Turing Machines - KeithSchwarz.comDesign Turing machines! Alternate problem session tomorrow night,...
Transcript of Turing Machines - KeithSchwarz.comDesign Turing machines! Alternate problem session tomorrow night,...
![Page 1: Turing Machines - KeithSchwarz.comDesign Turing machines! Alternate problem session tomorrow night, 7-8PM, in Gates 100. The Turing Machine A Turing machine consists of three parts:](https://reader034.fdocuments.net/reader034/viewer/2022042923/5f7027f5297422064c2da53e/html5/thumbnails/1.jpg)
Turing Machines
![Page 2: Turing Machines - KeithSchwarz.comDesign Turing machines! Alternate problem session tomorrow night, 7-8PM, in Gates 100. The Turing Machine A Turing machine consists of three parts:](https://reader034.fdocuments.net/reader034/viewer/2022042923/5f7027f5297422064c2da53e/html5/thumbnails/2.jpg)
Announcements
● Problem Set 7 due Friday, November 18● Stop by OH with questions.● Email [email protected] with questions.
● Problem session tonight, 7-8PM, in 370-370● Practice with PDAs!● Play with the pumping lemma!● Design Turing machines!
● Alternate problem session tomorrow night, 7-8PM, in Gates 100.
![Page 3: Turing Machines - KeithSchwarz.comDesign Turing machines! Alternate problem session tomorrow night, 7-8PM, in Gates 100. The Turing Machine A Turing machine consists of three parts:](https://reader034.fdocuments.net/reader034/viewer/2022042923/5f7027f5297422064c2da53e/html5/thumbnails/3.jpg)
The Turing Machine
● A Turing machine consists of three parts:● A finite-state control used to determine which actions to take,● an infinite tape serving as both input and scratch space, and● a tape head that can read and write the tape and move left or right.
● At each step, the Turing machine● Replaces the contents of the current cell with a new symbol (which
could optionally be the same symbol as before),● Changes state, and● Moves the tape head to the left or to the right.
● The Turing machine accepts if it enters a special accept state. It rejects if it enters a special reject state.
![Page 4: Turing Machines - KeithSchwarz.comDesign Turing machines! Alternate problem session tomorrow night, 7-8PM, in Gates 100. The Turing Machine A Turing machine consists of three parts:](https://reader034.fdocuments.net/reader034/viewer/2022042923/5f7027f5297422064c2da53e/html5/thumbnails/4.jpg)
A Simple Turing Machine
![Page 5: Turing Machines - KeithSchwarz.comDesign Turing machines! Alternate problem session tomorrow night, 7-8PM, in Gates 100. The Turing Machine A Turing machine consists of three parts:](https://reader034.fdocuments.net/reader034/viewer/2022042923/5f7027f5297422064c2da53e/html5/thumbnails/5.jpg)
A Simple Turing Machine
q0
qacc
qrej
q1
start
1 → B, R
1 → B, R
B → B, R
B → B, R
qacc
qrej
![Page 6: Turing Machines - KeithSchwarz.comDesign Turing machines! Alternate problem session tomorrow night, 7-8PM, in Gates 100. The Turing Machine A Turing machine consists of three parts:](https://reader034.fdocuments.net/reader034/viewer/2022042923/5f7027f5297422064c2da53e/html5/thumbnails/6.jpg)
A Simple Turing Machine
q0
qacc
qrej
q1
start
1 → B, R
1 → B, R
B → B, R
B → B, R
qacc
qrej
Each transition of the form
x y, D→
means “upon reading x, replace it with symbol y and move the tape head in direction D (which is either L or R).
The letter B represents a blank.
![Page 7: Turing Machines - KeithSchwarz.comDesign Turing machines! Alternate problem session tomorrow night, 7-8PM, in Gates 100. The Turing Machine A Turing machine consists of three parts:](https://reader034.fdocuments.net/reader034/viewer/2022042923/5f7027f5297422064c2da53e/html5/thumbnails/7.jpg)
A Simple Turing Machine
q0
qacc
qrej
q1
start
1 → B, R
1 → B, R
B → B, R
B → B, R
qacc
qrej
This special accept state causes the
machine to immediately accept.
Each transition of the form
x y, D→
means “upon reading x, replace it with symbol y and move the tape head in direction D (which is either L or R).
The letter B represents a blank.
![Page 8: Turing Machines - KeithSchwarz.comDesign Turing machines! Alternate problem session tomorrow night, 7-8PM, in Gates 100. The Turing Machine A Turing machine consists of three parts:](https://reader034.fdocuments.net/reader034/viewer/2022042923/5f7027f5297422064c2da53e/html5/thumbnails/8.jpg)
A Simple Turing Machine
q0
qacc
qrej
q1
start
1 → B, R
1 → B, R
B → B, R
B → B, R
qacc
qrej
Each transition of the form
x y, D→
means “upon reading x, replace it with symbol y and move the tape head in direction D (which is either L or R).
The letter B represents a blank.
This special accept state causes the
machine to immediately accept.
This special reject state causes the
machine to immediately reject.
![Page 9: Turing Machines - KeithSchwarz.comDesign Turing machines! Alternate problem session tomorrow night, 7-8PM, in Gates 100. The Turing Machine A Turing machine consists of three parts:](https://reader034.fdocuments.net/reader034/viewer/2022042923/5f7027f5297422064c2da53e/html5/thumbnails/9.jpg)
A Simple Turing Machine
q0
qacc
qrej
q1
start
1 → B, R
1 → B, R
B → B, R
B → B, R
qacc
qrej
![Page 10: Turing Machines - KeithSchwarz.comDesign Turing machines! Alternate problem session tomorrow night, 7-8PM, in Gates 100. The Turing Machine A Turing machine consists of three parts:](https://reader034.fdocuments.net/reader034/viewer/2022042923/5f7027f5297422064c2da53e/html5/thumbnails/10.jpg)
A Simple Turing Machine
q0
qacc
qrej
q1
start
1 → B, R
1 → B, R
B → B, R
B → B, R
qacc
qrej
1 1 1 1
![Page 11: Turing Machines - KeithSchwarz.comDesign Turing machines! Alternate problem session tomorrow night, 7-8PM, in Gates 100. The Turing Machine A Turing machine consists of three parts:](https://reader034.fdocuments.net/reader034/viewer/2022042923/5f7027f5297422064c2da53e/html5/thumbnails/11.jpg)
q0
q0
q1
A Simple Turing Machineq
acc
qrej
q1
start
1 → B, R
1 → B, R
B → B, R
B → B, R
qacc
qrej
1 1 1 1
![Page 12: Turing Machines - KeithSchwarz.comDesign Turing machines! Alternate problem session tomorrow night, 7-8PM, in Gates 100. The Turing Machine A Turing machine consists of three parts:](https://reader034.fdocuments.net/reader034/viewer/2022042923/5f7027f5297422064c2da53e/html5/thumbnails/12.jpg)
q0
q0
q1
A Simple Turing Machineq
acc
qrej
q1
start
1 → B, R
1 → B, R
B → B, R
B → B, R
qacc
qrej
1 1 1 1
![Page 13: Turing Machines - KeithSchwarz.comDesign Turing machines! Alternate problem session tomorrow night, 7-8PM, in Gates 100. The Turing Machine A Turing machine consists of three parts:](https://reader034.fdocuments.net/reader034/viewer/2022042923/5f7027f5297422064c2da53e/html5/thumbnails/13.jpg)
q0
q0
q1
A Simple Turing Machineq
acc
qrej
q1
start
1 → B, R
1 → B, R
B → B, R
B → B, R
qacc
qrej
1 1 1 1
![Page 14: Turing Machines - KeithSchwarz.comDesign Turing machines! Alternate problem session tomorrow night, 7-8PM, in Gates 100. The Turing Machine A Turing machine consists of three parts:](https://reader034.fdocuments.net/reader034/viewer/2022042923/5f7027f5297422064c2da53e/html5/thumbnails/14.jpg)
q0
q0
q1
A Simple Turing Machineq
acc
qrej
q1
start
1 → B, R
1 → B, R
B → B, R
B → B, R
qacc
qrej
1 1 1
![Page 15: Turing Machines - KeithSchwarz.comDesign Turing machines! Alternate problem session tomorrow night, 7-8PM, in Gates 100. The Turing Machine A Turing machine consists of three parts:](https://reader034.fdocuments.net/reader034/viewer/2022042923/5f7027f5297422064c2da53e/html5/thumbnails/15.jpg)
q0
q0
q1
A Simple Turing Machineq
acc
qrej
q1
start
1 → B, R
1 → B, R
B → B, R
B → B, R
qacc
qrej
1 1 1
![Page 16: Turing Machines - KeithSchwarz.comDesign Turing machines! Alternate problem session tomorrow night, 7-8PM, in Gates 100. The Turing Machine A Turing machine consists of three parts:](https://reader034.fdocuments.net/reader034/viewer/2022042923/5f7027f5297422064c2da53e/html5/thumbnails/16.jpg)
q1
q0
q0
q1
A Simple Turing Machineq
acc
qrej
start
1 → B, R
1 → B, R
B → B, R
B → B, R
qacc
qrej
1 1 1
![Page 17: Turing Machines - KeithSchwarz.comDesign Turing machines! Alternate problem session tomorrow night, 7-8PM, in Gates 100. The Turing Machine A Turing machine consists of three parts:](https://reader034.fdocuments.net/reader034/viewer/2022042923/5f7027f5297422064c2da53e/html5/thumbnails/17.jpg)
q1
q0
q0
q1
A Simple Turing Machineq
acc
qrej
start
1 → B, R
1 → B, R
B → B, R
B → B, R
qacc
qrej
1 1 1
![Page 18: Turing Machines - KeithSchwarz.comDesign Turing machines! Alternate problem session tomorrow night, 7-8PM, in Gates 100. The Turing Machine A Turing machine consists of three parts:](https://reader034.fdocuments.net/reader034/viewer/2022042923/5f7027f5297422064c2da53e/html5/thumbnails/18.jpg)
q1
q0
q0
q1
A Simple Turing Machineq
acc
qrej
start
1 → B, R
1 → B, R
B → B, R
B → B, R
qacc
qrej
1 1
![Page 19: Turing Machines - KeithSchwarz.comDesign Turing machines! Alternate problem session tomorrow night, 7-8PM, in Gates 100. The Turing Machine A Turing machine consists of three parts:](https://reader034.fdocuments.net/reader034/viewer/2022042923/5f7027f5297422064c2da53e/html5/thumbnails/19.jpg)
q1
q0
q0
q1
A Simple Turing Machineq
acc
qrej
start
1 → B, R
1 → B, R
B → B, R
B → B, R
qacc
qrej
1 1
![Page 20: Turing Machines - KeithSchwarz.comDesign Turing machines! Alternate problem session tomorrow night, 7-8PM, in Gates 100. The Turing Machine A Turing machine consists of three parts:](https://reader034.fdocuments.net/reader034/viewer/2022042923/5f7027f5297422064c2da53e/html5/thumbnails/20.jpg)
q0
q1
q1
q0
A Simple Turing Machineq
acc
qrej
start
1 → B, R
1 → B, R
B → B, R
B → B, R
qacc
qrej
1 1
![Page 21: Turing Machines - KeithSchwarz.comDesign Turing machines! Alternate problem session tomorrow night, 7-8PM, in Gates 100. The Turing Machine A Turing machine consists of three parts:](https://reader034.fdocuments.net/reader034/viewer/2022042923/5f7027f5297422064c2da53e/html5/thumbnails/21.jpg)
q0
q1
q1
q0
A Simple Turing Machineq
acc
qrej
start
1 → B, R
1 → B, R
B → B, R
B → B, R
qacc
qrej
1 1
![Page 22: Turing Machines - KeithSchwarz.comDesign Turing machines! Alternate problem session tomorrow night, 7-8PM, in Gates 100. The Turing Machine A Turing machine consists of three parts:](https://reader034.fdocuments.net/reader034/viewer/2022042923/5f7027f5297422064c2da53e/html5/thumbnails/22.jpg)
q0
q1
q1
q0
A Simple Turing Machineq
acc
qrej
start
1 → B, R
1 → B, R
B → B, R
B → B, R
qacc
qrej
1
![Page 23: Turing Machines - KeithSchwarz.comDesign Turing machines! Alternate problem session tomorrow night, 7-8PM, in Gates 100. The Turing Machine A Turing machine consists of three parts:](https://reader034.fdocuments.net/reader034/viewer/2022042923/5f7027f5297422064c2da53e/html5/thumbnails/23.jpg)
q0
q1
q1
q0
A Simple Turing Machineq
acc
qrej
start
1 → B, R
1 → B, R
B → B, R
B → B, R
qacc
qrej
1
![Page 24: Turing Machines - KeithSchwarz.comDesign Turing machines! Alternate problem session tomorrow night, 7-8PM, in Gates 100. The Turing Machine A Turing machine consists of three parts:](https://reader034.fdocuments.net/reader034/viewer/2022042923/5f7027f5297422064c2da53e/html5/thumbnails/24.jpg)
q1
q0
q0
q1
A Simple Turing Machineq
acc
qrej
start
1 → B, R
1 → B, R
B → B, R
B → B, R
qacc
qrej
1
![Page 25: Turing Machines - KeithSchwarz.comDesign Turing machines! Alternate problem session tomorrow night, 7-8PM, in Gates 100. The Turing Machine A Turing machine consists of three parts:](https://reader034.fdocuments.net/reader034/viewer/2022042923/5f7027f5297422064c2da53e/html5/thumbnails/25.jpg)
q1
q0
q0
q1
A Simple Turing Machineq
acc
qrej
start
1 → B, R
1 → B, R
B → B, R
B → B, R
qacc
qrej
1
![Page 26: Turing Machines - KeithSchwarz.comDesign Turing machines! Alternate problem session tomorrow night, 7-8PM, in Gates 100. The Turing Machine A Turing machine consists of three parts:](https://reader034.fdocuments.net/reader034/viewer/2022042923/5f7027f5297422064c2da53e/html5/thumbnails/26.jpg)
q1
q0
q0
q1
A Simple Turing Machineq
acc
qrej
start
1 → B, R
1 → B, R
B → B, R
B → B, R
qacc
qrej
![Page 27: Turing Machines - KeithSchwarz.comDesign Turing machines! Alternate problem session tomorrow night, 7-8PM, in Gates 100. The Turing Machine A Turing machine consists of three parts:](https://reader034.fdocuments.net/reader034/viewer/2022042923/5f7027f5297422064c2da53e/html5/thumbnails/27.jpg)
q1
q0
q0
q1
A Simple Turing Machineq
acc
qrej
start
1 → B, R
1 → B, R
B → B, R
B → B, R
qacc
qrej
![Page 28: Turing Machines - KeithSchwarz.comDesign Turing machines! Alternate problem session tomorrow night, 7-8PM, in Gates 100. The Turing Machine A Turing machine consists of three parts:](https://reader034.fdocuments.net/reader034/viewer/2022042923/5f7027f5297422064c2da53e/html5/thumbnails/28.jpg)
