bfe1emodule4turing machine

8/4/2019 Bfe1emodule4turing Machine

1/47

1

Turing Machines

Zeph Grunschlag


2/47

2

A

gendaTuring Machines Alan Turing

Motivation

Church-Turing Thesis

Definitions

Computation

TM Configuration

Recognizers vs. Deciders


3/47

3

A

lan TuringAlan Turing was one of the founding fathers of

CS.

His computer model the Turing Machinewas inspiration/premonition of the electroniccomputer that came two decades later

Was instrumental in cracking the Nazi

Enigma cryptosystem in WWIIInvented the Turing Test used in AI

Legacy: The Turing Award. Pre-eminentaward in Theoretical CS


4/47

4

A

Thinking MachineFirst Goal of Turings Machine: A model that

can compute anything that a human can

compute. Before invention of electroniccomputers the term computer actuallyreferred to a person whos line of work is tocalculate numerical quantities!

As this is a philosophical endeavor, it cantreally be proved.

Turings Thesis: Any algorithm can be carriedout by one of his machines


5/47

5

A

Thinking MachineSecond Goal of Turings Machine: A model

thats so simple, that can actually prove

interesting epistemological results. EyedHilberts 10th problem, as well as acomputational analog of GdelsIncompleteness Theorem in Logic.

Philosophy notwithstanding, Turings programsfor cracking the Enigma cryptosystem provethat he really was a true hacker! Turingsmachine is actually easily programmable, ifyou really get into it. Not practically useful,

though


6/47

6

A

Thinking MachineImagine a super-organized, obsessive-

compulsive human computer. The computer

wants to avoid mistakes so everything writtendown is completely specified oneletter/number at a time. The computerfollows a finite set of rules which are referredto every time another symbol is written

down. Rules are such that at any given time,only one rule is active so no ambiguity canarise. Each rule activates another ruledepending on what letter/number is currently

read, EG:


7/47

7

A Thinking MachineEG Successor Program

Sample Rules:

If read 1, write 0, go right, repeat.

If read 0, write 1, HALT!

If read , write 1, HALT!

Lets see how they are carried out on apiece of paper that contains thereverse binary representation of47:


8/47

8





1 1 1 1 0 1


9/47

9





0 1 1 1 0 1


10/47

10





0 0 1 1 0 1


11/47

11





0 0 0 1 0 1


12/47

12





0 0 0 0 0 1


13/47

13





0 0 0 0 1 1


14/47

14


So the successors output on 111101 was000011 which is the reverse binaryrepresentation of48.

Similarly, the successor of 127 should be128:


15/47

15





1 1 1 1 1 1 1


16/47

16





0 1 1 1 1 1 1


17/47

17





0 0 1 1 1 1 1


18/47

18





0 0 0 1 1 1 1


19/47

19





0 0 0 0 1 1 1


20/47

20





0 0 0 0 0 1 1


21/47

21





0 0 0 0 0 0 1


22/47

22





0 0 0 0 0 0 0


23/47

23





0 0 0 0 0 0 0 1


24/47

24

A Thinking Machine

It was hard for the ancients to believe that anyalgorithm could be carried out on such a

device. For us, its much easier to believe,especially if youve programmed in assembly!

However, ancients did finally believe Turingwhen Churchs lambda-calculus paradigm (on

which lisp programming is based) provedequivalent!


25/47

25

Turing Machines

A Turing Machine (TM) is a device with a finiteamount of read-only hard memory (states),and an unbounded1 amount of read/writetape-memory. There is no separate input.Rather, the input is assumed to reside on thetape at the time when the TM starts running.

Just as with Automata, TMs can either beinput/output machines (compare with FiniteState Transducers), or yes/no decisionmachines. Start with yes/no machines.


26/47

26

Comparison with PreviousModels

DeviceSeparateInput?

Read/Write DataStructure

Deterministicby default?

FA

PDA

TM


27/47

27





FA Yes None Yes

PDA

TM


28/47

28





FA Yes None Yes

PDA Yes LIFO Stack No

TM


29/47

29





FA Yes None Yes

PDA Yes LIFO Stack No

TM No1-way infinitetape. 1 cell

access per step.

Yes

(but will alsoallow crashes)


30/47

30

Turing MachineDecision Machine Example

First example (adding 1 bit to reverse binarystring) was basically something that a Finite

Transducer could have achieved (exceptwhen theres overflow). Lets give anexample from next step up in languagehierarchy.

