Turing Machines Wen-Guey Tzeng Computer Science Department - PowerPoint PPT Presentation

Turing Machines Wen-Guey Tzeng Computer Science Department National Chiao Tung University

Alan Turing • One of the first to conceive a machine that can run computation mechanically without human intervention. We call “ algorithmic computation ” now. • The first to conceive a universal 1912-1954 machine – that can run programs • Break German Enigma Machine (encryption) in WWII • Turing Award: the most prestigious award in CS • Bio movie: The Imitation Game, 2015 2017 Spring 2

TM Model 2017 Spring 3

Definition • A Turing machine (TM) M is defined by M=(Q,  ,  ,  , q 0 , ฀ , F) – Q: the set of internal states –  : the input alphabet –  : the tape alphabet –  : Q   Q  {L,R}, the transition function – ฀ : blank symbol – q 0  Q: the initial state – F  Q: the set of final states • Note: this TM is deterministic. 2017 Spring 4

Transition function  •  (q 0 , a)=(q 1 , d, R) 2017 Spring 5

Example: • M=({q 0 , q 1 }, {a,b}, {a, b, ฀ },  , q 0 , ฀ , {q 1 }) –  (q 0 , a)=(q 0 , b, R) –  (q 0 , b)=(q 0 , b, R) –  (q 0 , ฀ )=(q 1 , ฀ , L) 2017 Spring 6

Transition graph –  (q 0 , a)=(q 0 , b, R) –  (q 0 , b)=(q 0 , b, R) –  (q 0 , ฀ )=(q 1 , ฀ , L) 2017 Spring 7

TM runs forever • Check TM on input aab 2017 Spring 8

Instantaneous description • x 1 qx 2 = a 1 a 2 …a k-1 q a k a k+1 …a n • Also, called “configuration” 2017 Spring 9

Example: • M=({q 0 , q 1 }, {a,b}, {a, b, ฀ },  , q 0 , ฀ , {q 1 }) –  (q 0 , a)=(q 0 , b, R) –  (q 0 , b)=(q 0 , b, R) –  (q 0 , ฀ )=(q 1 , ฀ , L) • ID: q 0 aa, bq 0 a , bbq 0 ฀ , bq 1 b • Move: – q 0 aa ├ bq 0 a ├ bbq 0 ฀ ├ bq 1 b (halt) 2017 Spring 10

TM halts • A TM on input w “halts” if q 0 w reaches a configuration that  is not defined. – A TM on an input w may not halt. (Run forever) • Halting configuration: x 1 qx 2 (halt) – Halting does not imply acceptance • Accepting configuration: x 1 q f x 2 ( halt ) – Must halt and enter a final state q f – Acceptance implies halting • Infinite loop (non-stop): x 1 qx 2 ├ *  2017 Spring 11

A computation • A computation is a sequence of configurations that leads to a halt state. • q 0 aa ├ bq 1 b ├ bbq 1 ฀ ├ bq 1 b (halt) 2017 Spring 12

Standard TM • It is deterministic. • There is only one tape. • The tape is unbounded in both directions. • All other cells are filled with blanks. • No output devices. • The input is on the tape in the beginning. • The left string on the tape can be viewed as output. 2017 Spring 13

TM as • Language acceptors • Transducers 2017 Spring 14

TM as language acceptors • A string w is accepted by M if – M on input w enters a final state and halts. – Note: no need to read the entire input • The language accepted by M L(M)={w  + : q 0 w ├ * x 1 q f x 2 (halt), q f  F, x 1 , x 2  * } – x 1 q f x 2 (halt) is a halt configuration. • A language accepted by some TM is called a “ recursively enumerable (r.e.) ” set. 2017 Spring 15

Example: • Design a TM to accept the strings of form 00* • The transition function –  (q 0 , 0)=(q 1 , 0, R) –  (q 1 , 0)=(q 1 , 0, R) –  (q 1 , ฀ )=(q 2 , ฀ , R) • F={q 2 } 2017 Spring 16

Example: • Design a TM to accept L={a n b n : n  1} • 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 ฀ 2017 Spring 17