q1
q0
q1
q0
A Simple Turing Machineq
acc
qrej
start
1 → B, R
1 → B, R
B → B, R
B → B, R
qacc
qrej
![Page 29: Turing Machines - KeithSchwarz.comDesign Turing machines! Alternate problem session tomorrow night, 7-8PM, in Gates 100. The Turing Machine A Turing machine consists of three parts:](https://reader034.fdocuments.net/reader034/viewer/2022042923/5f7027f5297422064c2da53e/html5/thumbnails/29.jpg)
q1
q0
q1
q0
A Simple Turing Machineq
acc
qrej
start
1 → B, R
1 → B, R
B → B, R
B → B, R
qacc
qrej
![Page 30: Turing Machines - KeithSchwarz.comDesign Turing machines! Alternate problem session tomorrow night, 7-8PM, in Gates 100. The Turing Machine A Turing machine consists of three parts:](https://reader034.fdocuments.net/reader034/viewer/2022042923/5f7027f5297422064c2da53e/html5/thumbnails/30.jpg)
q1
q0
q1
q0
A Simple Turing Machineq
acc
qrej
start
1 → B, R
1 → B, R
B → B, R
B → B, R
qacc
qrej
![Page 31: Turing Machines - KeithSchwarz.comDesign Turing machines! Alternate problem session tomorrow night, 7-8PM, in Gates 100. The Turing Machine A Turing machine consists of three parts:](https://reader034.fdocuments.net/reader034/viewer/2022042923/5f7027f5297422064c2da53e/html5/thumbnails/31.jpg)
q0
q1
q0
q1
A Simple Turing Machineq
acc
qrej
start
1 → B, R
1 → B, R
B → B, R
B → B, R
qacc
qrej
![Page 32: Turing Machines - KeithSchwarz.comDesign Turing machines! Alternate problem session tomorrow night, 7-8PM, in Gates 100. The Turing Machine A Turing machine consists of three parts:](https://reader034.fdocuments.net/reader034/viewer/2022042923/5f7027f5297422064c2da53e/html5/thumbnails/32.jpg)
q0
q1
q0
q1
A Simple Turing Machineq
acc
qrej
start
1 → B, R
1 → B, R
B → B, R
B → B, R
qacc
qrej
![Page 33: Turing Machines - KeithSchwarz.comDesign Turing machines! Alternate problem session tomorrow night, 7-8PM, in Gates 100. The Turing Machine A Turing machine consists of three parts:](https://reader034.fdocuments.net/reader034/viewer/2022042923/5f7027f5297422064c2da53e/html5/thumbnails/33.jpg)
q0
q1
q0
q1
A Simple Turing Machineq
acc
qrej
start
1 → B, R
1 → B, R
B → B, R
B → B, R
qacc
qrej
![Page 34: Turing Machines - KeithSchwarz.comDesign Turing machines! Alternate problem session tomorrow night, 7-8PM, in Gates 100. The Turing Machine A Turing machine consists of three parts:](https://reader034.fdocuments.net/reader034/viewer/2022042923/5f7027f5297422064c2da53e/html5/thumbnails/34.jpg)
q0
q1
A Simple Turing Machine
q0
qacc
qrej
q1
start
1 → B, R
1 → B, R
B → B, R
B → B, R
qacc
qrej
1
![Page 35: Turing Machines - KeithSchwarz.comDesign Turing machines! Alternate problem session tomorrow night, 7-8PM, in Gates 100. The Turing Machine A Turing machine consists of three parts:](https://reader034.fdocuments.net/reader034/viewer/2022042923/5f7027f5297422064c2da53e/html5/thumbnails/35.jpg)
q0
q0
q1
A Simple Turing Machineq
acc
qrej
q1
start
1 → B, R
1 → B, R
B → B, R
B → B, R
qacc
qrej
1
![Page 36: Turing Machines - KeithSchwarz.comDesign Turing machines! Alternate problem session tomorrow night, 7-8PM, in Gates 100. The Turing Machine A Turing machine consists of three parts:](https://reader034.fdocuments.net/reader034/viewer/2022042923/5f7027f5297422064c2da53e/html5/thumbnails/36.jpg)
q0
q0
q1
A Simple Turing Machineq
acc
qrej
q1
start
1 → B, R
1 → B, R
B → B, R
B → B, R
qacc
qrej
1
![Page 37: Turing Machines - KeithSchwarz.comDesign Turing machines! Alternate problem session tomorrow night, 7-8PM, in Gates 100. The Turing Machine A Turing machine consists of three parts:](https://reader034.fdocuments.net/reader034/viewer/2022042923/5f7027f5297422064c2da53e/html5/thumbnails/37.jpg)
q0
q0
q1
A Simple Turing Machineq
acc
qrej
q1
start
1 → B, R
1 → B, R
B → B, R
B → B, R
qacc
qrej
1
![Page 38: Turing Machines - KeithSchwarz.comDesign Turing machines! Alternate problem session tomorrow night, 7-8PM, in Gates 100. The Turing Machine A Turing machine consists of three parts:](https://reader034.fdocuments.net/reader034/viewer/2022042923/5f7027f5297422064c2da53e/html5/thumbnails/38.jpg)
q0
q0
q1
A Simple Turing Machineq
acc
qrej
q1
start
1 → B, R
1 → B, R
B → B, R
B → B, R
qacc
qrej
![Page 39: Turing Machines - KeithSchwarz.comDesign Turing machines! Alternate problem session tomorrow night, 7-8PM, in Gates 100. The Turing Machine A Turing machine consists of three parts:](https://reader034.fdocuments.net/reader034/viewer/2022042923/5f7027f5297422064c2da53e/html5/thumbnails/39.jpg)
q0
q0
q1
A Simple Turing Machineq
acc
qrej
q1
start
1 → B, R
1 → B, R
B → B, R
B → B, R
qacc
qrej
![Page 40: Turing Machines - KeithSchwarz.comDesign Turing machines! Alternate problem session tomorrow night, 7-8PM, in Gates 100. The Turing Machine A Turing machine consists of three parts:](https://reader034.fdocuments.net/reader034/viewer/2022042923/5f7027f5297422064c2da53e/html5/thumbnails/40.jpg)
q0
q1
q0
q1
A Simple Turing Machineq
acc
qrej
start
1 → B, R
1 → B, R
B → B, R
B → B, R
qacc
qrej
![Page 41: Turing Machines - KeithSchwarz.comDesign Turing machines! Alternate problem session tomorrow night, 7-8PM, in Gates 100. The Turing Machine A Turing machine consists of three parts:](https://reader034.fdocuments.net/reader034/viewer/2022042923/5f7027f5297422064c2da53e/html5/thumbnails/41.jpg)
q0
q1
q0
q1
A Simple Turing Machineq
acc
qrej
start
1 → B, R
1 → B, R
B → B, R
B → B, R
qacc
qrej
![Page 42: Turing Machines - KeithSchwarz.comDesign Turing machines! Alternate problem session tomorrow night, 7-8PM, in Gates 100. The Turing Machine A Turing machine consists of three parts:](https://reader034.fdocuments.net/reader034/viewer/2022042923/5f7027f5297422064c2da53e/html5/thumbnails/42.jpg)
q0
q1
q0
q1
A Simple Turing Machineq
acc
qrej
start
1 → B, R
1 → B, R
B → B, R
B → B, R
qacc
qrej
![Page 43: Turing Machines - KeithSchwarz.comDesign Turing machines! Alternate problem session tomorrow night, 7-8PM, in Gates 100. The Turing Machine A Turing machine consists of three parts:](https://reader034.fdocuments.net/reader034/viewer/2022042923/5f7027f5297422064c2da53e/html5/thumbnails/43.jpg)
q0
q1
q0
q1
A Simple Turing Machineq
acc
qrej
start
1 → B, R
1 → B, R
B → B, R
B → B, R
qacc
qrej
![Page 44: Turing Machines - KeithSchwarz.comDesign Turing machines! Alternate problem session tomorrow night, 7-8PM, in Gates 100. The Turing Machine A Turing machine consists of three parts:](https://reader034.fdocuments.net/reader034/viewer/2022042923/5f7027f5297422064c2da53e/html5/thumbnails/44.jpg)
q0
q1
q0
q1
A Simple Turing Machineq
acc
qrej
start
1 → B, R
1 → B, R
B → B, R
B → B, R
qacc
qrej
![Page 45: Turing Machines - KeithSchwarz.comDesign Turing machines! Alternate problem session tomorrow night, 7-8PM, in Gates 100. The Turing Machine A Turing machine consists of three parts:](https://reader034.fdocuments.net/reader034/viewer/2022042923/5f7027f5297422064c2da53e/html5/thumbnails/45.jpg)
q0
q1
q0
q1
A Simple Turing Machineq
acc
qrej
start
1 → B, R
1 → B, R
B → B, R
B → B, R
qacc
qrej
![Page 46: Turing Machines - KeithSchwarz.comDesign Turing machines! Alternate problem session tomorrow night, 7-8PM, in Gates 100. The Turing Machine A Turing machine consists of three parts:](https://reader034.fdocuments.net/reader034/viewer/2022042923/5f7027f5297422064c2da53e/html5/thumbnails/46.jpg)
A More Powerful Turing Machine
● Let Σ = {0, 1} and let L = { w | w is a palindrome }
● We can build a TM for L as follows:● Look at the leftmost character of the string.● Scan across the tape until we find the end of the
string.● If the last character doesn't match, reject the input.● Sweep back to the left of the tape and repeat.● If every character becomes matched, accept.
![Page 47: Turing Machines - KeithSchwarz.comDesign Turing machines! Alternate problem session tomorrow night, 7-8PM, in Gates 100. The Turing Machine A Turing machine consists of three parts:](https://reader034.fdocuments.net/reader034/viewer/2022042923/5f7027f5297422064c2da53e/html5/thumbnails/47.jpg)
q0
q0
q1
q2
q3
q4
q5
qacc q
rej
q1
start qrej
q2
q3
q4
q5
B → B, R
0 → B, R
1 → B, R
0 → 0, R1 → 1, R
B → B, L
0 → 0, R 1 → 1, R
B → B, L
B → B, R
B → B, R
qacc
1 → 1, R
0 → 0, R
0 → B, L
1 → B, L
0 → 0, L 1 → 1, L
B → B, R
![Page 48: Turing Machines - KeithSchwarz.comDesign Turing machines! Alternate problem session tomorrow night, 7-8PM, in Gates 100. The Turing Machine A Turing machine consists of three parts:](https://reader034.fdocuments.net/reader034/viewer/2022042923/5f7027f5297422064c2da53e/html5/thumbnails/48.jpg)
q0
q0
q1
q2
q3
q4
q5
qacc q
rej
q1
start qrej
q2
q3
q4
q5
B → B, R
0 → B, R
1 → B, R
0 → 0, R1 → 1, R
B → B, L
0 → 0, R 1 → 1, R
B → B, L
B → B, R
B → B, R
qacc
1 → 1, R
0 → 0, R
0 → B, L
1 → B, L
0 → 0, L 1 → 1, L
B → B, R
0 1 1 0
![Page 49: Turing Machines - KeithSchwarz.comDesign Turing machines! Alternate problem session tomorrow night, 7-8PM, in Gates 100. The Turing Machine A Turing machine consists of three parts:](https://reader034.fdocuments.net/reader034/viewer/2022042923/5f7027f5297422064c2da53e/html5/thumbnails/49.jpg)
q0
q0
q1
q2
q3
q4
q5
qacc q
rej
q1
start qrej
q2
q3
q4
q5
B → B, R
0 → B, R
1 → B, R
0 → 0, R1 → 1, R
B → B, L
0 → 0, R 1 → 1, R
B → B, L
B → B, R
B → B, R
qacc
1 → 1, R
0 → 0, R
0 → B, L
1 → B, L
0 → 0, L 1 → 1, L
B → B, R
0 1 1 0
![Page 50: Turing Machines - KeithSchwarz.comDesign Turing machines! Alternate problem session tomorrow night, 7-8PM, in Gates 100. The Turing Machine A Turing machine consists of three parts:](https://reader034.fdocuments.net/reader034/viewer/2022042923/5f7027f5297422064c2da53e/html5/thumbnails/50.jpg)
q0
q0
q1
q2
q3
q4
q5
qacc q
rej
q1
start qrej
q2
q3
q4
q5
B → B, R
0 → B, R
1 → B, R
0 → 0, R1 → 1, R
B → B, L
0 → 0, R 1 → 1, R
B → B, L
B → B, R
B → B, R
qacc
1 → 1, R
0 → 0, R
0 → B, L
1 → B, L
0 → 0, L 1 → 1, L
B → B, R
0 1 1 0
![Page 51: Turing Machines - KeithSchwarz.comDesign Turing machines! Alternate problem session tomorrow night, 7-8PM, in Gates 100. The Turing Machine A Turing machine consists of three parts:](https://reader034.fdocuments.net/reader034/viewer/2022042923/5f7027f5297422064c2da53e/html5/thumbnails/51.jpg)
q0
q0
q1
q2
q3
q4
q5
qacc q
rej
q1
start qrej
q2
q3
q4
q5
B → B, R
0 → B, R
1 → B, R
0 → 0, R1 → 1, R
B → B, L
0 → 0, R 1 → 1, R
B → B, L
B → B, R
B → B, R
qacc
1 → 1, R
0 → 0, R
0 → B, L
1 → B, L
0 → 0, L 1 → 1, L
B → B, R
0 1 1 0
![Page 52: Turing Machines - KeithSchwarz.comDesign Turing machines! Alternate problem session tomorrow night, 7-8PM, in Gates 100. The Turing Machine A Turing machine consists of three parts:](https://reader034.fdocuments.net/reader034/viewer/2022042923/5f7027f5297422064c2da53e/html5/thumbnails/52.jpg)
q0
q0
q1
q2
q3
q4
q5
qacc q
rej
q1
start qrej
q2
q3
q4
q5
B → B, R
0 → B, R
1 → B, R
0 → 0, R1 → 1, R
B → B, L
0 → 0, R 1 → 1, R
B → B, L
B → B, R
B → B, R
qacc
1 → 1, R
0 → 0, R
0 → B, L
1 → B, L
0 → 0, L 1 → 1, L
B → B, R
1 1 0
![Page 53: Turing Machines - KeithSchwarz.comDesign Turing machines! Alternate problem session tomorrow night, 7-8PM, in Gates 100. The Turing Machine A Turing machine consists of three parts:](https://reader034.fdocuments.net/reader034/viewer/2022042923/5f7027f5297422064c2da53e/html5/thumbnails/53.jpg)
q1
q0
q0
q1
q2
q3
q4
q5
qacc q
rejstart q
rej
q2
q3
q4
q5
B → B, R
0 → B, R
1 → B, R
0 → 0, R1 → 1, R
B → B, L
0 → 0, R 1 → 1, R
B → B, L
B → B, R
B → B, R
qacc
1 → 1, R
0 → 0, R
0 → B, L
1 → B, L
0 → 0, L 1 → 1, L
B → B, R
1 1 0
![Page 54: Turing Machines - KeithSchwarz.comDesign Turing machines! Alternate problem session tomorrow night, 7-8PM, in Gates 100. The Turing Machine A Turing machine consists of three parts:](https://reader034.fdocuments.net/reader034/viewer/2022042923/5f7027f5297422064c2da53e/html5/thumbnails/54.jpg)
q1
q0
q0
q1
q2
q3
q4
q5
qacc q
rejstart q
rej
q2
q3
q4
q5
B → B, R
0 → B, R
1 → B, R
0 → 0, R1 → 1, R
B → B, L
0 → 0, R 1 → 1, R
B → B, L
B → B, R
B → B, R
qacc
1 → 1, R
0 → 0, R
0 → B, L
1 → B, L
0 → 0, L 1 → 1, L
B → B, R
1 1 0
![Page 55: Turing Machines - KeithSchwarz.comDesign Turing machines! Alternate problem session tomorrow night, 7-8PM, in Gates 100. The Turing Machine A Turing machine consists of three parts:](https://reader034.fdocuments.net/reader034/viewer/2022042923/5f7027f5297422064c2da53e/html5/thumbnails/55.jpg)
q1
q0
q0
q1
q2
q3
q4
q5
qacc q
rejstart q
rej
q2
q3
q4
q5
B → B, R
0 → B, R
1 → B, R
0 → 0, R1 → 1, R
B → B, L
0 → 0, R 1 → 1, R
B → B, L
B → B, R
B → B, R
qacc
1 → 1, R
0 → 0, R
0 → B, L
1 → B, L
0 → 0, L 1 → 1, L
B → B, R
1 1 0