{bit-strings with same number of 0s as 1s}a context free language:


31/47

31


This is a true Turing machine as:

Tape is semi-infinite (indicated by torn cell):

Input is prepared at beginning of tape

No intrinsic way to detect left tape end similar to empty stack detection problem for PDAs

similar trick used introduce $ as the end symbolAll rules must include a move direction (R/L)

Situations that cant happen arent dealt with(technically under-deterministic)


32/47

32


{bit-strings with same number of 0s as 1s}:

Pseudocode:

while (there is a 0 and a 1)

cross these out

if (everything crossed out)

accept

else

reject


33/47

33

TM ExampleInstructions Set

0. if read , go right (dummy move), ACCEPT

if read 0, write $, go right, goto 1 // $ detects start of tape

if read 1, write $, go right, goto 2

1. if read , go right, REJECT

if read 0 or X, go right, repeat (= goto 1) // look for a 1if read 1, write X, go left, goto 3

2. if read , go right, REJECT

if read 1 or X, go right, repeat // look for a 0

if read 0, write X, go left, goto 3

3. if read $, go right, goto 4 // look for start of tapeelse, go left, repeat

4. if read 0, write X, go right, goto 1 // similar to step 0

if read 1, write X, go right, goto 2

if read X, go right, repeat

if read , go right, ACCEPT


34/47

34

TM ExampleState Diagram

These instructions can be expressed by a familiarlooking flow diagram:

0

1

rej

0$,R

acc

R

21$,R

0|XR

1|XR

3

R

0X,L

1X,L 0|1|XL

4

$R

XR

0X,R

1X,R

R


35/47

35

TM Transition NotationAn edge from the state p to the state q labeled

by

ab,D means if in state p and tape head

reading a, replace a by b and move in thedirection D, and into state q

aD means if in state p and tape headreading a, dont change a and move in the

direction D, and into state qa|b||z means that given that thetape head is reading any of the pipeseparated symbols, take same action onany of the symbols


36/47

36

TM Configuration NotationA TMs next action is completely determined by

current state and symbol read, so canpredict all of future actions if know:

1. current state2. current tape contents3. current position of TMs reading headHandy notation lists all of these in a single

string. A symbol representing current state,is sandwiched between content of tape toleft of head, and content of tape to right(including tape head). The part of tape

which is blank ad-infinitum is ignored.


37/47

37

TM Configuration NotationFor example

Is denoted by:$xxx1q3010

Reading rule 3


38/47

38

TM ExampleCrazy Web-Page

The following link shows how theexample machine accepts 01101010

and how the tape configuration notationchanges step by step.


39/47

39

TM Formal DefinitionStatic Picture

DEF: ATuring machine (TM) consists ofa 7-tuple M = (Q, 7, +, H, q0, qacc, qrej). Q, 7,and q0, are the same as for an FA. +is the

tape alphabet which necessarily containsthe blank symbol , as well as the inputalphabet7. H is as follows:

Therefore given a non-halt state p, and a tapesymbol x, Hp,x= (q,y,D) means that TM goesinto state q, replaces x by y, and the tapehead moves in direction D.

}RL,{}),{-(: rejacc v+vp+v QQ qq


40/47

40

TM Dynamic Picture

A string x is acceptedby M if after beingput on the tape with the Turing

machine head set to the left-mostposition, and letting M run, Meventually enters the accept state. In

this case w is an element of L(M) thelanguage accepted by M.We can formalize this notion as follows:


41/47

41

TM Formal DefinitionDynamic Picture

Suppose TMs configuration at time t is given by uapxvwhere p is the current state, ua is whats to the left ofthe head, x is whats being read, and v is whats to theright of the head.

IfHp,x= (q,y,R) then write:uapxv uaypv

With resulting configuration uaypv at time t+1. If,Hp,x= (q,y,L) instead, then write:

uapxv upayvThere are also two special cases: head is forging new ground pad with the blank symbol

head is stuck at left end by def. head stays put (only case)

is read as yields


42/47

42

TM Formal DefinitionDynamic Picture

As with context free grammars, one can consider thereflexive, transitive closure * of . I.e. this isthe relation between strings recursively defined by:

if u = v then u * v

if u v then u * vif u *v and v * w, then u *w

* is read as computesto

A string x is said to be accepted by M if the start

configuration q0 x computes to some acceptingconfiguration y i.e., a configuration containing qacc.

The language accepted byM is the set of all acceptedstrings. I.e:

L(M) = { x 7* | accepting config. y, q0 x * y}


43/47

43

TM Acceptors vs. DecidersThree possibilities occur on a given input w :1. The TM M eventually enters qacc and

therefore halts and accepts. (w L(M) )2. The TM M eventually enters q

rej

or crashessomewhere. M rejectsw . (w L(M) )

3. Neither occurs! I.e., M never halts itscomputation and is caught up in an infiniteloop, never reaching qacc or qrej. In this case

w is neither accepted nor rejected. However,any string not explicitly accepted isconsidered to be outside the acceptedlanguage. (w L(M) )


44/47

44

TM Acceptors vs. DecidersAny Turing Machines is said to be a recognizerand

recognizesL(M); if in addition, M never enters aninfinite loop, M is called a deciderand is said todecide L(M).

Q: Is the above M an recognizer? A decider? What isL(M)?

0

1

rej acc

2

R

1R

0R

1R 0R

0R1L


45/47

45

TM Acceptors vs. DecidersA: M is an recognizer but not a decider because 101

causes an infinite loop.L(M) = 1+0+

Q: Is L(M ) decidable ?

0

1

rej acc

2

R

1R

0R

1R 0R

0R1L


46/47

46

TM Acceptors vs. DecidersA: Yes. All regular languages are decidable

because can always convert a DFA into aTM without infinite loops.


47/47

47

Constructive Example

Heres a document showing how modulardesign can help you write down a TM

decider for {anbncn}. The example isnon-context free.

bfe1emodule4turing machine

Documents