Turing Machines

Wen-Guey Tzeng

Department of Computer Science

National Chiao Tung University

1

Model

2

Definition

• A Turing machine M is defined by M=(Q, , , , q0, , F) – Q: the set of internal states

– : the input alphabet

– : the tape alphabet

– : QQ{L,R}, the transition function

– : blank symbol

– q0Q: the initial state

– FQ: the set of final states

3

Transition function

4

• (q0, a)=(q1, d, R)

Example

5

• M=({q0, q1}, {a,b}, {a, b, }, , q0, , {q1}) – (q0, a)=(q0, b, R)

– (q0, b)=(q0, b, R)

– (q0, )=(q1, , L)

Transition graph

6

– (q0, a)=(q0, b, R)

– (q0, b)=(q0, b, R)

– (q0, )=(q1, , L)

TM runs forever

7

• Check TM on input aab

TM halts

8

• A TM “halts” if it reaches a configuration that is not defined.

– A TM on an input w may not halt. (Run forever)

– The halting of a TM on some input w does not mean w is accepted.

Standard TM

• The tape is unbounded in both directions.

• All other cells are filled with blanks.

• It is deterministic.

• The input is on the tape in the beginning.

• No output devices. But, the left string on the tape can be viewed as output.

9

Instantaneous description (configuration)

• x1qx2=a1a2…ak-1 q akak+1…an

10

Example

11

• M=({q0, q1}, {a,b}, {a, b, }, , q0, , {q1}) – (q0, a)=(q0, b, R)

– (q0, b)=(q0, b, R)

– (q0, )=(q1, , L)

• ID: q0aa, bq0a, bbq0, bq1b

Move

• q0aa ├ bq1b ├ bbq1├ bq1b (halt)

12

A computation

• A sequence of configurations leading to a halt state.

– q0aa ├ bq1b ├ bbq1├ bq1b (halt)

• x1qxq ├ (inf)

– The configuration leads to no halting states

13

TM as language acceptors

• The language accepted by M L(M)={w+: q0w├* x1qfx2, qfF, x1, x2* }

• A language accepted by some TM is “recursively enumerable”.

14

Example

• Design a TM to accept the strings of form 00*

• The transition function

– (q0, a)=(q0, 0, R)

– (q0, )=(q1, , R)

15

• Design a TM to accept L={anbn : n1}

• Idea:

– Loop

• Mark leftmost ‘a’ as ‘x’ in the tape

• Move all the way right to mark ‘b’ as ‘y’

– After marking all a’s, move all the way right to find

16

• Mark leftmost ‘a’ as ‘x’ in the tape

– (q0, a)=(q1, x, R)

• Move all the way right to mark ‘b’ as ‘y’

– (q1, a)=(q1, a, R)

– (q1, y)=(q1, y, R)

– (q1, b)=(q2, y, L)

17

• Move back to mark leftmost ‘a’ as ‘x’ – (q2, y)=(q2, y, L)

– (q2, a)=(q2, a, L)

– (q2, x)=(q0, x, R)

• After marking all a’s, move all the way right to find – (q0, y)=(q3, y, R)

– (q3, y)=(q3, y, R)

– (q3, )=(q4, , R) – If in the middle ‘b’ is encountered, TM stops.

18

• Run q0aabb

19

TM as transducers

• To compute a function f(w)=w’, where w, w’ are strings

– q0w├* qf w’

• A function f:DR is TM-computable if there is a TM M=(Q, , , , q0, , F) such that q0w├* qf w’ for all wD

20

Example

• Given two positive integer x and y, compute x+y

– x and y are unary-represented

– Eg. X=3, w(x)=111

– Input: w(x)0w(y)

– Output: w(x+y)0

– That is, q0w(x)0w(y)├qfw(x+y)0

21

• The transition function

• (q0, 1)=(q0, 1, R)

• (q0, 0)=(q1, 1, R)

• (q1, 1)=(q1, 1, R)

• (q1, )=(q2, , R)

• (q2, 1)=(q3, 0, R)

• (q3, 1)=(q3, 1, L)

• (q3, )=(q4, , R)

• Run q0111011

22

Example

• Design a TM to compute q0w├* qfww, w=1+

• Steps

1. Replace every 1 by x

2. Find the rightmost x and replace it with 1

3. Travel to the right end and create 1 (in replace of )

4. Repeat 2 and 3 until there are no x’s

23

24

Run q011

Example

• Design a TM to compute q0w(x)0w(y)├* qyw(x)0w(y) if xy q0w(x)0w(y)├* qNw(x)0w(y) if x<y

• Design a TM to compute q0w(x)0w(y)├* qfw(xy)

25

Combining TM for complicated tasks

• Design a TM to compute f(x,y)=x+y if xy f(x,y)=0 if x<y

26

• qC,0w(x)0w(y) ├* qA,0w(x)0w(y) if xy ├* qE,0w(x)0w(y) if x<y

• qA,0w(x)0w(y) ├* qA,fw(x+y)0

• qE,0w(x)0w(y) ├* qE,f 0

27

Example

• if a then qj else qk

– (qi, a)=(qj0, a, R), for qiQ

– (qj0, c)=(qj, c, L), for all c

– (qi, b)=(qk0, b, R) for b-{a}

– (qk0, c)=(qk, c, L), for all c

28

Example

• Design a TM for q0w(x)0w(y)├* qfw(xy)

• Steps

– Loop until x contains no more 1’s

• Find a 1 in x and replace it with another a

• Replace leftmost 0 by 0y

– Replace all a’s with 1’s

29

30

Macroinstructionssubprogram

• A calls B

31

Turing’s Thesis

• Mechanical computation

– Computation without intervention of humans

• Turing Thesis: any computation that can be carried out by mechanical means can be performed by some Turing machine.

• Some call it “Church-Turing Thesis”

32

• Reasoning:

– Anything that can be done by an existing digital computer can be done by a TM

– No one has yet been able to suggest a problem, solvable by algorithms, that cannot be solved by some TM

– Other models are proposed. But, they are no more powerful than TM’s

33

Algorithms

• An algorithm for a functin f:DR is a Turing machine such that q0d├* qff(d) for all dD

34