![Page 56: Turing Machines - KeithSchwarz.comDesign Turing machines! Alternate problem session tomorrow night, 7-8PM, in Gates 100. The Turing Machine A Turing machine consists of three parts:](https://reader034.fdocuments.net/reader034/viewer/2022042923/5f7027f5297422064c2da53e/html5/thumbnails/56.jpg)
q1
q0
q0
q1
q2
q3
q4
q5
qacc q
rejstart q
rej
q2
q3
q4
q5
B → B, R
0 → B, R
1 → B, R
0 → 0, R1 → 1, R
B → B, L
0 → 0, R 1 → 1, R
B → B, L
B → B, R
B → B, R
qacc
1 → 1, R
0 → 0, R
0 → B, L
1 → B, L
0 → 0, L 1 → 1, L
B → B, R
1 1 0
![Page 57: Turing Machines - KeithSchwarz.comDesign Turing machines! Alternate problem session tomorrow night, 7-8PM, in Gates 100. The Turing Machine A Turing machine consists of three parts:](https://reader034.fdocuments.net/reader034/viewer/2022042923/5f7027f5297422064c2da53e/html5/thumbnails/57.jpg)
q1
q0
q0
q1
q2
q3
q4
q5
qacc q
rejstart q
rej
q2
q3
q4
q5
B → B, R
0 → B, R
1 → B, R
0 → 0, R1 → 1, R
B → B, L
0 → 0, R 1 → 1, R
B → B, L
B → B, R
B → B, R
qacc
1 → 1, R
0 → 0, R
0 → B, L
1 → B, L
0 → 0, L 1 → 1, L
B → B, R
1 1 0
![Page 58: Turing Machines - KeithSchwarz.comDesign Turing machines! Alternate problem session tomorrow night, 7-8PM, in Gates 100. The Turing Machine A Turing machine consists of three parts:](https://reader034.fdocuments.net/reader034/viewer/2022042923/5f7027f5297422064c2da53e/html5/thumbnails/58.jpg)
q1
q0
q0
q1
q2
q3
q4
q5
qacc q
rejstart q
rej
q2
q3
q4
q5
B → B, R
0 → B, R
1 → B, R
0 → 0, R1 → 1, R
B → B, L
0 → 0, R 1 → 1, R
B → B, L
B → B, R
B → B, R
qacc
1 → 1, R
0 → 0, R
0 → B, L
1 → B, L
0 → 0, L 1 → 1, L
B → B, R
1 1 0
![Page 59: Turing Machines - KeithSchwarz.comDesign Turing machines! Alternate problem session tomorrow night, 7-8PM, in Gates 100. The Turing Machine A Turing machine consists of three parts:](https://reader034.fdocuments.net/reader034/viewer/2022042923/5f7027f5297422064c2da53e/html5/thumbnails/59.jpg)
q1
q0
q0
q1
q2
q3
q4
q5
qacc q
rejstart q
rej
q2
q3
q4
q5
B → B, R
0 → B, R
1 → B, R
0 → 0, R1 → 1, R
B → B, L
0 → 0, R 1 → 1, R
B → B, L
B → B, R
B → B, R
qacc
1 → 1, R
0 → 0, R
0 → B, L
1 → B, L
0 → 0, L 1 → 1, L
B → B, R
1 1 0
![Page 60: Turing Machines - KeithSchwarz.comDesign Turing machines! Alternate problem session tomorrow night, 7-8PM, in Gates 100. The Turing Machine A Turing machine consists of three parts:](https://reader034.fdocuments.net/reader034/viewer/2022042923/5f7027f5297422064c2da53e/html5/thumbnails/60.jpg)
q1
q0
q0
q1
q2
q3
q4
q5
qacc q
rejstart q
rej
q2
q3
q4
q5
B → B, R
0 → B, R
1 → B, R
0 → 0, R1 → 1, R
B → B, L
0 → 0, R 1 → 1, R
B → B, L
B → B, R
B → B, R
qacc
1 → 1, R
0 → 0, R
0 → B, L
1 → B, L
0 → 0, L 1 → 1, L
B → B, R
1 1 0
![Page 61: Turing Machines - KeithSchwarz.comDesign Turing machines! Alternate problem session tomorrow night, 7-8PM, in Gates 100. The Turing Machine A Turing machine consists of three parts:](https://reader034.fdocuments.net/reader034/viewer/2022042923/5f7027f5297422064c2da53e/html5/thumbnails/61.jpg)
q1
q0
q0
q1
q2
q3
q4
q5
qacc q
rejstart q
rej
q2
q3
q4
q5
B → B, R
0 → B, R
1 → B, R
0 → 0, R1 → 1, R
B → B, L
0 → 0, R 1 → 1, R
B → B, L
B → B, R
B → B, R
qacc
1 → 1, R
0 → 0, R
0 → B, L
1 → B, L
0 → 0, L 1 → 1, L
B → B, R
1 1 0
![Page 62: Turing Machines - KeithSchwarz.comDesign Turing machines! Alternate problem session tomorrow night, 7-8PM, in Gates 100. The Turing Machine A Turing machine consists of three parts:](https://reader034.fdocuments.net/reader034/viewer/2022042923/5f7027f5297422064c2da53e/html5/thumbnails/62.jpg)
q1
q0
q0
q1
q2
q3
q4
q5
qacc q
rejstart q
rej
q2
q3
q4
q5
B → B, R
0 → B, R
1 → B, R
0 → 0, R1 → 1, R
B → B, L
0 → 0, R 1 → 1, R
B → B, L
B → B, R
B → B, R
qacc
1 → 1, R
0 → 0, R
0 → B, L
1 → B, L
0 → 0, L 1 → 1, L
B → B, R
1 1 0
![Page 63: Turing Machines - KeithSchwarz.comDesign Turing machines! Alternate problem session tomorrow night, 7-8PM, in Gates 100. The Turing Machine A Turing machine consists of three parts:](https://reader034.fdocuments.net/reader034/viewer/2022042923/5f7027f5297422064c2da53e/html5/thumbnails/63.jpg)
q1
q0
q0
q1
q2
q3
q4
q5
qacc q
rejstart q
rej
q2
q3
q4
q5
B → B, R
0 → B, R
1 → B, R
0 → 0, R1 → 1, R
B → B, L
0 → 0, R 1 → 1, R
B → B, L
B → B, R
B → B, R
qacc
1 → 1, R
0 → 0, R
0 → B, L
1 → B, L
0 → 0, L 1 → 1, L
B → B, R
1 1 0
![Page 64: Turing Machines - KeithSchwarz.comDesign Turing machines! Alternate problem session tomorrow night, 7-8PM, in Gates 100. The Turing Machine A Turing machine consists of three parts:](https://reader034.fdocuments.net/reader034/viewer/2022042923/5f7027f5297422064c2da53e/html5/thumbnails/64.jpg)
q1
q0
q0
q1
q2
q3
q4
q5
qacc q
rejstart q
rej
q2
q3
q4
q5
B → B, R
0 → B, R
1 → B, R
0 → 0, R1 → 1, R
B → B, L
0 → 0, R 1 → 1, R
B → B, L
B → B, R
B → B, R
qacc
1 → 1, R
0 → 0, R
0 → B, L
1 → B, L
0 → 0, L 1 → 1, L
B → B, R
1 1 0
![Page 65: Turing Machines - KeithSchwarz.comDesign Turing machines! Alternate problem session tomorrow night, 7-8PM, in Gates 100. The Turing Machine A Turing machine consists of three parts:](https://reader034.fdocuments.net/reader034/viewer/2022042923/5f7027f5297422064c2da53e/html5/thumbnails/65.jpg)
q3
q1
q1
q0
q0
q2
q3
q4
q5
qacc q
rejstart q
rej
q2
q4
q5
B → B, R
0 → B, R
1 → B, R
0 → 0, R1 → 1, R
B → B, L
0 → 0, R 1 → 1, R
B → B, L
B → B, R
B → B, R
qacc
1 → 1, R
0 → 0, R
0 → B, L
1 → B, L
0 → 0, L 1 → 1, L
B → B, R
1 1 0
![Page 66: Turing Machines - KeithSchwarz.comDesign Turing machines! Alternate problem session tomorrow night, 7-8PM, in Gates 100. The Turing Machine A Turing machine consists of three parts:](https://reader034.fdocuments.net/reader034/viewer/2022042923/5f7027f5297422064c2da53e/html5/thumbnails/66.jpg)
q3
q1
q1
q0
q0
q2
q3
q4
q5
qacc q
rejstart q
rej
q2
q4
q5
B → B, R
0 → B, R
1 → B, R
0 → 0, R1 → 1, R
B → B, L
0 → 0, R 1 → 1, R
B → B, L
B → B, R
B → B, R
qacc
1 → 1, R
0 → 0, R
0 → B, L
1 → B, L
0 → 0, L 1 → 1, L
B → B, R
1 1 0
![Page 67: Turing Machines - KeithSchwarz.comDesign Turing machines! Alternate problem session tomorrow night, 7-8PM, in Gates 100. The Turing Machine A Turing machine consists of three parts:](https://reader034.fdocuments.net/reader034/viewer/2022042923/5f7027f5297422064c2da53e/html5/thumbnails/67.jpg)
q3
q1
q1
q0
q0
q2
q3
q4
q5
qacc q
rejstart q
rej
q2
q4
q5
B → B, R
0 → B, R
1 → B, R
0 → 0, R1 → 1, R
B → B, L
0 → 0, R 1 → 1, R
B → B, L
B → B, R
B → B, R
qacc
1 → 1, R
0 → 0, R
0 → B, L
1 → B, L
0 → 0, L 1 → 1, L
B → B, R
1 1
![Page 68: Turing Machines - KeithSchwarz.comDesign Turing machines! Alternate problem session tomorrow night, 7-8PM, in Gates 100. The Turing Machine A Turing machine consists of three parts:](https://reader034.fdocuments.net/reader034/viewer/2022042923/5f7027f5297422064c2da53e/html5/thumbnails/68.jpg)
q5
q3
q3
q1
q1
q0
q0
q2
q4
q5
qacc q
rejstart q
rej
q2
q4
B → B, R
0 → B, R
1 → B, R
0 → 0, R1 → 1, R
B → B, L
0 → 0, R 1 → 1, R
B → B, L
B → B, R
B → B, R
qacc
1 → 1, R
0 → 0, R
0 → B, L
1 → B, L
0 → 0, L 1 → 1, L
B → B, R
1 1
![Page 69: Turing Machines - KeithSchwarz.comDesign Turing machines! Alternate problem session tomorrow night, 7-8PM, in Gates 100. The Turing Machine A Turing machine consists of three parts:](https://reader034.fdocuments.net/reader034/viewer/2022042923/5f7027f5297422064c2da53e/html5/thumbnails/69.jpg)
q5
q3
q3
q1
q1
q0
q0
q2
q4
q5
qacc q
rejstart q
rej
q2
q4
B → B, R
0 → B, R
1 → B, R
0 → 0, R1 → 1, R
B → B, L
0 → 0, R 1 → 1, R
B → B, L
B → B, R
B → B, R
qacc
1 → 1, R
0 → 0, R
0 → B, L
1 → B, L
0 → 0, L 1 → 1, L
B → B, R
1 1
![Page 70: Turing Machines - KeithSchwarz.comDesign Turing machines! Alternate problem session tomorrow night, 7-8PM, in Gates 100. The Turing Machine A Turing machine consists of three parts:](https://reader034.fdocuments.net/reader034/viewer/2022042923/5f7027f5297422064c2da53e/html5/thumbnails/70.jpg)
q5
q3
q3
q1
q1
q0
q0
q2
q4
q5
qacc q
rejstart q
rej
q2
q4
B → B, R
0 → B, R
1 → B, R
0 → 0, R1 → 1, R
B → B, L
0 → 0, R 1 → 1, R
B → B, L
B → B, R
B → B, R
qacc
1 → 1, R
0 → 0, R
0 → B, L
1 → B, L
0 → 0, L 1 → 1, L
B → B, R
1 1
![Page 71: Turing Machines - KeithSchwarz.comDesign Turing machines! Alternate problem session tomorrow night, 7-8PM, in Gates 100. The Turing Machine A Turing machine consists of three parts:](https://reader034.fdocuments.net/reader034/viewer/2022042923/5f7027f5297422064c2da53e/html5/thumbnails/71.jpg)
q5
q3
q3
q1
q1
q0
q0
q2
q4
q5
qacc q
rejstart q
rej
q2
q4
B → B, R
0 → B, R
1 → B, R
0 → 0, R1 → 1, R
B → B, L
0 → 0, R 1 → 1, R
B → B, L
B → B, R
B → B, R
qacc
1 → 1, R
0 → 0, R
0 → B, L
1 → B, L
0 → 0, L 1 → 1, L
B → B, R
1 1
![Page 72: Turing Machines - KeithSchwarz.comDesign Turing machines! Alternate problem session tomorrow night, 7-8PM, in Gates 100. The Turing Machine A Turing machine consists of three parts:](https://reader034.fdocuments.net/reader034/viewer/2022042923/5f7027f5297422064c2da53e/html5/thumbnails/72.jpg)
q5
q3
q3
q1
q1
q0
q0
q2
q4
q5
qacc q
rejstart q
rej
q2
q4
B → B, R
0 → B, R
1 → B, R
0 → 0, R1 → 1, R
B → B, L
0 → 0, R 1 → 1, R
B → B, L
B → B, R
B → B, R
qacc
1 → 1, R
0 → 0, R
0 → B, L
1 → B, L
0 → 0, L 1 → 1, L
B → B, R
1 1
![Page 73: Turing Machines - KeithSchwarz.comDesign Turing machines! Alternate problem session tomorrow night, 7-8PM, in Gates 100. The Turing Machine A Turing machine consists of three parts:](https://reader034.fdocuments.net/reader034/viewer/2022042923/5f7027f5297422064c2da53e/html5/thumbnails/73.jpg)
q5
q3
q3
q1
q1
q0
q0
q2
q4
q5
qacc q
rejstart q
rej
q2
q4
B → B, R
0 → B, R
1 → B, R
0 → 0, R1 → 1, R
B → B, L
0 → 0, R 1 → 1, R
B → B, L
B → B, R
B → B, R
qacc
1 → 1, R
0 → 0, R
0 → B, L
1 → B, L
0 → 0, L 1 → 1, L
B → B, R
1 1
![Page 74: Turing Machines - KeithSchwarz.comDesign Turing machines! Alternate problem session tomorrow night, 7-8PM, in Gates 100. The Turing Machine A Turing machine consists of three parts:](https://reader034.fdocuments.net/reader034/viewer/2022042923/5f7027f5297422064c2da53e/html5/thumbnails/74.jpg)
q5
q3
q3
q1
q1
q0
q0
q2
q4
q5
qacc q
rejstart q
rej
q2
q4
B → B, R
0 → B, R
1 → B, R
0 → 0, R1 → 1, R
B → B, L
0 → 0, R 1 → 1, R
B → B, L
B → B, R
B → B, R
qacc
1 → 1, R
0 → 0, R
0 → B, L
1 → B, L
0 → 0, L 1 → 1, L
B → B, R
1 1
![Page 75: Turing Machines - KeithSchwarz.comDesign Turing machines! Alternate problem session tomorrow night, 7-8PM, in Gates 100. The Turing Machine A Turing machine consists of three parts:](https://reader034.fdocuments.net/reader034/viewer/2022042923/5f7027f5297422064c2da53e/html5/thumbnails/75.jpg)
q5
q3
q3
q1
q1
q0
q0
q2
q4
q5
qacc q
rejstart q
rej
q2
q4
B → B, R
0 → B, R
1 → B, R
0 → 0, R1 → 1, R
B → B, L
0 → 0, R 1 → 1, R
B → B, L
B → B, R
B → B, R
qacc
1 → 1, R
0 → 0, R
0 → B, L
1 → B, L
0 → 0, L 1 → 1, L
B → B, R
1 1
![Page 76: Turing Machines - KeithSchwarz.comDesign Turing machines! Alternate problem session tomorrow night, 7-8PM, in Gates 100. The Turing Machine A Turing machine consists of three parts:](https://reader034.fdocuments.net/reader034/viewer/2022042923/5f7027f5297422064c2da53e/html5/thumbnails/76.jpg)
q0
q5
q5
q3
q3
q1
q1
q0
q2
q4
qacc q
rejstart q
rej
q2
q4
B → B, R
0 → B, R
1 → B, R
0 → 0, R1 → 1, R
B → B, L
0 → 0, R 1 → 1, R
B → B, L
B → B, R
B → B, R
qacc
1 → 1, R
0 → 0, R
0 → B, L
1 → B, L
0 → 0, L 1 → 1, L
B → B, R
1 1
![Page 77: Turing Machines - KeithSchwarz.comDesign Turing machines! Alternate problem session tomorrow night, 7-8PM, in Gates 100. The Turing Machine A Turing machine consists of three parts:](https://reader034.fdocuments.net/reader034/viewer/2022042923/5f7027f5297422064c2da53e/html5/thumbnails/77.jpg)
q0
q5
q5
q3
q3
q1
q1
q0
q2
q4
qacc q
rejstart q
rej
q2
q4
B → B, R
0 → B, R
1 → B, R
0 → 0, R1 → 1, R
B → B, L
0 → 0, R 1 → 1, R
B → B, L
B → B, R
B → B, R
qacc
1 → 1, R
0 → 0, R
0 → B, L
1 → B, L
0 → 0, L 1 → 1, L
B → B, R
1 1
![Page 78: Turing Machines - KeithSchwarz.comDesign Turing machines! Alternate problem session tomorrow night, 7-8PM, in Gates 100. The Turing Machine A Turing machine consists of three parts:](https://reader034.fdocuments.net/reader034/viewer/2022042923/5f7027f5297422064c2da53e/html5/thumbnails/78.jpg)
q0
q5
q5
q3
q3
q1
q1
q0
q2
q4
qacc q
rejstart q
rej
q2
q4
B → B, R
0 → B, R
1 → B, R
0 → 0, R1 → 1, R
B → B, L
0 → 0, R 1 → 1, R
B → B, L
B → B, R
B → B, R
qacc
1 → 1, R
0 → 0, R
0 → B, L
1 → B, L
0 → 0, L 1 → 1, L
B → B, R
1
![Page 79: Turing Machines - KeithSchwarz.comDesign Turing machines! Alternate problem session tomorrow night, 7-8PM, in Gates 100. The Turing Machine A Turing machine consists of three parts:](https://reader034.fdocuments.net/reader034/viewer/2022042923/5f7027f5297422064c2da53e/html5/thumbnails/79.jpg)
q0
q2
q0
q5
q5
q3
q3
q1
q1
q2
q4
qacc q
rejstart q
rej
q4
B → B, R
0 → B, R
1 → B, R
0 → 0, R1 → 1, R
B → B, L
0 → 0, R 1 → 1, R
B → B, L
B → B, R
B → B, R
qacc
1 → 1, R
0 → 0, R
0 → B, L
