( r ; X ) = ([ q ; X ] ; X ; R ) for all X in . In - - PDF document

SLIDE 1 Stupid T uring Mac hine T ric ks Often it is useful to think

• f

the state and the tap e sym b

• l

as ha ving structure.

• The

comp

• nen

ts

• f

a tap e sym b

• l

are tr acks.

• Usually
• ne

comp

• nen

t

• f

the state is the c

• ntr
• l,

resp

• nsible

for running the \program"

• f

the TM;

• ther

comp

• nen

ts hold data. Example Let M = (Q; ; ;

• ;

q ; B ; F ). Supp

• se

the program

• f

M needs to sw ap the con ten ts

• f

adjacen t tap e cells sometimes.

• Some
• f

the states in Q will b e

• f

the form [q ; X ] and [p; X ], where X is an y sym b

• l

in . W e also need states r to b egin and s to end.

• In

state r , M will pic k up the sym b

• l

scanned in to the data p

• rtion
• f

the state.

✦

• (r

; X ) = ([q ; X ]; X ; R) for all X in .

• In

con trol state q , M dep

• sits

the sym b

• l

in its data comp

• nen

t and pic ks up the sym b

• l

that w as there, going to con trol state p.

✦

• ([q

; X ]; Y ) = ([p; Y ]; X ; L) for all X and Y in .

• In

con trol state p, M dep

• sits

its data and en ters state s, mo ving righ t.

✦

• ([p;

X ]; Y ) = (s; X ; R) for all X and Y in . Example: Multiple T rac ks A common use for m ultiple trac ks is to use

• ne

trac k for data the

• ther

for a single \mark."

• Sym

b

• ls
• f
• are

pairs [A; X ], where X is the \real" sym b

• l,

and A is either B (blank)

• r

.

✦

Input sym b

• l

a is iden tied with [B ; a].

✦

The blank is [B ; B ].

• Here's

a program to nd the , assuming it is somewhere to the left

• f

the presen t p

• sition.

1.

• (q

; [B ; X ]) = (q ; [B ; X ]; L) 2.

• (q

; [; X ]) = (p; [B ; X ]; R) Other TM Mo dels While regular

• r

CF languages are classes

• f

languages that w e dened b y con v enien t notations (RE's, CF G's, etc.), no

• ne

supp

• sed

that they represen ted \ev erything w e can compute." 1

SLIDE 2

• The

purp

• se
• f

the TM w as to dene \ev erything w e can compute."

✦

F

• r

con v enience, w e use recognition

• f

languages as the space

• f

p

• ssibly

computable things;

• ther

spaces, e.g., computing arithmetic functions, yield the same conclusions.

• Th

us, it w

• uld

b e a wkw ard if w e could nd another notion

• f

\ev erything" that w as dieren t from the TM.

✦

A real computer is an imp

• rtan

t sp ecial case. Do real computers and TM's dene the same set

• f

computable things?

• Our

next steps are to consider p

• ten

tially more p

• w

erful notions

• f

computing and see that the TM mo del w e dened can sim ulate them.

✦

These mo dels are: m ultitap e TM, nondeterministic TM, m ultistac k mac hines, coun ter mac hines, \real" computers. Multitap e TM's Allo w the TM to ha v e some nite n um b er

• f

tap es k , with a head for eac h tap e.

• Mo

v e is a function

• f

the state and the sym b

• l

scanned b y eac h tap e head.

• Action

= new state, new sym b

• l

for eac h tap e, and a head motion (L, R,

• r

S, for \stationary").

• First

tap e holds the input,

• ther

tap es are initially blank. Man y T ap es to One T ap e Sim ulati

• n

T

• sim

ulate k tap es, use

• ne

tap e with 2k trac ks.

• One

trac k holds the con ten ts

• f

eac h tap e.

