1. CSCI 2670 Introduction to Theory of Computing
October 12, 2004
October 12, 2004

2. Agenda Last week This week Turing machines Sections 3.2 & 3.3
Variants of Turing machines and the definition of algorithm
October 12, 2004

3. Announcements Homework due next Tuesday Tutorial sessions are back
3.3, 3.8 a (low-level description), 3.9, 3.12, 3.13
Tutorial sessions are back
Office hours return to normal
Tuesday 3:00 – 4:00
Wednesday 3:00 – 4:00
October 12, 2004

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 12, 2004

5. TM design Read first symbol Move left to first x or y
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 12, 2004

6. 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 12, 2004

7. States Start state q1 Move right Move left to first x or y
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 12, 2004

8. Copy machine a→ x,R b→ y,R ~ → L qaccept a → R b → R a → L b → L
x → L
y → L
x → a,R
y → b,R
a,b,x,y → R
~ → a,R
~ → b,R
qreject
October 12, 2004

9. Variants of Turing machines
Robustness of model
Varying the model does not change the power
Example, making finite automata nondeterministic
Simple variant of TM model
Add “stay put” direction
Other variants
More tapes
Nondeterministic
October 12, 2004

10. 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 12, 2004

11. Equivalence of machines
Theorem: Every multitape Turing machine has an equivalent single tape Turing machine
Proof method: construction
October 12, 2004

12. Equivalent machines M 0 1 ~ ~ ~ ~ ~ ~ a a a ~ ~ ~ ~ ~ a b ~ ~ ~ ~ ~ ~
0 1 ~ ~ ~ ~ ~ ~
a a a ~ ~ ~ ~ ~
a b ~ ~ ~ ~ ~ ~
# # a a a # a b # ~ ~ S
October 12, 2004

13. 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 12, 2004

14. Corollary
Corollary: A language is Turing-recognizable if and only if some multitape Turing machine recognizes it.
October 12, 2004

15. Example Using 2-tape Turing machine, write a copy machine
Copy tape 1 to tape 2
Move tape 1 to beginning
Accept
October 12, 2004

16. Copy machine qreject {a,~}→ {a,a},{R,R} {b,~}→ {b,b},{R,R}
{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}
October 12, 2004

17. Nondeterministic Turing machines
Same as standard Turing machines, but may have one of several choices at any point
δ : Q × Г → P(Q × Г × {L,R})
October 12, 2004

18. Equivalence of machines
Theorem: Every nondeterministic Turing machine has an equivalent deterministic Turing machine
Proof method: construction
Proof idea: Use a 3-tape Turing machine to deterministically simulate the nondeterministic TM. First tape keeps copy of input, second tape is computation tape, third tape keeps track of choices.
October 12, 2004