1 → B, L
0 → 0, L 1 → 1, L
B → B, R
1
![Page 80: Turing Machines - KeithSchwarz.comDesign Turing machines! Alternate problem session tomorrow night, 7-8PM, in Gates 100. The Turing Machine A Turing machine consists of three parts:](https://reader034.fdocuments.net/reader034/viewer/2022042923/5f7027f5297422064c2da53e/html5/thumbnails/80.jpg)
q0
q2
q0
q5
q5
q3
q3
q1
q1
q2
q4
qacc q
rejstart q
rej
q4
B → B, R
0 → B, R
1 → B, R
0 → 0, R1 → 1, R
B → B, L
0 → 0, R 1 → 1, R
B → B, L
B → B, R
B → B, R
qacc
1 → 1, R
0 → 0, R
0 → B, L
1 → B, L
0 → 0, L 1 → 1, L
B → B, R
1
![Page 81: Turing Machines - KeithSchwarz.comDesign Turing machines! Alternate problem session tomorrow night, 7-8PM, in Gates 100. The Turing Machine A Turing machine consists of three parts:](https://reader034.fdocuments.net/reader034/viewer/2022042923/5f7027f5297422064c2da53e/html5/thumbnails/81.jpg)
q0
q2
q0
q5
q5
q3
q3
q1
q1
q2
q4
qacc q
rejstart q
rej
q4
B → B, R
0 → B, R
1 → B, R
0 → 0, R1 → 1, R
B → B, L
0 → 0, R 1 → 1, R
B → B, L
B → B, R
B → B, R
qacc
1 → 1, R
0 → 0, R
0 → B, L
1 → B, L
0 → 0, L 1 → 1, L
B → B, R
1
![Page 82: Turing Machines - KeithSchwarz.comDesign Turing machines! Alternate problem session tomorrow night, 7-8PM, in Gates 100. The Turing Machine A Turing machine consists of three parts:](https://reader034.fdocuments.net/reader034/viewer/2022042923/5f7027f5297422064c2da53e/html5/thumbnails/82.jpg)
q0
q2
q0
q5
q5
q3
q3
q1
q1
q2
q4
qacc q
rejstart q
rej
q4
B → B, R
0 → B, R
1 → B, R
0 → 0, R1 → 1, R
B → B, L
0 → 0, R 1 → 1, R
B → B, L
B → B, R
B → B, R
qacc
1 → 1, R
0 → 0, R
0 → B, L
1 → B, L
0 → 0, L 1 → 1, L
B → B, R
1
![Page 83: Turing Machines - KeithSchwarz.comDesign Turing machines! Alternate problem session tomorrow night, 7-8PM, in Gates 100. The Turing Machine A Turing machine consists of three parts:](https://reader034.fdocuments.net/reader034/viewer/2022042923/5f7027f5297422064c2da53e/html5/thumbnails/83.jpg)
q0
q2
q0
q5
q5
q3
q3
q1
q1
q2
q4
qacc q
rejstart q
rej
q4
B → B, R
0 → B, R
1 → B, R
0 → 0, R1 → 1, R
B → B, L
0 → 0, R 1 → 1, R
B → B, L
B → B, R
B → B, R
qacc
1 → 1, R
0 → 0, R
0 → B, L
1 → B, L
0 → 0, L 1 → 1, L
B → B, R
1
![Page 84: Turing Machines - KeithSchwarz.comDesign Turing machines! Alternate problem session tomorrow night, 7-8PM, in Gates 100. The Turing Machine A Turing machine consists of three parts:](https://reader034.fdocuments.net/reader034/viewer/2022042923/5f7027f5297422064c2da53e/html5/thumbnails/84.jpg)
q0
q2
q0
q5
q5
q3
q3
q1
q1
q2
q4
qacc q
rejstart q
rej
q4
B → B, R
0 → B, R
1 → B, R
0 → 0, R1 → 1, R
B → B, L
0 → 0, R 1 → 1, R
B → B, L
B → B, R
B → B, R
qacc
1 → 1, R
0 → 0, R
0 → B, L
1 → B, L
0 → 0, L 1 → 1, L
B → B, R
1
![Page 85: Turing Machines - KeithSchwarz.comDesign Turing machines! Alternate problem session tomorrow night, 7-8PM, in Gates 100. The Turing Machine A Turing machine consists of three parts:](https://reader034.fdocuments.net/reader034/viewer/2022042923/5f7027f5297422064c2da53e/html5/thumbnails/85.jpg)
q2
q4
q0
q2
q0
q5
q5
q3
q3
q1
q1
q4
qacc q
rejstart q
rej
B → B, R
0 → B, R
1 → B, R
0 → 0, R1 → 1, R
B → B, L
0 → 0, R 1 → 1, R
B → B, L
B → B, R
B → B, R
qacc
1 → 1, R
0 → 0, R
0 → B, L
1 → B, L
0 → 0, L 1 → 1, L
B → B, R
1
![Page 86: Turing Machines - KeithSchwarz.comDesign Turing machines! Alternate problem session tomorrow night, 7-8PM, in Gates 100. The Turing Machine A Turing machine consists of three parts:](https://reader034.fdocuments.net/reader034/viewer/2022042923/5f7027f5297422064c2da53e/html5/thumbnails/86.jpg)
q2
q4
q0
q2
q0
q5
q5
q3
q3
q1
q1
q4
qacc q
rejstart q
rej
B → B, R
0 → B, R
1 → B, R
0 → 0, R1 → 1, R
B → B, L
0 → 0, R 1 → 1, R
B → B, L
B → B, R
B → B, R
qacc
1 → 1, R
0 → 0, R
0 → B, L
1 → B, L
0 → 0, L 1 → 1, L
B → B, R
1
![Page 87: Turing Machines - KeithSchwarz.comDesign Turing machines! Alternate problem session tomorrow night, 7-8PM, in Gates 100. The Turing Machine A Turing machine consists of three parts:](https://reader034.fdocuments.net/reader034/viewer/2022042923/5f7027f5297422064c2da53e/html5/thumbnails/87.jpg)
q2
q4
q0
q2
q0
q5
q5
q3
q3
q1
q1
q4
qacc q
rejstart q
rej
B → B, R
0 → B, R
1 → B, R
0 → 0, R1 → 1, R
B → B, L
0 → 0, R 1 → 1, R
B → B, L
B → B, R
B → B, R
qacc
1 → 1, R
0 → 0, R
0 → B, L
1 → B, L
0 → 0, L 1 → 1, L
B → B, R
![Page 88: Turing Machines - KeithSchwarz.comDesign Turing machines! Alternate problem session tomorrow night, 7-8PM, in Gates 100. The Turing Machine A Turing machine consists of three parts:](https://reader034.fdocuments.net/reader034/viewer/2022042923/5f7027f5297422064c2da53e/html5/thumbnails/88.jpg)
q5
q4
q2
q4
q0
q2
q0
q5
q3
q3
q1
q1
qacc q
rejstart q
rej
B → B, R
0 → B, R
1 → B, R
0 → 0, R1 → 1, R
B → B, L
0 → 0, R 1 → 1, R
B → B, L
B → B, R
B → B, R
qacc
1 → 1, R
0 → 0, R
0 → B, L
1 → B, L
0 → 0, L 1 → 1, L
B → B, R
![Page 89: Turing Machines - KeithSchwarz.comDesign Turing machines! Alternate problem session tomorrow night, 7-8PM, in Gates 100. The Turing Machine A Turing machine consists of three parts:](https://reader034.fdocuments.net/reader034/viewer/2022042923/5f7027f5297422064c2da53e/html5/thumbnails/89.jpg)
q5
q4
q2
q4
q0
q2
q0
q5
q3
q3
q1
q1
qacc q
rejstart q
rej
B → B, R
0 → B, R
1 → B, R
0 → 0, R1 → 1, R
B → B, L
0 → 0, R 1 → 1, R
B → B, L
B → B, R
B → B, R
qacc
1 → 1, R
0 → 0, R
0 → B, L
1 → B, L
0 → 0, L 1 → 1, L
B → B, R
![Page 90: Turing Machines - KeithSchwarz.comDesign Turing machines! Alternate problem session tomorrow night, 7-8PM, in Gates 100. The Turing Machine A Turing machine consists of three parts:](https://reader034.fdocuments.net/reader034/viewer/2022042923/5f7027f5297422064c2da53e/html5/thumbnails/90.jpg)
q5
q4
q2
q4
q0
q2
q0
q5
q3
q3
q1
q1
qacc q
rejstart q
rej
B → B, R
0 → B, R
1 → B, R
0 → 0, R1 → 1, R
B → B, L
0 → 0, R 1 → 1, R
B → B, L
B → B, R
B → B, R
qacc
1 → 1, R
0 → 0, R
0 → B, L
1 → B, L
0 → 0, L 1 → 1, L
B → B, R
![Page 91: Turing Machines - KeithSchwarz.comDesign Turing machines! Alternate problem session tomorrow night, 7-8PM, in Gates 100. The Turing Machine A Turing machine consists of three parts:](https://reader034.fdocuments.net/reader034/viewer/2022042923/5f7027f5297422064c2da53e/html5/thumbnails/91.jpg)
q0
q5
q5
q4
q2
q4
q0
q2
q3
q3
q1
q1
qacc q
rejstart q
rej
B → B, R
0 → B, R
1 → B, R
0 → 0, R1 → 1, R
B → B, L
0 → 0, R 1 → 1, R
B → B, L
B → B, R
B → B, R
qacc
1 → 1, R
0 → 0, R
0 → B, L
1 → B, L
0 → 0, L 1 → 1, L
B → B, R
![Page 92: Turing Machines - KeithSchwarz.comDesign Turing machines! Alternate problem session tomorrow night, 7-8PM, in Gates 100. The Turing Machine A Turing machine consists of three parts:](https://reader034.fdocuments.net/reader034/viewer/2022042923/5f7027f5297422064c2da53e/html5/thumbnails/92.jpg)
q0
q5
q5
q4
q2
q4
q0
q2
q3
q3
q1
q1
qacc q
rejstart q
rej
B → B, R
0 → B, R
1 → B, R
0 → 0, R1 → 1, R
B → B, L
0 → 0, R 1 → 1, R
B → B, L
B → B, R
B → B, R
qacc
1 → 1, R
0 → 0, R
0 → B, L
1 → B, L
0 → 0, L 1 → 1, L
B → B, R
![Page 93: Turing Machines - KeithSchwarz.comDesign Turing machines! Alternate problem session tomorrow night, 7-8PM, in Gates 100. The Turing Machine A Turing machine consists of three parts:](https://reader034.fdocuments.net/reader034/viewer/2022042923/5f7027f5297422064c2da53e/html5/thumbnails/93.jpg)
q0
q5
q5
q4
q2
q4
q0
q2
q3
q3
q1
q1
qacc q
rejstart q
rej
B → B, R
0 → B, R
1 → B, R
0 → 0, R1 → 1, R
B → B, L
0 → 0, R 1 → 1, R
B → B, L
B → B, R
B → B, R
qacc
1 → 1, R
0 → 0, R
0 → B, L
1 → B, L
0 → 0, L 1 → 1, L
B → B, R
![Page 94: Turing Machines - KeithSchwarz.comDesign Turing machines! Alternate problem session tomorrow night, 7-8PM, in Gates 100. The Turing Machine A Turing machine consists of three parts:](https://reader034.fdocuments.net/reader034/viewer/2022042923/5f7027f5297422064c2da53e/html5/thumbnails/94.jpg)
q0
q0
q5
q5
q4
q2
q4
q2
q3
q3
q1
q1
qacc q
rejstart q
rej
B → B, R
0 → B, R
1 → B, R
0 → 0, R1 → 1, R
B → B, L
0 → 0, R 1 → 1, R
B → B, L
B → B, R
B → B, R
qacc
1 → 1, R
0 → 0, R
0 → B, L
1 → B, L
0 → 0, L 1 → 1, L
B → B, R
![Page 95: Turing Machines - KeithSchwarz.comDesign Turing machines! Alternate problem session tomorrow night, 7-8PM, in Gates 100. The Turing Machine A Turing machine consists of three parts:](https://reader034.fdocuments.net/reader034/viewer/2022042923/5f7027f5297422064c2da53e/html5/thumbnails/95.jpg)
q0
q0
q5
q5
q4
q2
q4
q2
q3
q3
q1
q1
qacc q
rejstart q
rej
B → B, R
0 → B, R
1 → B, R
0 → 0, R1 → 1, R
B → B, L
0 → 0, R 1 → 1, R
B → B, L
B → B, R
B → B, R
qacc
1 → 1, R
0 → 0, R
0 → B, L
1 → B, L
0 → 0, L 1 → 1, L
B → B, R
![Page 96: Turing Machines - KeithSchwarz.comDesign Turing machines! Alternate problem session tomorrow night, 7-8PM, in Gates 100. The Turing Machine A Turing machine consists of three parts:](https://reader034.fdocuments.net/reader034/viewer/2022042923/5f7027f5297422064c2da53e/html5/thumbnails/96.jpg)
q0
q0
q1
q2
q3
q4
q5
qacc q
rej
q1
start qrej
q2
q3
q4
q5
B → B, R
0 → B, R
1 → B, R
0 → 0, R1 → 1, R
B → B, L
0 → 0, R 1 → 1, R
B → B, L
B → B, R
B → B, R
qacc
1 → 1, R
0 → 0, R
0 → B, L
1 → B, L
0 → 0, L 1 → 1, L
B → B, R
![Page 97: Turing Machines - KeithSchwarz.comDesign Turing machines! Alternate problem session tomorrow night, 7-8PM, in Gates 100. The Turing Machine A Turing machine consists of three parts:](https://reader034.fdocuments.net/reader034/viewer/2022042923/5f7027f5297422064c2da53e/html5/thumbnails/97.jpg)
q0
q0
q1
q2
q3
q4
q5
qacc q
rej
q1
start qrej
q2
q3
q4
q5
B → B, R
0 → B, R
1 → B, R
0 → 0, R1 → 1, R
B → B, L
0 → 0, R 1 → 1, R
B → B, L
B → B, R
B → B, R
qacc
1 → 1, R
0 → 0, R
0 → B, L
1 → B, L
0 → 0, L 1 → 1, L
B → B, R
1 0 0 0
![Page 98: Turing Machines - KeithSchwarz.comDesign Turing machines! Alternate problem session tomorrow night, 7-8PM, in Gates 100. The Turing Machine A Turing machine consists of three parts:](https://reader034.fdocuments.net/reader034/viewer/2022042923/5f7027f5297422064c2da53e/html5/thumbnails/98.jpg)
q0
q0
q1
q2
q3
q4
q5
qacc q
rej
q1
start qrej
q2
q3
q4
q5
B → B, R
0 → B, R
1 → B, R
0 → 0, R1 → 1, R
B → B, L
0 → 0, R 1 → 1, R
B → B, L
B → B, R
B → B, R
qacc
1 → 1, R
0 → 0, R
0 → B, L
1 → B, L
0 → 0, L 1 → 1, L
B → B, R
1 0 0 0
![Page 99: Turing Machines - KeithSchwarz.comDesign Turing machines! Alternate problem session tomorrow night, 7-8PM, in Gates 100. The Turing Machine A Turing machine consists of three parts:](https://reader034.fdocuments.net/reader034/viewer/2022042923/5f7027f5297422064c2da53e/html5/thumbnails/99.jpg)
q0
q0
q1
q2
q3
q4
q5
qacc q
rej
q1
start qrej
q2
q3
q4
q5
B → B, R
0 → B, R
1 → B, R
0 → 0, R1 → 1, R
B → B, L
0 → 0, R 1 → 1, R
B → B, L
B → B, R
B → B, R
qacc
1 → 1, R
0 → 0, R
0 → B, L
1 → B, L
0 → 0, L 1 → 1, L
B → B, R
1 0 0 0
![Page 100: Turing Machines - KeithSchwarz.comDesign Turing machines! Alternate problem session tomorrow night, 7-8PM, in Gates 100. The Turing Machine A Turing machine consists of three parts:](https://reader034.fdocuments.net/reader034/viewer/2022042923/5f7027f5297422064c2da53e/html5/thumbnails/100.jpg)
q0
q0
q1
q2
q3
q4
q5
qacc q
rej
q1
start qrej
q2
q3
q4
q5
B → B, R
0 → B, R
1 → B, R
0 → 0, R1 → 1, R
B → B, L
0 → 0, R 1 → 1, R
B → B, L
B → B, R
B → B, R
qacc
1 → 1, R
0 → 0, R
0 → B, L
1 → B, L
0 → 0, L 1 → 1, L
B → B, R
1 0 0 0
![Page 101: Turing Machines - KeithSchwarz.comDesign Turing machines! Alternate problem session tomorrow night, 7-8PM, in Gates 100. The Turing Machine A Turing machine consists of three parts:](https://reader034.fdocuments.net/reader034/viewer/2022042923/5f7027f5297422064c2da53e/html5/thumbnails/101.jpg)
q0
q0
q1
q2
q3
q4
q5
qacc q
rej
q1
start qrej
q2
q3
q4
q5
B → B, R
0 → B, R
1 → B, R
0 → 0, R1 → 1, R
B → B, L
0 → 0, R 1 → 1, R
B → B, L
B → B, R
B → B, R
qacc
1 → 1, R
0 → 0, R
0 → B, L
1 → B, L
0 → 0, L 1 → 1, L
B → B, R
0 0 0
![Page 102: Turing Machines - KeithSchwarz.comDesign Turing machines! Alternate problem session tomorrow night, 7-8PM, in Gates 100. The Turing Machine A Turing machine consists of three parts:](https://reader034.fdocuments.net/reader034/viewer/2022042923/5f7027f5297422064c2da53e/html5/thumbnails/102.jpg)
q0
q2
q0
q1
q2
q3
q4
q5
qacc q
rej
q1
start qrej
q3
q4
q5
B → B, R
0 → B, R
1 → B, R
0 → 0, R1 → 1, R
B → B, L
0 → 0, R 1 → 1, R
B → B, L
B → B, R
B → B, R
qacc
1 → 1, R
0 → 0, R
0 → B, L
1 → B, L
0 → 0, L 1 → 1, L
B → B, R
0 0 0
![Page 103: Turing Machines - KeithSchwarz.comDesign Turing machines! Alternate problem session tomorrow night, 7-8PM, in Gates 100. The Turing Machine A Turing machine consists of three parts:](https://reader034.fdocuments.net/reader034/viewer/2022042923/5f7027f5297422064c2da53e/html5/thumbnails/103.jpg)
q0
q2
q0
q1
q2
q3
q4
q5
qacc q
rej
q1
start qrej
q3
q4
q5
B → B, R
0 → B, R
1 → B, R
