Decidability - Complexity Laure Gonnord (UCB Lyon1) CAPES - MI MEEF 2 / 25

3 Motivation Un peu en vrac Informatique fondamentale Formalisation de ce qu est un calcul (mais avant les ordinateurs) Notion d impossibilité, de problème difficile Laure Gonnord (UCB Lyon1) CAPES - MI MEEF 3 / 25

4 Motivation Notion de Modèle de calcul Les classes UML - les BD Les Graphcet Les automates Les machines de Turing Laure Gonnord (UCB Lyon1) CAPES - MI MEEF 4 / 25

5 1 Motivation 2 Turing Machines 3 Decidability - Complexity Laure Gonnord (UCB Lyon1) CAPES - MI MEEF 5 / 25

6 Turing Machine, def - 1 Turing Machine, Turing, 1935 A tape (infinite in both sides) : the memory. (it has a working alphabet which contains a blank symbol). A head : read/write symbols on the tape. A finite transition table (or function) image credits : Wikipedia.org Laure Gonnord (UCB Lyon1) CAPES - MI MEEF 6 / 25

7 Turing Machine, def - 2 Turing Machine elements States (Q), initial state (q 0 ), final (accepting) states (F). One alphabet Γ for the tape. B Γ the blank letter. Transitions are finitely many : read the letter under the head, and then : Erase a symbol or write one under the head Move the head (Left, Right, or Stay) The state can be modified. The transition function is : Q Γ Q Γ {L, R, S} Laure Gonnord (UCB Lyon1) CAPES - MI MEEF 7 / 25

8 An Example TM that decides if x is odd (impair) : (final State : q 4 ) State/char 0 1 B q 1 (q 1,0,R) (q 1,1,R) (q 2,B,L) q 2 (q 3,0,L) (q 4,1,L) Play on BBBBBBB11BBBBB and BBBBBBB10BBBBB A configuration is a tuple (word on the tape, position of the head, state). Adapted from : slide 4 Laure Gonnord (UCB Lyon1) CAPES - MI MEEF 8 / 25

9 Another Example Demo : Laure Gonnord (UCB Lyon1) CAPES - MI MEEF 9 / 25

10 General results 1 There exists Turing machines for the following languages : palindromes a n b n c n (non algebraic language) a i with i prime (non algebraic) a n2, with n 0 Turing machines are more powerful than all other models (we have seen yet) Decidable Languages A language that is recognised by a Turing Machine is said to be decidable. Laure Gonnord (UCB Lyon1) CAPES - MI MEEF 10 / 25

11 Accepting or computing A TM can also compute functions TM that writes 1 if x is odd, 0 else (q 6 is final state) : State/char 0 1 B q 1 (q 1,0,R) (q 1,1,R) (q 2,B,L) q 2 (q 3,B,L) (q 4,B,L) q 3 (q 3,B,L) (q 3,B,L) (q 5,1,R) q 4 (q 4,B,L) (q 4,B,L) (q 5,0,R) q 5 (q 6,B,R) Play on BBBBBBB11BBBBB and BBBBBBB10BBBBB Laure Gonnord (UCB Lyon1) CAPES - MI MEEF 11 / 25

12 Another Example Demo of a TM computing the addition. Laure Gonnord (UCB Lyon1) CAPES - MI MEEF 12 / 25

13 General results 2 - extensions infinite tape in both directions more than one tape many heads per tape bidimensional tape non determinism random access All these extensions can be simulated by a standard machine. Laure Gonnord (UCB Lyon1) CAPES - MI MEEF 13 / 25

14 Example : Infinite tape (copyright S.Brandel) Laure Gonnord (UCB Lyon1) CAPES - MI MEEF 14 / 25

15 Example : Infinite tape Laure Gonnord (UCB Lyon1) CAPES - MI MEEF 15 / 25

16 Example : Infinite tape Laure Gonnord (UCB Lyon1) CAPES - MI MEEF 16 / 25

17 Example : Infinite tape Laure Gonnord (UCB Lyon1) CAPES - MI MEEF 17 / 25

18 General results 3 There exists Turing machines for the computation of : x x + 1 (x, y) x + y x x 2 all the functions you are able to write on computers Computable functions A function (defined for all its input) that is computable by a Turing Machine is said to be TM computable A Model of what can be computed with machines. (Church Thesis - see later) Laure Gonnord (UCB Lyon1) CAPES - MI MEEF 18 / 25

19 Important result 1 - Universal MT Laure Gonnord (UCB Lyon1) CAPES - MI MEEF 19 / 25

20 Church Laure Gonnord (UCB Lyon1) CAPES - MI MEEF 20 / 25

21 Decidability - Complexity 1 Motivation 2 Turing Machines 3 Decidability - Complexity Laure Gonnord (UCB Lyon1) CAPES - MI MEEF 21 / 25

22 Decidability - Complexity Decidable - Semi-decidable languages If L is a language, L is decidable if there exists a Turing Machine (or an algorithm) that outputs for all w : 1 if w L else 0. Semi-decidable : 1 if w L else does not terminate equivalent definition for problems. Laure Gonnord (UCB Lyon1) CAPES - MI MEEF 22 / 25

23 Decidability - Complexity Complexity Link with computational complexity : The number of steps in the execution of a TM gives the complexity in time, The number of seen squares in the execution of a TM gives the complexity in space. Laure Gonnord (UCB Lyon1) CAPES - MI MEEF 23 / 25

24 Decidability - Complexity Examples of undecidable problems The halting problem for TM or counter automata (for more that 2 counters) Given a program, does it loops? Is a given algebraic expression (with log, *, exp, sin, abs) equal to 0? (Richardson, 1968) The 10 th Hilbert Problem (Diophantine equations) (23 non solved pbs 2017). Garbage Collecting Absence of bugs Laure Gonnord (UCB Lyon1) CAPES - MI MEEF 24 / 25

25 Conclusion Summary Turing Machines : A model closed to programming languages. Non deterministic. Acceptors for decidable languages. But some languages are not decidable! (equivalently) Computes solutions to problems. But... Important fact Some common problems are undecidable! Laure Gonnord (UCB Lyon1) CAPES - MI MEEF 25 / 25

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