• Another

trac k holds a mark represen ting the head p

• sition
• f

that tap e, as

• W

X Y Z

• T
• sim

ulate

• ne

mo v e

• f

the m ultitap e TM, the

• ne-tap

e TM m ust remem b er ho w man y *'s are to its left. 2

SLIDE 3 1. Mo v e left, then righ t, visiting all the *'s to see what eac h tap e head is scanning. 2. Decide

• n

the m ultitap e TM's mo v e, based

• n

the scanned sym b

• ls

and its state (remem b ered in the state

• f

the

• ne-

tap e TM). 3. Visit eac h * again, making the necessary adjustmen ts: c hange sym b

• ls

and mo v e *'s

• ne

cell left

• r

righ t, as needed.

• Imp
• rtan

t

• bserv

ation for when w e study p

• lynomial

time TM's: If the m ultitap e TM mak es T (n) mo v es when the input is

• f

length n, then the

• ne-tap

e TM mak es O

• T

2 (n)

• mo

v es.

✦

Th us, if the m ultitap e TM tak es p

• lynomial

time, so do es the

• ne-tap

e TM.

✦

Key p

• in

t in pro

• f:

The *'s can't get more than T (n) cells apart, so

• ne

mo v e is sim ulated in O

• T

(n)

• mo

v es

• f

the

• ne-

tap e TM. Nondetermini st ic TM Let the TM ha v e a nite set

• f

c hoices

• f

mo v e.

• As

with the (nondeterministic) PD A, there is no \mix-and-m a tc h"; if (p; X ; L) and (q ; Y ; R) are c hoices, w e cannot go to state q , prin t X

• n

the cell and mo v e righ t, e.g., unless (q ; X ; R) is another c hoice. Nondetermini st ic to Determini sti c Sim ulati

• n

Let the NTM ha v e

• ne

tap e, but rst sim ulate with a m ultitap e DTM; later con v ert the m ultitap e DTM to a

• ne-tap

e DTM.

• Use
• ne

tap e

• f

DTM to hold a queue

• f

ID's

• f

the NTM, separated b y sp ecial mark ers (*).

• When

an ID reac hes the fron t

• f

the queue, nd all its next ID's, and add them to the bac k

• f

the queue.

• Accept

if y

• u

ev er reac h an ID with an accepting state.

✦

Note that the queue discipline is imp

• rtan

t, so that the DTM ev en tually reac hes ev ery ID that the NTM can en ter.

✦

In con trast, if w e used a stac k discipline, the NTM migh t reac h acceptance, but the 3

SLIDE 4 DTM w

• uld

go

• n

some innite c hase

• f

ID's and nev er reac h the accepting ID

• f

the NTM. MultiStac k Mac hines Lik e PD A, but with more than

• ne

stac k.

• One

stac k is not enough to sim ulate a TM; y

• u

get

• nly

CFL's.

✦

While w e ha v en't emphasized the p

• in

t, all the examples in Section 7.2.3

• f

non- CFL's are recognized b y TM's.

• But

2 stac ks is enough!

• Key

idea: use

• ne

stac k to hold what is to the left

• f

the tap e head, use the

• ther

to hold what is to the righ t. Coun ter Mac hines Tw

• equiv

alen t w a ys to think

• f

a coun ter: 1. A stac k with a b

• ttom-mark

er, sa y Z , and

• ne
• ther

sym b

• l,

sa y X , that can b e placed

• n

the stac k.

✦

Th us, stac k alw a ys lo

• ks

lik e X X

• X

Z . 2. A device that holds a nonnegativ e in teger, with the

• p

erations add 1, subtract 1, and test-if-0.

• 1

coun ter = subset

• f

CFL's, including all regular languages and some nonregular languages lik e f0 n 1 n j n

• 1g.
• 2

coun ters = TM!

✦

Pro

• f

in t w

• stages:

3 coun ters sim ulate 2 stac ks, then 2 coun ters sim ulate 3 coun ters. 2 Stac ks to 3 Coun ters Supp

• se

a stac k has r

• 1

sym b

• ls.