0 → 0, R1 → 1, R
B → B, L
0 → 0, R 1 → 1, R
B → B, L
B → B, R
B → B, R
qacc
1 → 1, R
0 → 0, R
0 → B, L
1 → B, L
0 → 0, L 1 → 1, L
B → B, R
0 0 0
![Page 104: Turing Machines - KeithSchwarz.comDesign Turing machines! Alternate problem session tomorrow night, 7-8PM, in Gates 100. The Turing Machine A Turing machine consists of three parts:](https://reader034.fdocuments.net/reader034/viewer/2022042923/5f7027f5297422064c2da53e/html5/thumbnails/104.jpg)
q0
q2
q0
q1
q2
q3
q4
q5
qacc q
rej
q1
start qrej
q3
q4
q5
B → B, R
0 → B, R
1 → B, R
0 → 0, R1 → 1, R
B → B, L
0 → 0, R 1 → 1, R
B → B, L
B → B, R
B → B, R
qacc
1 → 1, R
0 → 0, R
0 → B, L
1 → B, L
0 → 0, L 1 → 1, L
B → B, R
0 0 0
![Page 105: Turing Machines - KeithSchwarz.comDesign Turing machines! Alternate problem session tomorrow night, 7-8PM, in Gates 100. The Turing Machine A Turing machine consists of three parts:](https://reader034.fdocuments.net/reader034/viewer/2022042923/5f7027f5297422064c2da53e/html5/thumbnails/105.jpg)
q0
q2
q0
q1
q2
q3
q4
q5
qacc q
rej
q1
start qrej
q3
q4
q5
B → B, R
0 → B, R
1 → B, R
0 → 0, R1 → 1, R
B → B, L
0 → 0, R 1 → 1, R
B → B, L
B → B, R
B → B, R
qacc
1 → 1, R
0 → 0, R
0 → B, L
1 → B, L
0 → 0, L 1 → 1, L
B → B, R
0 0 0
![Page 106: Turing Machines - KeithSchwarz.comDesign Turing machines! Alternate problem session tomorrow night, 7-8PM, in Gates 100. The Turing Machine A Turing machine consists of three parts:](https://reader034.fdocuments.net/reader034/viewer/2022042923/5f7027f5297422064c2da53e/html5/thumbnails/106.jpg)
q0
q2
q0
q1
q2
q3
q4
q5
qacc q
rej
q1
start qrej
q3
q4
q5
B → B, R
0 → B, R
1 → B, R
0 → 0, R1 → 1, R
B → B, L
0 → 0, R 1 → 1, R
B → B, L
B → B, R
B → B, R
qacc
1 → 1, R
0 → 0, R
0 → B, L
1 → B, L
0 → 0, L 1 → 1, L
B → B, R
0 0 0
![Page 107: Turing Machines - KeithSchwarz.comDesign Turing machines! Alternate problem session tomorrow night, 7-8PM, in Gates 100. The Turing Machine A Turing machine consists of three parts:](https://reader034.fdocuments.net/reader034/viewer/2022042923/5f7027f5297422064c2da53e/html5/thumbnails/107.jpg)
q0
q2
q0
q1
q2
q3
q4
q5
qacc q
rej
q1
start qrej
q3
q4
q5
B → B, R
0 → B, R
1 → B, R
0 → 0, R1 → 1, R
B → B, L
0 → 0, R 1 → 1, R
B → B, L
B → B, R
B → B, R
qacc
1 → 1, R
0 → 0, R
0 → B, L
1 → B, L
0 → 0, L 1 → 1, L
B → B, R
0 0 0
![Page 108: Turing Machines - KeithSchwarz.comDesign Turing machines! Alternate problem session tomorrow night, 7-8PM, in Gates 100. The Turing Machine A Turing machine consists of three parts:](https://reader034.fdocuments.net/reader034/viewer/2022042923/5f7027f5297422064c2da53e/html5/thumbnails/108.jpg)
q0
q2
q0
q1
q2
q3
q4
q5
qacc q
rej
q1
start qrej
q3
q4
q5
B → B, R
0 → B, R
1 → B, R
0 → 0, R1 → 1, R
B → B, L
0 → 0, R 1 → 1, R
B → B, L
B → B, R
B → B, R
qacc
1 → 1, R
0 → 0, R
0 → B, L
1 → B, L
0 → 0, L 1 → 1, L
B → B, R
0 0 0
![Page 109: Turing Machines - KeithSchwarz.comDesign Turing machines! Alternate problem session tomorrow night, 7-8PM, in Gates 100. The Turing Machine A Turing machine consists of three parts:](https://reader034.fdocuments.net/reader034/viewer/2022042923/5f7027f5297422064c2da53e/html5/thumbnails/109.jpg)
q0
q2
q0
q1
q2
q3
q4
q5
qacc q
rej
q1
start qrej
q3
q4
q5
B → B, R
0 → B, R
1 → B, R
0 → 0, R1 → 1, R
B → B, L
0 → 0, R 1 → 1, R
B → B, L
B → B, R
B → B, R
qacc
1 → 1, R
0 → 0, R
0 → B, L
1 → B, L
0 → 0, L 1 → 1, L
B → B, R
0 0 0
![Page 110: Turing Machines - KeithSchwarz.comDesign Turing machines! Alternate problem session tomorrow night, 7-8PM, in Gates 100. The Turing Machine A Turing machine consists of three parts:](https://reader034.fdocuments.net/reader034/viewer/2022042923/5f7027f5297422064c2da53e/html5/thumbnails/110.jpg)
q2
q4
q0
q2
q0
q1
q3
q4
q5
qacc q
rej
q1
start qrej
q3
q5
B → B, R
0 → B, R
1 → B, R
0 → 0, R1 → 1, R
B → B, L
0 → 0, R 1 → 1, R
B → B, L
B → B, R
B → B, R
qacc
1 → 1, R
0 → 0, R
0 → B, L
1 → B, L
0 → 0, L 1 → 1, L
B → B, R
0 0 0
![Page 111: Turing Machines - KeithSchwarz.comDesign Turing machines! Alternate problem session tomorrow night, 7-8PM, in Gates 100. The Turing Machine A Turing machine consists of three parts:](https://reader034.fdocuments.net/reader034/viewer/2022042923/5f7027f5297422064c2da53e/html5/thumbnails/111.jpg)
q2
q4
q0
q2
q0
q1
q3
q4
q5
qacc q
rej
q1
start qrej
q3
q5
B → B, R
0 → B, R
1 → B, R
0 → 0, R1 → 1, R
B → B, L
0 → 0, R 1 → 1, R
B → B, L
B → B, R
B → B, R
qacc
1 → 1, R
0 → 0, R
0 → B, L
1 → B, L
0 → 0, L 1 → 1, L
B → B, R
0 0 0
![Page 112: Turing Machines - KeithSchwarz.comDesign Turing machines! Alternate problem session tomorrow night, 7-8PM, in Gates 100. The Turing Machine A Turing machine consists of three parts:](https://reader034.fdocuments.net/reader034/viewer/2022042923/5f7027f5297422064c2da53e/html5/thumbnails/112.jpg)
q2
q4
q0
q2
q0
q1
q3
q4
q5
qacc q
rej
q1
start qrej
q3
q5
B → B, R
0 → B, R
1 → B, R
0 → 0, R1 → 1, R
B → B, L
0 → 0, R 1 → 1, R
B → B, L
B → B, R
B → B, R
qacc
1 → 1, R
0 → 0, R
0 → B, L
1 → B, L
0 → 0, L 1 → 1, L
B → B, R
0 0 0
![Page 113: Turing Machines - KeithSchwarz.comDesign Turing machines! Alternate problem session tomorrow night, 7-8PM, in Gates 100. The Turing Machine A Turing machine consists of three parts:](https://reader034.fdocuments.net/reader034/viewer/2022042923/5f7027f5297422064c2da53e/html5/thumbnails/113.jpg)
q4
q2
q4
q0
q2
q0
q1
q3
q5
qacc q
rej
q1
start qrej
q3
q5
B → B, R
0 → B, R
1 → B, R
0 → 0, R1 → 1, R
B → B, L
0 → 0, R 1 → 1, R
B → B, L
B → B, R
B → B, R
qacc
1 → 1, R
0 → 0, R
0 → B, L
1 → B, L
0 → 0, L 1 → 1, L
B → B, R
0 0 0
![Page 114: Turing Machines - KeithSchwarz.comDesign Turing machines! Alternate problem session tomorrow night, 7-8PM, in Gates 100. The Turing Machine A Turing machine consists of three parts:](https://reader034.fdocuments.net/reader034/viewer/2022042923/5f7027f5297422064c2da53e/html5/thumbnails/114.jpg)
q4
q2
q4
q0
q2
q0
q1
q3
q5
qacc q
rej
q1
start qrej
q3
q5
B → B, R
0 → B, R
1 → B, R
0 → 0, R1 → 1, R
B → B, L
0 → 0, R 1 → 1, R
B → B, L
B → B, R
B → B, R
qacc
1 → 1, R
0 → 0, R
0 → B, L
1 → B, L
0 → 0, L 1 → 1, L
B → B, R
0 0 0
![Page 115: Turing Machines - KeithSchwarz.comDesign Turing machines! Alternate problem session tomorrow night, 7-8PM, in Gates 100. The Turing Machine A Turing machine consists of three parts:](https://reader034.fdocuments.net/reader034/viewer/2022042923/5f7027f5297422064c2da53e/html5/thumbnails/115.jpg)
0 1 Bq
0
q1
q2
q3
q4
q5
A More Sane Representation
0 1 Bq
0
q1
q2
q3
q4
q5
B R q1 B R q
2 B R qacc
0 R q1 1 R q
1 B L q3
0 R q2 1 R q
2 B L q4
B L q5 1 R q
rej B R qacc
0 R qrej B R q
accB L q5
0 L q5 1 L q
5 B R q0
![Page 116: Turing Machines - KeithSchwarz.comDesign Turing machines! Alternate problem session tomorrow night, 7-8PM, in Gates 100. The Turing Machine A Turing machine consists of three parts:](https://reader034.fdocuments.net/reader034/viewer/2022042923/5f7027f5297422064c2da53e/html5/thumbnails/116.jpg)
0 1 Bq
0
q1
q2
q3
q4
q5
A More Sane Representation
0 1 Bq
0
q1
q2
q3
q4
q5
B R q1 B R q
2 B R qacc
0 R q1 1 R q
1 B L q3
0 R q2 1 R q
2 B L q4
B L q5 1 R q
rej B R qacc
0 R qrej B R q
accB L q5
0 L q5 1 L q
5 B R q0
0 0 1 0 0
![Page 117: Turing Machines - KeithSchwarz.comDesign Turing machines! Alternate problem session tomorrow night, 7-8PM, in Gates 100. The Turing Machine A Turing machine consists of three parts:](https://reader034.fdocuments.net/reader034/viewer/2022042923/5f7027f5297422064c2da53e/html5/thumbnails/117.jpg)
q0
0 1 Bq
0
q1
q2
q3
q4
q5
A More Sane Representation
0 1 B
q1
q2
q3
q4
q5
B R q1 B R q
2 B R qacc
0 R q1 1 R q
1 B L q3
0 R q2 1 R q
2 B L q4
B L q5 1 R q
rej B R qacc
0 R qrej B R q
accB L q5
0 L q5 1 L q
5 B R q0
0 0 1 0 0
![Page 118: Turing Machines - KeithSchwarz.comDesign Turing machines! Alternate problem session tomorrow night, 7-8PM, in Gates 100. The Turing Machine A Turing machine consists of three parts:](https://reader034.fdocuments.net/reader034/viewer/2022042923/5f7027f5297422064c2da53e/html5/thumbnails/118.jpg)
q0
0 1 Bq
0
q1
q2
q3
q4
q5
A More Sane Representation
0 1 B
q1
q2
q3
q4
q5
B R q1 B R q
2 B R qacc
0 R q1 1 R q
1 B L q3
0 R q2 1 R q
2 B L q4
B L q5 1 R q
rej B R qacc
0 R qrej B R q
accB L q5
0 L q5 1 L q
5 B R q0
0 0 1 0 0
![Page 119: Turing Machines - KeithSchwarz.comDesign Turing machines! Alternate problem session tomorrow night, 7-8PM, in Gates 100. The Turing Machine A Turing machine consists of three parts:](https://reader034.fdocuments.net/reader034/viewer/2022042923/5f7027f5297422064c2da53e/html5/thumbnails/119.jpg)
0q
0
0 1 Bq
0
q1
q2
q3
q4
q5
A More Sane Representation
1 B
q1
q2
q3
q4
q5
B R q1 B R q
2 B R qacc
0 R q1 1 R q
1 B L q3
0 R q2 1 R q
2 B L q4
B L q5 1 R q
rej B R qacc
0 R qrej B R q
accB L q5
0 L q5 1 L q
5 B R q0
0 0 1 0 0
![Page 120: Turing Machines - KeithSchwarz.comDesign Turing machines! Alternate problem session tomorrow night, 7-8PM, in Gates 100. The Turing Machine A Turing machine consists of three parts:](https://reader034.fdocuments.net/reader034/viewer/2022042923/5f7027f5297422064c2da53e/html5/thumbnails/120.jpg)
q1
0q
0
0q
0
1 B
q1
q2
q3
q4
q5
A More Sane Representation
1 B
q2
q3
q4
q5
B R q1 B R q
2 B R qacc
0 R q1 1 R q
1 B L q3
0 R q2 1 R q
2 B L q4
B L q5 1 R q
rej B R qacc
0 R qrej B R q
accB L q5
0 L q5 1 L q
5 B R q0
0 1 0 0
![Page 121: Turing Machines - KeithSchwarz.comDesign Turing machines! Alternate problem session tomorrow night, 7-8PM, in Gates 100. The Turing Machine A Turing machine consists of three parts:](https://reader034.fdocuments.net/reader034/viewer/2022042923/5f7027f5297422064c2da53e/html5/thumbnails/121.jpg)
0
q1
0q
0q
0
1 B
q1
q2
q3
q4
q5
A More Sane Representation
1 B
q2
q3
q4
q5
B R q1 B R q
2 B R qacc
0 R q1 1 R q
1 B L q3
0 R q2 1 R q
2 B L q4
B L q5 1 R q
rej B R qacc
0 R qrej B R q
accB L q5
0 L q5 1 L q
5 B R q0
0 1 0 0
![Page 122: Turing Machines - KeithSchwarz.comDesign Turing machines! Alternate problem session tomorrow night, 7-8PM, in Gates 100. The Turing Machine A Turing machine consists of three parts:](https://reader034.fdocuments.net/reader034/viewer/2022042923/5f7027f5297422064c2da53e/html5/thumbnails/122.jpg)
q1
0
q1
0q
0q
0
1 B
q2
q3
q4
q5
A More Sane Representation
1 B
q2
q3
q4
q5
B R q1 B R q
2 B R qacc
0 R q1 1 R q
1 B L q3
0 R q2 1 R q
2 B L q4
B L q5 1 R q
rej B R qacc
0 R qrej B R q
accB L q5
0 L q5 1 L q
5 B R q0
0 1 0 0
![Page 123: Turing Machines - KeithSchwarz.comDesign Turing machines! Alternate problem session tomorrow night, 7-8PM, in Gates 100. The Turing Machine A Turing machine consists of three parts:](https://reader034.fdocuments.net/reader034/viewer/2022042923/5f7027f5297422064c2da53e/html5/thumbnails/123.jpg)
q1
10
q1
0q
0q
0
1 B
q2
q3
q4
q5
A More Sane Representation
B
q2
q3
q4
q5
B R q1 B R q
2 B R qacc
0 R q1 1 R q
1 B L q3
0 R q2 1 R q
2 B L q4
B L q5 1 R q
rej B R qacc
0 R qrej B R q
accB L q5
0 L q5 1 L q
5 B R q0
0 1 0 0
![Page 124: Turing Machines - KeithSchwarz.comDesign Turing machines! Alternate problem session tomorrow night, 7-8PM, in Gates 100. The Turing Machine A Turing machine consists of three parts:](https://reader034.fdocuments.net/reader034/viewer/2022042923/5f7027f5297422064c2da53e/html5/thumbnails/124.jpg)
q1
q1
1100q
0q
0
B
q2
q3
q4
q5
A More Sane Representation
B
q2
q3
q4
q5
B R q1 B R q
2 B R qacc
0 R q1 1 R q
1 B L q3
0 R q2 1 R q
2 B L q4
B L q5 1 R q
rej B R qacc
0 R qrej B R q
accB L q5
0 L q5 1 L q
5 B R q0
0 1 0 0
![Page 125: Turing Machines - KeithSchwarz.comDesign Turing machines! Alternate problem session tomorrow night, 7-8PM, in Gates 100. The Turing Machine A Turing machine consists of three parts:](https://reader034.fdocuments.net/reader034/viewer/2022042923/5f7027f5297422064c2da53e/html5/thumbnails/125.jpg)
q1
0
q1
110q
0q
0
B
q2
q3
q4
q5
A More Sane Representation
B
q2
q3
q4
q5
B R q1 B R q
2 B R qacc
0 R q1 1 R q
1 B L q3
0 R q2 1 R q
2 B L q4
B L q5 1 R q
rej B R qacc
0 R qrej B R q
accB L q5
0 L q5 1 L q
5 B R q0
0 1 0 0
![Page 126: Turing Machines - KeithSchwarz.comDesign Turing machines! Alternate problem session tomorrow night, 7-8PM, in Gates 100. The Turing Machine A Turing machine consists of three parts:](https://reader034.fdocuments.net/reader034/viewer/2022042923/5f7027f5297422064c2da53e/html5/thumbnails/126.jpg)
0
q1
0
q1
11q
0q
0
B
q2
q3
q4
q5
A More Sane Representation
B
q2
q3
q4
q5
B R q1 B R q
2 B R qacc
0 R q1 1 R q
1 B L q3
0 R q2 1 R q
2 B L q4
B L q5 1 R q
rej B R qacc
0 R qrej B R q
accB L q5
0 L q5 1 L q
5 B R q0
0 1 0 0
![Page 127: Turing Machines - KeithSchwarz.comDesign Turing machines! Alternate problem session tomorrow night, 7-8PM, in Gates 100. The Turing Machine A Turing machine consists of three parts:](https://reader034.fdocuments.net/reader034/viewer/2022042923/5f7027f5297422064c2da53e/html5/thumbnails/127.jpg)
