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}
