CSCI 2670 Introduction to Theory of Computing

1

CSCI 2670Introduction to Theory of

Computing

October 4, 2005

October 4, 2005 2

Agenda

• Last week– Turing machines

• This week– Sections 3.2 & 3.3

• Variants of Turing machines and the definition of algorithm

October 4, 2005 3

Announcements

• Homework due next Tuesday– 3.7, 3.15 b & c, 3.16 d

• 3.7 and 3.15 are same in both texts• 3.16 d – Show the collection of Turing-

recognizable languages is closed under the operation of intersection.

• Quiz tomorrow– Turing machine definition, calculation,

and design

October 4, 2005 4

Group project 1

• Design a Turing machine to accept any string in {a,b}* after making a copy of it on the tape– The tape will start with w – After TM processes the string, the

tape should read ww

October 4, 2005 5

Group project 2

• Write a Turing machine that accepts the language {w {a,b}* | |w| is even}

October 4, 2005 6

Group project 3

• Write a Turing machine that accepts the language {anbm | nm and nm}

October 4, 2005 7

Group project 4

• Write a Turing machine that accepts the language {anbman+m | n0 and m1}

October 4, 2005 8

Group project 5

• Write a Turing machine that accepts the language {wwR | w{a,b}*}

October 4, 2005 9

Group project 6

• Design a Turing machine that accepts the language {w{a,b}* | w has more a’s than

b’s}

October 4, 2005 10

Group project 1

• Design a Turing machine to accept any string in {a,b}* after making a copy of it on the tape– The tape will start with w – After TM processes the string, the

tape should read ww

October 4, 2005 11

TM design

• Read first symbol– If a, replace with x, move to end and

write x– If b, replace with y, move to end and

write y

• Move left to first x or y• Move left to next x or y

– If x, replace with a and move right– If y, replace with b and move right

October 4, 2005 12

TM design (cont.)

• Read symbol at tape head– If x, replace with a and accept– If y, replace with b and accept– If neither, loop to beginning

October 4, 2005 13

States

• Start state q1

• Move right– Write a at end, write b at end, write x

at end, write y at end

• Move left to first x or y• Move left to second x or y &

replace with a or b• Replace a or b with x or y

October 4, 2005 14

Copy machine

a→ x,R

b→ y,R

~ → L

qaccept

a → Rb → R

a → Rb → R

a → Lb → L~ → L

~ → x,R

~ → y,R

x → Ly → L

a → Lb → L

x → a,Ry → b,R a→ x,R

b→ y,R

a,b,x,y → R

a,b,x,y → R

~ → a,R

~ → b,R x → a,Ry → b,R

qreject

October 4, 2005 15

Variants of Turing machines

• Robustness of model– Varying the model does not change

the power• Unlike finite automata (making finite

automata nondeterministic changes the class of languages accepted)

• Simple variant of TM model– Add “stay put” direction

• Other variants– More tapes– Nondeterministic

October 4, 2005 16

Multitape Turing machines

• Same as standard Turing machine, but have several tapes

• TM definition changes only in definition of δ

δ : Q × Гk → Q × Гk × {L,R}k

October 4, 2005 17

Equivalence of machines

Theorem: Every multitape Turing machine has an equivalent single tape Turing machine

Proof method: construction

October 4, 2005 18

Equivalent machines

M

0 1 ~ ~ ~ ~ ~ ~

a a a ~ ~ ~ ~ ~

a b ~ ~ ~ ~ ~ ~

# 0 1 # a a a # a b # ~ ~S

October 4, 2005 19

Simulating k-tape behavior

• Single tape start string is#w#~#...#~#

• Each move proceeds as follows:– Start at leftmost slot– Scan right to (k+1)st # to find symbol

at each virtual tape head– Make second pass making updates

indicated by k-tape transition function– When a virtual head moves onto a #,

shift string to right

October 4, 2005 20

Corollary

Corollary: A language is Turing-recognizable if and only if some multitape Turing machine recognizes it.

October 4, 2005 21

Example

• Using 2-tape Turing machine, write a copy machine

• Copy tape 1 to tape 2• Move tape 1 to beginning• Copy tape 1 to tape 2• Accept

October 4, 2005 22

Copy machine

{a,~}→ {x,a},{R,R}

{b,~}→ {y,b},{R,R}

{~,~} → {L,L}

qaccept

~,~ → {L,S}{x,~} → {a,a},{R,R}{y,~} → {b,b},{R,R}

{~, ,~}→ {R,R}

qreject{a,~}→ {a,a},{R,R}

{b,~}→ {b,b},{R,R}

{a,~}→ {L,S}

{b,~}→ {L,S}

{a,~}→ {a,a},{R,R}

{b,~}→ {b,b},{R,R}