00
q1
q1
11q
0q
0
B
q2
q3
q4
q5
A More Sane Representation
B
q2
q3
q4
q5
B R q1 B R q
2 B R qacc
0 R q1 1 R q
1 B L q3
0 R q2 1 R q
2 B L q4
B L q5 1 R q
rej B R qacc
0 R qrej B R q
accB L q5
0 L q5 1 L q
5 B R q0
0 1 0 0
![Page 128: Turing Machines - KeithSchwarz.comDesign Turing machines! Alternate problem session tomorrow night, 7-8PM, in Gates 100. The Turing Machine A Turing machine consists of three parts:](https://reader034.fdocuments.net/reader034/viewer/2022042923/5f7027f5297422064c2da53e/html5/thumbnails/128.jpg)
00
q1
q1
11q
0q
0
B
q2
q3
q4
q5
A More Sane Representation
B
q2
q3
q4
q5
B R q1 B R q
2 B R qacc
0 R q1 1 R q
1 B L q3
0 R q2 1 R q
2 B L q4
B L q5 1 R q
rej B R qacc
0 R qrej B R q
accB L q5
0 L q5 1 L q
5 B R q0
0 1 0 0
![Page 129: Turing Machines - KeithSchwarz.comDesign Turing machines! Alternate problem session tomorrow night, 7-8PM, in Gates 100. The Turing Machine A Turing machine consists of three parts:](https://reader034.fdocuments.net/reader034/viewer/2022042923/5f7027f5297422064c2da53e/html5/thumbnails/129.jpg)
B00
q1
q1
11q
0q
0
B
q2
q3
q4
q5
A More Sane Representation
q2
q3
q4
q5
B R q1 B R q
2 B R qacc
0 R q1 1 R q
1 B L q3
0 R q2 1 R q
2 B L q4
B L q5 1 R q
rej B R qacc
0 R qrej B R q
accB L q5
0 L q5 1 L q
5 B R q0
0 1 0 0
![Page 130: Turing Machines - KeithSchwarz.comDesign Turing machines! Alternate problem session tomorrow night, 7-8PM, in Gates 100. The Turing Machine A Turing machine consists of three parts:](https://reader034.fdocuments.net/reader034/viewer/2022042923/5f7027f5297422064c2da53e/html5/thumbnails/130.jpg)
q1
B
q3
B00
q1
11q
0q
0
q2
q3
q4
q5
A More Sane Representation
q2
q4
q5
B R q1 B R q
2 B R qacc
0 R q1 1 R q
1 B L q3
0 R q2 1 R q
2 B L q4
B L q5 1 R q
rej B R qacc
0 R qrej B R q
accB L q5
0 L q5 1 L q
5 B R q0
0 1 0 0
![Page 131: Turing Machines - KeithSchwarz.comDesign Turing machines! Alternate problem session tomorrow night, 7-8PM, in Gates 100. The Turing Machine A Turing machine consists of three parts:](https://reader034.fdocuments.net/reader034/viewer/2022042923/5f7027f5297422064c2da53e/html5/thumbnails/131.jpg)
0
q1
B
q3
B0
q1
11q
0q
0
q2
q3
q4
q5
A More Sane Representation
q2
q4
q5
B R q1 B R q
2 B R qacc
0 R q1 1 R q
1 B L q3
0 R q2 1 R q
2 B L q4
B L q5 1 R q
rej B R qacc
0 R qrej B R q
accB L q5
0 L q5 1 L q
5 B R q0
0 1 0 0
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0
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A More Sane Representation
q2
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B R q1 B R q
2 B R qacc
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00
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A More Sane Representation
q2
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B R q1 B R q
2 B R qacc
0 R q1 1 R q
1 B L q3
0 R q2 1 R q
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00
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A More Sane Representation
q2
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B R q1 B R q
2 B R qacc
0 R q1 1 R q
1 B L q3
0 R q2 1 R q
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100
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A More Sane Representation
q2
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B R q1 B R q
2 B R qacc
0 R q1 1 R q
1 B L q3
0 R q2 1 R q
2 B L q4
B L q5 1 R q
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1100
q3
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B
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A More Sane Representation
q2
q4
B R q1 B R q
2 B R qacc
0 R q1 1 R q
1 B L q3
0 R q2 1 R q
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0 110
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B
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A More Sane Representation
q2
q4
B R q1 B R q
2 B R qacc
0 R q1 1 R q
1 B L q3
0 R q2 1 R q
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00 11
q3
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B
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A More Sane Representation
q2
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B R q1 B R q
2 B R qacc
0 R q1 1 R q
1 B L q3
0 R q2 1 R q
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B00 11
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q5
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B
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A More Sane Representation
q2
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B R q1 B R q
2 B R qacc
0 R q1 1 R q
1 B L q3
0 R q2 1 R q
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Bq
0
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A More Sane Representation
q2
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B R q1 B R q
2 B R qacc
0 R q1 1 R q
1 B L q3
0 R q2 1 R q
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0 Bq
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A More Sane Representation
q2
q4
B R q1 B R q
2 B R qacc
0 R q1 1 R q
1 B L q3
0 R q2 1 R q
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0
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0 Bq
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B11
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A More Sane Representation
q2
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B R q1 B R q
2 B R qacc
0 R q1 1 R q
1 B L q3
0 R q2 1 R q
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10
q1
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0 Bq
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B1
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A More Sane Representation
q2
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B R q1 B R q
2 B R qacc
0 R q1 1 R q
1 B L q3
0 R q2 1 R q
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110
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0 Bq
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A More Sane Representation
q2
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B R q1 B R q
2 B R qacc
0 R q1 1 R q
1 B L q3
0 R q2 1 R q
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0 110
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0
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B
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A More Sane Representation
q2
q4
B R q1 B R q
2 B R qacc
0 R q1 1 R q
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0 R q2 1 R q
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00 11
q1
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B
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A More Sane Representation
q2
q4
B R q1 B R q
2 B R qacc
0 R q1 1 R q
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0 R q2 1 R q
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B00 11
q1
q0
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0
q5
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A More Sane Representation
q2
q4
B R q1 B R q
2 B R qacc
0 R q1 1 R q
1 B L q3
0 R q2 1 R q
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B
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A More Sane Representation
q2
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B R q1 B R q
2 B R qacc
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0 B
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A More Sane Representation
q2
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B R q1 B R q
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0
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0 B
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A More Sane Representation
q2
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B R q1 B R q
2 B R qacc
0 R q1 1 R q
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10
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0 B
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A More Sane Representation
q2
q4
B R q1 B R q
2 B R qacc
0 R q1 1 R q
1 B L q3
0 R q2 1 R q
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110
q3
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0 B
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B
q1
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A More Sane Representation
q2
q4
B R q1 B R q
2 B R qacc
0 R q1 1 R q
1 B L q3
0 R q2 1 R q
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B110
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0 B
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A More Sane Representation
q2
q4
B R q1 B R q
2 B R qacc
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Bq
0
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B110
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0
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q1
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A More Sane Representation
q2
q4
B R q1 B R q
2 B R qacc
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1 Bq
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B10
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0
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A More Sane Representation
q2
q4
B R q1 B R q
2 B R qacc
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1q
0
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0
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0
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A More Sane Representation
q4
B R q1 B R q
2 B R qacc
0 R q1 1 R q
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B1q
0
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0
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0
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0
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A More Sane Representation
q4
B R q1 B R q
2 B R qacc
0 R q1 1 R q
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B
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B1q
0
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0
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0
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A More Sane Representation
B R q1 B R q
2 B R qacc
0 R q1 1 R q
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0 R q2 1 R q
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BB
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0
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0
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0
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0
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A More Sane Representation
B R q1 B R q
2 B R qacc
0 R q1 1 R q
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0 R q2 1 R q
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B
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B
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0
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0
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0
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A More Sane Representation
B R q1 B R q
2 B R qacc
0 R q1 1 R q
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0 R q2 1 R q
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B
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B
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0
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0
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A More Sane Representation
B R q1 B R q
2 B R qacc
0 R q1 1 R q
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0
q3
q5
0
q1
q3
q1
A More Sane Representation
B R q1 B R q
2 B R qacc
0 R q1 1 R q
1 B L q3
0 R q2 1 R q
2 B L q4
B L q5 1 R q
rej B R qacc
0 R qrej B R q
accB L q5
0 L q5 1 L q
5 B R q0
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Accept
Reject
Reject Accept
Accept
B
q4
B
q2
q4
1q
0
q2
1q
0
q5
0
q3
q5
0
q1
q3
q1
A More Sane Representation
B R q1 B R q
2
0 R q1 1 R q
1 B L q3
0 R q2 1 R q
2 B L q4
B L q5 Reject
B L q5
0 L q5 1 L q
5 B R q0
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Accept
Reject
Reject Accept
Accept
B
q4
B
q2
q4
1q
0
q2
1q
0
q5
0
q3
q5
0
q1
q3
q1
A More Sane Representation
B R q1 B R q
2
0 R q1 1 R q
1 B L q3
0 R q2 1 R q
2 B L q4
B L q5 Reject
B L q5
0 L q5 1 L q
5 B R q0
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Turing Machines, Formally
● A Turing machine is an 8-tuple (Q, Σ, Γ, δ, q0, q
acc, q
rej, B),
where
Q is a finite set of states,
Σ is a finite input alphabet,
Γ is a finite tape alphabet, with Σ Γ,⊆
δ : Q × Γ → Q × Γ × {R, L} is the transition function,
q0 Q is the ∈ start state,
qacc
Q is the ∈ accept state,
qrej
Q, q∈rej
≠ qacc
, is the reject state, and
B Γ is the ∈ blank symbol.