Think

• f

the stac k con ten ts as a base-r n um b er, with the sym b

• ls

as digits 1 through r

• 1.
• Use
• ne

coun ter for eac h stac k, plus

• ne

\scratc h" coun ter.

• Multiply

and divide b y r using t w

• coun

ters.

✦

Subtract r from

• ne,

add 1 to the

• ther,
• r

vice-v ersa.

• Push

X = m ultiply b y r , then add digit represen ted b y X . 4

SLIDE 5

• P
• p

= divide b y r , thro w a w a y the remainder.

• Read

top sym b

• l

= mo v e from

• ne

coun ter to the

• ther,

coun ting in the state mo dulo r to determine the remainder. 3 Coun ters to 2 Coun ters Key idea, represen t coun ters i, j , and k b y the in teger 2 i 3 j 5 k .

• Store

this n um b er

• n
• ne

coun ter, use the

• ther

coun ter as scratc h.

• T

est if i = b y mo ving coun t from

• ne

coun ter to the

• ther,

coun ting mo dulo 2 in the state.

✦

i = if and

• nly

if the n um b er is not divisible b y 2.

• T

ests for j = and k = analogous.

• Adding

to i, j , k are m ultiplicati

• ns
• f

the coun t b y 2, 3, 5, resp ectiv ely .

• Subtractions

are similarly divisions. Real Computers In

• ne

sense, a real computer has a nite n um b er

• f

states, and th us is we aker than a TM.

• W

e ha v e to p

• stulate

an innite supply

• f

tap es, disks,

• r

some p eriferal storage device to sim ulate an innite TM tap e.

• Assume

h uman

• p

erator to moun t disks, k eep them stac k ed neatly

• n

the sides

• f

the computer. TM to Real Computer Computer can sim ulate nite con trol, and moun t

• ne

disk that holds the region

• f

the TM tap e around the tap e head.

• When

the tap e head mo v es

• this

region, the computer prin ts an

• rder

to ha v e its disk mo v ed to the top

• f

the left

• r

righ t pile, and the top

• f

the

• ther

pile moun ted. Real Computer to TM Sim ulation is at the lev el

• f

stored instructions and w

• rds
• f

memory .

• TM

has

• ne

tap e that holds all the used memory lo cations, and their con ten ts. 5

SLIDE 6

• Other

TM tap es hold the instruction coun ter, memory address, computer input le, and scratc h.

• Instruction

cycle

• f

computer sim ulated b y: 1. Find the w

• rd

indicated b y the instruction coun ter

• n

the memory tap e. 2. Examine the instruction co de (a nite set

• f
• ptions),

and get the con ten ts

• f

an y memory w

• rds

men tioned in the instruction, using the scratc h tap e. 3. P erform the instruction, c hanging an y w

• rds'

v alues as needed, and adding new address-v alue pairs to the memory tap e, if needed. Comparison

• f

Running Times

• If

the computer can do a m ultiplicatio n

• f

w

• rds

whose length is not limited (e.g., to 64 bits, as

• n

most computers), then the length

• f

the longest v alue can double at eac h step, and it tak es O (2 T (n) ) steps

• f

the TM to sim ulate T (n) steps

• f

the computer.

• Ho

w ev er, if w e limit the length

• f

w

• rds

to, sa y , 64,

• r

w e allo w arbitrarily long w

• rds

but

• nly

instructions that add at most 1 to the length in

• ne

step (e.g., addition), then O

• T

3 (n)

• steps
• f

the TM suce to sim ulate T (n) computer steps.

✦

Th us, p

• lynomial
• tim

e computer program b ecomes p

• lynomial-ti

m e TM.

✦

Wh y? Memory tap e can

• nly

gro w to O

• T

2 (n)

• .

Th us,

• ne

step tak es O

• T

2 (n)

• n

the TM, and T (n) steps tak e O

• T

3 (n)

• .

6