Turing s original paper (1936)

Transcription

1 Turing s original paper (1936)

2 Turing s original paper (1936) There exist problems that cannot be solved mechanically

3 Turing s original paper (1936) There exist problems that cannot be solved mechanically automatic machines

4 Turing s original paper (1936) There exist problems that cannot be solved mechanically automatic machines All mechanical computations

5 Turing s original paper (1936) There exist problems that cannot be solved mechanically automatic machines All mechanical computations ex)

6 Turing s original paper (1936) There exist problems that cannot be solved mechanically automatic machines All mechanical computations ex)

7 Turing s original paper (1936) There exist problems that cannot be solved mechanically automatic machines All mechanical computations ex) addition, copy,

8 Turing s original paper (1936) There exist problems that cannot be solved mechanically automatic machines All mechanical computations ex) addition, copy, universal machine

9 Universal Turing Machine

10 Can we go beyond TMs? Trials to extend the Turing machines: tay-option Multiple tapes Nondeterminism

11 TM with a tay-option (Q,,,,q 0,B,F) : Q! Q {L, R, }

12 TM with a tay-option e.g., (Q,,,,q 0,B,F) : Q! Q {L, R, } (q 0, 0) = (q 1, 1,) B B B q0

13 TM with a tay-option e.g., (Q,,,,q 0,B,F) : Q! Q {L, R, } (q 0, 0) = (q 1, 1,) B B B q1

14 Equivalence A Language is accepted by a TM iff it is accepted by a TM/ Replace (q i,a)=(q j,b,) by (q i,a)=(q k,b,r) (q k,c)=(q j,c,l)

15 Multitape Turing Machines B 1 0 B B 0 1 B q0

16 Multitape Turing Machines B 1 0 B B 0 1 B q0 (Q,,,,q 0,B,F) : Q n! Q n {L, R} n

17 Multitape Turing Machines B 1 0 B B 0 1 B q0 (Q,,,,q 0,B,F) : Q n! Q n {L, R} n! e.g., (q 0, 1, 1) = (q 1, 0, 1,L,R)

18 Multitape Turing Machines B 0 0 B B 0 1 B q1 (Q,,,,q 0,B,F) : Q n! Q n {L, R} n! e.g., (q 0, 1, 1) = (q 1, 0, 1,L,R)

19 Equivalence Any MTM can be simulated by a standard TM with multiple tracks B 0 0 B B * B B B 0 1 B B B * B

20 cf) Efficiency of MTMs MTMs can be more efficient than standard TMs

21 cf) Efficiency of MTMs MTMs can be more efficient than standard TMs the standard TM. Example Design a multitape Turing machine that accepts L = {a n b n n 1}. { } In standard TM, repeated back-and-forth movements are required. In MTM, copy all a s to tape 2 and then match b s on tape 1 against a s on tape 2

22 Nondeterministic TMs (Q,,,,q 0,B,F) Q {L,R} : Q! 2 E.g., (q 0,a)={(q 1,b,R), (q 2,c,L)} ANTMacceptsw if there is a sequence s.t. with q f 2 F. till, equivalent. q 0 w ` x 1 q f x 2

23 cf) Efficiency of NTM The equivalent, deterministic TM is exponentially slower than NTM. Is this exponential slowdown inevitable? Unknown (P = NP?)

24 Turing machines are very powerful.

25 Computable problems are what can be solved by Turing machines Turing

26 Computable problems are what can be solved by Turing machines Turing Computable problems are what can be defined by Lambda calculus Church

27 Computable problems are what can be solved by Turing machines Turing Turing-Church Thesis Computable problems are what can be defined by Lambda calculus Church

28 * proof of the existence of incomputable problems:

29 Halting Problem P X halt / not halt

30 Halting Problem P X halt / not halt Does such H exist?

31 Halting Problem P X halt / not halt Does such H exist? No, logically impossible.

32 uppose such H exists: P X halt / not halt Two simple programs: P P P halt not halt run forever :)

33 Construct the program:

34 Construct the program:

35 Construct the program:

36 Construct the program: halt

37 Construct the program: halt run forever

38 Construct the program: halt not halt run forever

39 Construct the program: halt not halt run forever :)

40 ummary Computable problems are what can be solved by Turing machines There exist incomputable problems