![Page 166: Turing Machines - KeithSchwarz.comDesign Turing machines! Alternate problem session tomorrow night, 7-8PM, in Gates 100. The Turing Machine A Turing machine consists of three parts:](https://reader034.fdocuments.net/reader034/viewer/2022042923/5f7027f5297422064c2da53e/html5/thumbnails/166.jpg)
Turing Machines, Formally
● A Turing machine is an 8-tuple (Q, Σ, Γ, δ, q0, q
acc, q
rej, B),
where● Q is a finite set of states,
Σ is a finite input alphabet,
Γ is a finite tape alphabet, with Σ Γ,⊆
δ : Q × Γ → Q × Γ × {R, L} is the transition function,
q0 Q is the ∈ start state,
qacc
Q is the ∈ accept state,
qrej
Q, q∈rej
≠ qacc
, is the reject state, and
B Γ is the ∈ blank symbol.
![Page 167: Turing Machines - KeithSchwarz.comDesign Turing machines! Alternate problem session tomorrow night, 7-8PM, in Gates 100. The Turing Machine A Turing machine consists of three parts:](https://reader034.fdocuments.net/reader034/viewer/2022042923/5f7027f5297422064c2da53e/html5/thumbnails/167.jpg)
Turing Machines, Formally
● A Turing machine is an 8-tuple (Q, Σ, Γ, δ, q0, q
acc, q
rej, B),
where● Q is a finite set of states,● Σ is a finite input alphabet,
Γ is a finite tape alphabet, with Σ Γ,⊆
δ : Q × Γ → Q × Γ × {R, L} is the transition function,
q0 Q is the ∈ start state,
qacc
Q is the ∈ accept state,
qrej
Q, q∈rej
≠ qacc
, is the reject state, and
B Γ is the ∈ blank symbol.
![Page 168: Turing Machines - KeithSchwarz.comDesign Turing machines! Alternate problem session tomorrow night, 7-8PM, in Gates 100. The Turing Machine A Turing machine consists of three parts:](https://reader034.fdocuments.net/reader034/viewer/2022042923/5f7027f5297422064c2da53e/html5/thumbnails/168.jpg)
Turing Machines, Formally
● A Turing machine is an 8-tuple (Q, Σ, Γ, δ, q0, q
acc, q
rej, B),
where● Q is a finite set of states,● Σ is a finite input alphabet,● Γ is a finite tape alphabet, with Σ Γ⊆ ,
δ : Q × Γ → Q × Γ × {R, L} is the transition function,
q0 Q is the ∈ start state,
qacc
Q is the ∈ accept state,
qrej
Q, q∈rej
≠ qacc
, is the reject state, and
B Γ is the ∈ blank symbol.
![Page 169: Turing Machines - KeithSchwarz.comDesign Turing machines! Alternate problem session tomorrow night, 7-8PM, in Gates 100. The Turing Machine A Turing machine consists of three parts:](https://reader034.fdocuments.net/reader034/viewer/2022042923/5f7027f5297422064c2da53e/html5/thumbnails/169.jpg)
Turing Machines, Formally
● A Turing machine is an 8-tuple (Q, Σ, Γ, δ, q0, q
acc, q
rej, B),
where● Q is a finite set of states,● Σ is a finite input alphabet,● Γ is a finite tape alphabet, with Σ Γ⊆ ,
δ : Q × Γ → Q × Γ × {R, L} is the transition function,
q0 Q is the ∈ start state,
qacc
Q is the ∈ accept state,
qrej
Q, q∈rej
≠ qacc
, is the reject state, and
B Γ is the ∈ blank symbol.
Why must ?Σ ⊆ Γ
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Turing Machines, Formally
● A Turing machine is an 8-tuple (Q, Σ, Γ, δ, q0, q
acc, q
rej, B),
where● Q is a finite set of states,● Σ is a finite input alphabet,● Γ is a finite tape alphabet, with Σ Γ,⊆● δ : Q × Γ → Q × Γ × {R, L} is the transition function,
q0 Q is the ∈ start state,
qacc
Q is the ∈ accept state,
qrej
Q, q∈rej
≠ qacc
, is the reject state, and
B Γ is the ∈ blank symbol.
![Page 171: Turing Machines - KeithSchwarz.comDesign Turing machines! Alternate problem session tomorrow night, 7-8PM, in Gates 100. The Turing Machine A Turing machine consists of three parts:](https://reader034.fdocuments.net/reader034/viewer/2022042923/5f7027f5297422064c2da53e/html5/thumbnails/171.jpg)
Turing Machines, Formally
● A Turing machine is an 8-tuple (Q, Σ, Γ, δ, q0, q
acc, q
rej, B),
where● Q is a finite set of states,● Σ is a finite input alphabet,● Γ is a finite tape alphabet, with Σ Γ,⊆● δ : Q × Γ → Q × Γ × {R, L} is the transition function,
q0 Q is the ∈ start state,
qacc
Q is the ∈ accept state,
qrej
Q, q∈rej
≠ qacc
, is the reject state, and
B Γ is the ∈ blank symbol.
Each transition is based on the current tape symbol and state. Each transition maps to a new
state, a new tape symbol, and a direction (either left or right).
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Turing Machines, Formally
● A Turing machine is an 8-tuple (Q, Σ, Γ, δ, q0, q
acc, q
rej, B),
where● Q is a finite set of states,● Σ is a finite input alphabet,● Γ is a finite tape alphabet, with Σ Γ,⊆● δ : Q × Γ → Q × Γ × {R, L} is the transition function,
● q0 Q is the ∈ start state,
qacc
Q is the ∈ accept state,
qrej
Q, q∈rej
≠ qacc
, is the reject state, and
B Γ is the ∈ blank symbol.
![Page 173: Turing Machines - KeithSchwarz.comDesign Turing machines! Alternate problem session tomorrow night, 7-8PM, in Gates 100. The Turing Machine A Turing machine consists of three parts:](https://reader034.fdocuments.net/reader034/viewer/2022042923/5f7027f5297422064c2da53e/html5/thumbnails/173.jpg)
Turing Machines, Formally
● A Turing machine is an 8-tuple (Q, Σ, Γ, δ, q0, q
acc, q
rej, B),
where● Q is a finite set of states,● Σ is a finite input alphabet,● Γ is a finite tape alphabet, with Σ Γ,⊆● δ : Q × Γ → Q × Γ × {R, L} is the transition function,
● q0 Q is the ∈ start state,
● qacc
Q is the ∈ accept state,
qrej
Q, q∈rej
≠ qacc
, is the reject state, and
B Γ is the ∈ blank symbol.
![Page 174: Turing Machines - KeithSchwarz.comDesign Turing machines! Alternate problem session tomorrow night, 7-8PM, in Gates 100. The Turing Machine A Turing machine consists of three parts:](https://reader034.fdocuments.net/reader034/viewer/2022042923/5f7027f5297422064c2da53e/html5/thumbnails/174.jpg)
Turing Machines, Formally
● A Turing machine is an 8-tuple (Q, Σ, Γ, δ, q0, q
acc, q
rej, B),
where● Q is a finite set of states,● Σ is a finite input alphabet,● Γ is a finite tape alphabet, with Σ Γ,⊆● δ : Q × Γ → Q × Γ × {R, L} is the transition function,
● q0 Q is the ∈ start state,
● qacc
Q is the ∈ accept state,
● qrej
Q, q∈rej
≠ qacc
, is the reject state, and
B Γ is the ∈ blank symbol.
![Page 175: Turing Machines - KeithSchwarz.comDesign Turing machines! Alternate problem session tomorrow night, 7-8PM, in Gates 100. The Turing Machine A Turing machine consists of three parts:](https://reader034.fdocuments.net/reader034/viewer/2022042923/5f7027f5297422064c2da53e/html5/thumbnails/175.jpg)
Turing Machines, Formally
● A Turing machine is an 8-tuple (Q, Σ, Γ, δ, q0, q
acc, q
rej, B),
where● Q is a finite set of states,● Σ is a finite input alphabet,● Γ is a finite tape alphabet, with Σ Γ,⊆● δ : Q × Γ → Q × Γ × {R, L} is the transition function,
● q0 Q is the ∈ start state,
● qacc
Q is the ∈ accept state,
● qrej
Q, q∈rej
≠ qacc
, is the reject state, and
● B Γ – Σ is the ∈ blank symbol.
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The Language of a Turing Machine
● The language of a TM M is the set
L(M) = { w | M enters qacc when run on w }
● If there is a TM M such that L(M) = L, we say that L is Turing-recognizable.● These languages are sometimes called recursively
enumerable.
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Describing POWER2
● Recall: POWER2 is the language
● In Problem Set 6, you saw that POWER2 is not
regular.● In Problem Set 7, you'll prove that POWER2 is
not context-free.● However, POWER2 is Turing-recognizable.● How would we build a TM to recognize POWER2?
{12n | n∈ℕ }
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Describing Turing Machines
● Specify TM explicitly using the transition table.● Incontrovertibly shows there's a TM for a problem.● Almost impossible to decipher.
● Specify TM's tape operations in natural language.● Describes what the tape head needs to do rather than how it's
done.● Might miss an important edge case.
● Specify TM's entire behavior at a high level.● Provides a clear description of how the TM tackles the
problem.● Almost impossible to translate into a machine.
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POWER2 Intuition
1 1 1 1 1 1 1 1
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POWER2 Intuition
1 1 1 1 1 1 1 1
x 1 x 1 x 1 x 1
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POWER2 Intuition
1 1 1 1 1 1 1 1
x 1 x 1 x 1 x 1
x x x 1 x x x 1
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POWER2 Intuition
1 1 1 1 1 1 1 1
x 1 x 1 x 1 x 1
x x x 1 x x x 1
x x x x x x x 1
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A Very High-Level Description
● M = “On input w:● If w = ε, reject.● Otherwise, repeatedly cross out pairs of 1s in w
until the number of 1s is odd.● If there is exactly one 1 left, accept.● Otherwise, reject.”
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A Tape-Operation Description
● M = “On input w:● If the initial cell contains a blank, reject.● If there is exactly one 1 in the input, accept.● Otherwise:
– Scan across the tape crossing off pairs of 1s. If an odd number of 1s is found, reject.
– Scan back to the start of the tape.– If we see exactly one 1 when scanning back, accept.”