• Example, w=aabb • Mark leftmost ‘a’ as ‘x’ in the tape –  (q 0 , a)=(q 1 , x, R) • Move all the way right to mark ‘b’ as ‘y’ –  (q 1 , a)=(q 1 , a, R) –  (q 1 , y)=(q 1 , y, R) –  (q 1 , b)=(q 2 , y, L) ฀ ฀ . . . a a b b . . . 2017 Spring 18

• Move back to find leftmost ‘a’ –  (q 2 , y)=(q 2 , y, L) –  (q 2 , a)=(q 2 , a, L) –  (q 2 , x)=(q 0 , x, R) • Repeat until all a’s are marked as x’s ฀ ฀ . . . x a y b . . . 2017 Spring 19

• After marking all a’s as x’s, move all the way right to find ฀ –  (q 0 , y)=(q 3 , y, R) –  (q 3 , y)=(q 3 , y, R) –  (q 3 , ฀ )=(q 4 , ฀ , R) – If ‘b’ is encountered during the action of moving right, TM halts on a non-final state. ฀ ฀ . . . x x y y . . . 2017 Spring 20

Example: • Design a TM to accept L={w  {a,b} + : n a (w)=n b (w)} • Idea: – Loop • Pose at the beginning of the string • Mark leftmost ‘a’ as ‘x’ in the tape • Move all the way right to find the first b and mark it as – After marking all symbols as X, enter the final state 2017 Spring 21

TM as transducers • To compute a function f(w)=w’, where w, w’ are strings – q 0 w ├ * q f w’ (halt) • A function f:D  R is TM-computable if there is a TM M=(Q,  ,  ,  , q 0 , ฀ , F) such that q 0 w ├ * q f f(w) (halt) for all w  D • We will show in the following that TM can compute complicated functions 2017 Spring 22

Example: (integer adder) • 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) 0 w(y) – Output: w(x+y) 0 – That is, q 0 w(x)0w(y) ├ q f w(x+y)0 2017 Spring 23

• The transition function •  (q 0 , 1)=(q 0 , 1, R) •  (q 0 , 0)=(q 1 , 1, R) •  (q 1 , 1)=(q 1 , 1, R) •  (q 1 , ฀ )=(q 2 , ฀ , L) •  (q 2 , 1)=(q 3 , 0, L) •  (q 3 , 1)=(q 3 , 1, L) •  (q 3 , ฀ )=(q 4 , ฀ , R) • Run q 0 111011 2017 Spring 24

Example: (string copier) • Design a TM to compute q 0 w ├ * q f ww, 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 2017 Spring 25

Run q 0 11 2017 Spring 26

Examples: (comparer) • Design a TM to compute q 0 w(x)0w(y) ├ * q Y w(x)0w(y) if x  y q 0 w(x)0w(y) ├ * q N w(x)0w(y) if x<y 2017 Spring 27

Combining TM for complicated tasks • Design a TM to compute f(x,y)=x+y if x  y, and f(x,y)=0 if x<y 2017 Spring 28

• q C,0 w(x)0w(y) ├ * q A,0 w(x)0w(y) if x  y ├ * q E,0 w(x)0w(y) if x<y • q A,0 w(x)0w(y) ├ * q A,f w(x+y)0 • q E,0 w(x)0w(y) ├ * q E,f 0 2017 Spring 29

Example: (if-then-else) • if a then q j else q k –  (q i , a)=(q j0 , a, R), for q i  Q –  (q j0 , c)=(q j , c, L), for all c  –  (q i , b)=(q k0 , b, R) for b  -{a} –  (q k0 , c)=(q k , c, L), for all c  2017 Spring 30

Example: (integer multiplier) • Design a TM for q 0 w(x)0w(y) ├ * q f w(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 2017 Spring 31

2017 Spring 32

Macroinstructions  subprogram • A calls B 2017 Spring 33

Turing’s Thesis • Mechanical computation – Computation without human intervention • Turing Thesis: any computation that can be carried out by mechanical means can be carried out by some Turing machine. • Also called “Church-Turing Thesis” 2017 Spring 34

• 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, but cannot be solved by some TM – Other models are proposed. But, they are no more powerful than TM’s. • Even quantum computers are no more powerful than TM’s in terms of the power language acceptance 2017 Spring 35

Algorithms • An algorithm for a functin f:D  R is a TM such that q 0 d ├ * q f f(d) for all d  D 2017 Spring 36