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q0
q0
BB11 xx
q6
q6
rejectq
1
q2
q3
q4
q5
A Low-Level Description
q1
q2
q3
q4
q5
B R q1 reject
1 R q2 reject accept
x R q3 x R q
2 B L q4
1 R q2 x R q
3 reject
1 L q5 rejectx L q
4
1 L q6 x L q
5 accept
1 L q6 x L q
6 B R q2
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q0
q0
BB11 xx
q6
q6
rejectq
1
q2
q3
q4
q5
A Low-Level Description
q1
q2
q3
q4
q5
B R q1 reject
1 R q2 reject accept
x R q3 x R q
2 B L q4
1 R q2 x R q
3 reject
1 L q5 rejectx L q
4
1 L q6 x L q
5 accept
1 1 1 1 1 1 1 1
1 L q6 x L q
6 B R q2
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q0
q0
BB11 xx
q6
q6
rejectq
1
q2
q3
q4
q5
A Low-Level Description
q1
q2
q3
q4
q5
B R q1 reject
1 R q2 reject accept
x R q3 x R q
2 B L q4
1 R q2 x R q
3 reject
1 L q5 rejectx L q
4
1 L q6 x L q
5 accept
1 1 1 1 1 1 1 1
1 L q6 x L q
6 B R q2
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q0
q0
BB11 xx
q6
q6
rejectq
1
q2
q3
q4
q5
A Low-Level Description
q1
q2
q3
q4
q5
B R q1 reject
1 R q2 reject accept
x R q3 x R q
2 B L q4
1 R q2 x R q
3 reject
1 L q5 rejectx L q
4
1 L q6 x L q
5 accept
1 1 1 1 1 1 1 1
1 L q6 x L q
6 B R q2
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1q
0q
0
BB1 xx
q6
q6
rejectq
1
q2
q3
q4
q5
A Low-Level Description
q1
q2
q3
q4
q5
B R q1 reject
1 R q2 reject accept
x R q3 x R q
2 B L q4
1 R q2 x R q
3 reject
1 L q5 rejectx L q
4
1 L q6 x L q
5 accept
1 1 1 1 1 1 1 1
1 L q6 x L q
6 B R q2
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q0
q1
11q
0
BBxx
q6
q6
rejectq
1
q2
q3
q4
q5
A Low-Level Description
q2
q3
q4
q5
B R q1 reject
1 R q2 reject accept
x R q3 x R q
2 B L q4
1 R q2 x R q
3 reject
1 L q5 rejectx L q
4
1 L q6 x L q
5 accept
1 1 1 1 1 1 1
1 L q6 x L q
6 B R q2
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1q
0
q1
1q
0
BBxx
q6
q6
rejectq
1
q2
q3
q4
q5
A Low-Level Description
q2
q3
q4
q5
B R q1 reject
1 R q2 reject accept
x R q3 x R q
2 B L q4
1 R q2 x R q
3 reject
1 L q5 rejectx L q
4
1 L q6 x L q
5 accept
1 1 1 1 1 1 1
1 L q6 x L q
6 B R q2
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1
q1
q2
1q
0
q1
q0
BBxx
q6
q6
reject
q2
q3
q4
q5
A Low-Level Description
q3
q4
q5
B R q1 reject
1 R q2 reject accept
x R q3 x R q
2 B L q4
1 R q2 x R q
3 reject
1 L q5 rejectx L q
4
1 L q6 x L q
5 accept
1 1 1 1 1 1 1
1 L q6 x L q
6 B R q2
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11
q1
q2
q0
q1
q0
BBxx
q6
q6
reject
q2
q3
q4
q5
A Low-Level Description
q3
q4
q5
B R q1 reject
1 R q2 reject accept
x R q3 x R q
2 B L q4
1 R q2 x R q
3 reject
1 L q5 rejectx L q
4
1 L q6 x L q
5 accept
1 1 1 1 1 1 1
1 L q6 x L q
6 B R q2
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1
q2
q3
1
q1
q2
q0
q1
q0
BBxx
q6
q6
reject
q3
q4
q5
A Low-Level Description
q4
q5
B R q1 reject
1 R q2 reject accept
x R q3 x R q
2 B L q4
1 R q2 x R q
3 reject
1 L q5 rejectx L q
4
1 L q6 x L q
5 accept
1 x 1 1 1 1 1
1 L q6 x L q
6 B R q2
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11
q2
q3
q1
q2
q0
q1
q0
BBxx
q6
q6
reject
q3
q4
q5
A Low-Level Description
q4
q5
B R q1 reject
1 R q2 reject accept
x R q3 x R q
2 B L q4
1 R q2 x R q
3 reject
1 L q5 rejectx L q
4
1 L q6 x L q
5 accept
1 x 1 1 1 1 1
1 L q6 x L q
6 B R q2
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A Low-Level Description
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A Low-Level Description
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A Low-Level Description
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1 R q2 reject accept
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A Low-Level Description
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A Low-Level Description
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1 R q2 reject accept
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A Low-Level Description
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A Low-Level Description
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A Low-Level Description
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A Low-Level Description
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A Low-Level Description
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A Low-Level Description
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3 reject
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4
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B R q1 reject
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3 reject
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4
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A Low-Level Description
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3 reject
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4
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A Low-Level Description
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3 reject
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4
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A Low-Level Description
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x R q3 x R q
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3 reject
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4
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A Low-Level Description
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1 R q2 reject accept
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3 reject
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4
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A Low-Level Description
B R q1 reject
1 R q2 reject accept
x R q3 x R q
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3 reject
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4
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1 xx1
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A Low-Level Description
B R q1 reject
1 R q2 reject accept
x R q3 x R q
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3 reject
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4
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11 xx
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A Low-Level Description
B R q1 reject
1 R q2 reject accept
x R q3 x R q
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3 reject
1 L q5 rejectx L q
4
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5 accept
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A Low-Level Description
B R q1 reject
1 R q2 reject accept
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2 B L q4
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3 reject
1 L q5 rejectx L q
4
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5 accept
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A Low-Level Description
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1 R q2 reject accept
x R q3 x R q
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3 reject
1 L q5 rejectx L q
4
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A Low-Level Description
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3 reject
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A Low-Level Description
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3 reject
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A Low-Level Description
B R q1 reject
1 R q2 reject accept
x R q3 x R q
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3 reject
1 L q5 rejectx L q
4
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5 accept
x x 1 x 1 x 1
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1 B
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q6
B
q6
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q0 reject
A Low-Level Description
B R q1 reject
1 R q2 reject accept
x R q3 x R q
2 B L q4
1 R q2 x R q
3 reject
1 L q5 rejectx L q
4
1 L q6 x L q
5 accept
x x 1 x 1 x 1
1 L q6 x L q
6 B R q2
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1 xx1
q2
q3
B
q2
q6
B
q6
q5
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q4
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q1
q0
q1
q0 reject
A Low-Level Description
B R q1 reject
1 R q2 reject accept
x R q3 x R q
2 B L q4
1 R q2 x R q
3 reject
1 L q5 rejectx L q
4
1 L q6 x L q
5 accept
x x 1 x 1 x 1
1 L q6 x L q
6 B R q2
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1 xx
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q3
B
q6
B
q6
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q5
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q1
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q0 reject
A Low-Level Description
B R q1 reject
1 R q2 reject accept
x R q3 x R q
2 B L q4
1 R q2 x R q
3 reject
1 L q5 rejectx L q
4
1 L q6 x L q
5 accept
x x 1 x 1 x 1
1 L q6 x L q
6 B R q2
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1 x
q2
q3
B
q6
B
q6
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q4
q1
q0
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q0 reject
A Low-Level Description
B R q1 reject
1 R q2 reject accept
x R q3 x R q
2 B L q4
1 R q2 x R q
3 reject
1 L q5 rejectx L q
4
1 L q6 x L q
5 accept
x x 1 x 1 x 1
1 L q6 x L q
6 B R q2
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xx1
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1
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q3
B
q6
B
q6
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q1
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q1
q0 reject
A Low-Level Description
B R q1 reject
1 R q2 reject accept
x R q3 x R q
2 B L q4
1 R q2 x R q
3 reject
1 L q5 rejectx L q
4
1 L q6 x L q
5 accept
x x 1 x 1 x 1
1 L q6 x L q
6 B R q2
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1 xx1
q2
q3
q2
q3
B
q6
B
q6
q5
q4
q5
q4
q1
q0
q1
q0 reject
A Low-Level Description
B R q1 reject
1 R q2 reject accept
x R q3 x R q
2 B L q4
1 R q2 x R q
3 reject
1 L q5 rejectx L q
4
1 L q6 x L q
5 accept
x x 1 x 1 x 1
1 L q6 x L q
6 B R q2
![Page 232: Turing Machines - KeithSchwarz.comDesign Turing machines! Alternate problem session tomorrow night, 7-8PM, in Gates 100. The Turing Machine A Turing machine consists of three parts:](https://reader034.fdocuments.net/reader034/viewer/2022042923/5f7027f5297422064c2da53e/html5/thumbnails/232.jpg)
1
q2
q3
1 xx
q2
q3
B
q6
B
q6
q5
q4
q5
q4
q1
q0
q1
q0 reject
A Low-Level Description
B R q1 reject
1 R q2 reject accept
x R q3 x R q
2 B L q4
1 R q2 x R q
3 reject
1 L q5 rejectx L q
4
1 L q6 x L q
5 accept
x x 1 x x x 1
1 L q6 x L q
6 B R q2
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x
x
1
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q3
1 x
q2
q3
B
q6
B
q6
q5
q4
q5
q4
q1
q0
q1
q0 reject
A Low-Level Description
B R q1 reject
1 R q2 reject accept
x R q3 x R q
2 B L q4
1 R q2 x R q
3 reject
1 L q5 rejectx L q
4
1 L q6 x L q
5 accept
x x 1 x x 1
1 L q6 x L q
6 B R q2
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xx
x
1
q2
q3
1
q2
q3
B
q6
B
q6
q5
q4
q5
q4
q1
q0
q1
q0 reject
A Low-Level Description
B R q1 reject
1 R q2 reject accept
x R q3 x R q
2 B L q4
1 R q2 x R q
3 reject
1 L q5 rejectx L q
4
1 L q6 x L q
5 accept
x x 1 x x 1
1 L q6 x L q
6 B R q2
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1 xx
x
1
q2
q3
q2
q3
B
q6
B
q6
q5
q4
q5
q4
q1
q0
q1
q0 reject
A Low-Level Description
B R q1 reject
1 R q2 reject accept
x R q3 x R q
2 B L q4
1 R q2 x R q
3 reject
1 L q5 rejectx L q
4
1 L q6 x L q
5 accept
x x 1 x x 1
1 L q6 x L q
6 B R q2
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1
q2
q3
1 xx
x
q2
q3
B
q6
B
q6
q5
q4
q5
q4
q1
q0
q1
q0 reject
A Low-Level Description
B R q1 reject
1 R q2 reject accept
x R q3 x R q
2 B L q4
1 R q2 x R q
3 reject
1 L q5 rejectx L q
4
1 L q6 x L q
5 accept
x x 1 x x 1
1 L q6 x L q
6 B R q2
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B1
q2
q3
1 xx
x
q2
q3
B
q6
q6
q5
q4
q5
q4
q1
q0
q1
q0 reject
A Low-Level Description
B R q1 reject
1 R q2 reject accept
x R q3 x R q
2 B L q4
1 R q2 x R q
3 reject
1 L q5 rejectx L q
4
1 L q6 x L q
5 accept
x x 1 x x 1
1 L q6 x L q
6 B R q2
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q2
B
q4
B1
q2
q3
1 xx
x
q3
q6
q6
q5
q4
q5
q1
q0
q1
q0 reject
A Low-Level Description
B R q1 reject
1 R q2 reject accept
x R q3 x R q
2 B L q4
1 R q2 x R q
3 reject
1 L q5 rejectx L q
4
1 L q6 x L q
5 accept
x x 1 x x 1
1 L q6 x L q
6 B R q2
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1
q2
B
q4
B1
q2
q3
xx
x
q3
q6
q6
q5
q4
q5
q1
q0
q1
q0 reject
A Low-Level Description
B R q1 reject
1 R q2 reject accept
x R q3 x R q
2 B L q4
1 R q2 x R q
3 reject
1 L q5 rejectx L q
4
1 L q6 x L q
5 accept
x x 1 x x 1
1 L q6 x L q
6 B R q2
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1
q4
q5
1
q2
B
q4
B
q2
q3
xx
x
q3
q6
q6
q5
q1
q0
q1
q0 reject
A Low-Level Description
B R q1 reject
1 R q2 reject accept
x R q3 x R q
2 B L q4
1 R q2 x R q
3 reject
1 L q5 rejectx L q
4
1 L q6 x L q
5 accept
x x 1 x x 1
1 L q6 x L q
6 B R q2
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x1
q4
q5
1
q2
B
q4
B
q2
q3
x
x
q3
q6
q6
q5
q1
q0
q1
q0 reject
A Low-Level Description
B R q1 reject
1 R q2 reject accept
x R q3 x R q
2 B L q4
1 R q2 x R q
3 reject
1 L q5 rejectx L q
4
1 L q6 x L q
5 accept
x x 1 x x 1
1 L q6 x L q
6 B R q2
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xx1
q4
q5
1
q2
B
q4
B
q2
q3
x
q3
q6
q6
q5
q1
q0
q1
q0 reject
A Low-Level Description
B R q1 reject
1 R q2 reject accept
x R q3 x R q
2 B L q4
1 R q2 x R q
3 reject
1 L q5 rejectx L q
4
1 L q6 x L q
5 accept
x x 1 x x 1
1 L q6 x L q
6 B R q2
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xx1
q4
q5
1
q2
B
q4
B
q2
q3
x
q3
q6
q6
q5
q1
q0
q1
q0 reject
A Low-Level Description
B R q1 reject
1 R q2 reject accept
x R q3 x R q
2 B L q4
1 R q2 x R q
3 reject
1 L q5 rejectx L q
4
1 L q6 x L q
5 accept
x x 1 x x 1
1 L q6 x L q
6 B R q2
![Page 244: Turing Machines - KeithSchwarz.comDesign Turing machines! Alternate problem session tomorrow night, 7-8PM, in Gates 100. The Turing Machine A Turing machine consists of three parts:](https://reader034.fdocuments.net/reader034/viewer/2022042923/5f7027f5297422064c2da53e/html5/thumbnails/244.jpg)
xx1
q4
q5
1
q2
B
q4
B
q2
q3
x
q3
q6
q6
q5
q1
q0
q1
q0 reject
A Low-Level Description
B R q1 reject
1 R q2 reject accept
x R q3 x R q
2 B L q4
1 R q2 x R q
3 reject
1 L q5 rejectx L q
4
1 L q6 x L q
5 accept
x x 1 x x 1
1 L q6 x L q
6 B R q2
![Page 245: Turing Machines - KeithSchwarz.comDesign Turing machines! Alternate problem session tomorrow night, 7-8PM, in Gates 100. The Turing Machine A Turing machine consists of three parts:](https://reader034.fdocuments.net/reader034/viewer/2022042923/5f7027f5297422064c2da53e/html5/thumbnails/245.jpg)
xx1
q4
q5
1
q2
B
q4
B
q2
q3
x
q3
q6
q6
q5
q1
q0
q1
q0 reject
A Low-Level Description
B R q1 reject
1 R q2 reject accept
x R q3 x R q
2 B L q4
1 R q2 x R q
3 reject
1 L q5 rejectx L q
4
1 L q6 x L q
5 accept
x x 1 x x 1
1 L q6 x L q
6 B R q2
![Page 246: Turing Machines - KeithSchwarz.comDesign Turing machines! Alternate problem session tomorrow night, 7-8PM, in Gates 100. The Turing Machine A Turing machine consists of three parts:](https://reader034.fdocuments.net/reader034/viewer/2022042923/5f7027f5297422064c2da53e/html5/thumbnails/246.jpg)
xx1
q4
q5
1
q2
B
q4
B
q2
q3
x
q3
q6
q6
q5
q1
q0
q1
q0 reject
A Low-Level Description
B R q1 reject
1 R q2 reject accept
x R q3 x R q
2 B L q4
1 R q2 x R q
3 reject
1 L q5 rejectx L q
4
1 L q6 x L q
5 accept
x x 1 x x 1
1 L q6 x L q
6 B R q2
![Page 247: Turing Machines - KeithSchwarz.comDesign Turing machines! Alternate problem session tomorrow night, 7-8PM, in Gates 100. The Turing Machine A Turing machine consists of three parts:](https://reader034.fdocuments.net/reader034/viewer/2022042923/5f7027f5297422064c2da53e/html5/thumbnails/247.jpg)
1 xx1
q4
q5
q2
B
q4
B
q2
q3
x
q3
q6
q6
q5
q1
q0
q1
q0 reject
A Low-Level Description
B R q1 reject
1 R q2 reject accept
x R q3 x R q
2 B L q4
1 R q2 x R q
3 reject
1 L q5 rejectx L q
4
1 L q6 x L q
5 accept
x x 1 x x 1
1 L q6 x L q
6 B R q2
![Page 248: Turing Machines - KeithSchwarz.comDesign Turing machines! Alternate problem session tomorrow night, 7-8PM, in Gates 100. The Turing Machine A Turing machine consists of three parts:](https://reader034.fdocuments.net/reader034/viewer/2022042923/5f7027f5297422064c2da53e/html5/thumbnails/248.jpg)
1
q6
q5
1 xx
q4
q5
q2
B
q4
B
q2
q3
x
q3
q6
q1
q0
q1
q0 reject
A Low-Level Description
B R q1 reject
1 R q2 reject accept
x R q3 x R q
2 B L q4
1 R q2 x R q
3 reject
1 L q5 rejectx L q
4
1 L q6 x L q
5 accept
x x 1 x x 1
1 L q6 x L q
6 B R q2
![Page 249: Turing Machines - KeithSchwarz.comDesign Turing machines! Alternate problem session tomorrow night, 7-8PM, in Gates 100. The Turing Machine A Turing machine consists of three parts:](https://reader034.fdocuments.net/reader034/viewer/2022042923/5f7027f5297422064c2da53e/html5/thumbnails/249.jpg)
x1
q6
q5
1 x
q4
q5
q2
B
q4
B
q2
q3
x
q3
q6
q1
q0
q1
q0 reject
A Low-Level Description
B R q1 reject
1 R q2 reject accept
x R q3 x R q
2 B L q4
1 R q2 x R q
3 reject
1 L q5 rejectx L q
4
1 L q6 x L q
5 accept
x x 1 x x 1
1 L q6 x L q
6 B R q2
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xx1
q6
q5
1
q4
q5
q2
B
q4
B
q2
q3
x
q3
q6
q1
q0
q1
q0 reject
A Low-Level Description
B R q1 reject
1 R q2 reject accept
x R q3 x R q
2 B L q4
1 R q2 x R q
3 reject
1 L q5 rejectx L q
4
1 L q6 x L q
5 accept
x x 1 x x 1
1 L q6 x L q
6 B R q2
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xx1
q6
q5
1
q4
q5
q2
B
q4
B
q2
q3
x
q3
q6
q1
q0
q1
q0 reject
A Low-Level Description
B R q1 reject
1 R q2 reject accept
x R q3 x R q
2 B L q4
1 R q2 x R q
3 reject
1 L q5 rejectx L q
4
1 L q6 x L q
5 accept
x x 1 x x 1
1 L q6 x L q
6 B R q2
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xx1
q6
q5
1
q4
q5
q2
B
q4
B
q2
q3
x
q3
q6
q1
q0
q1
q0 reject
A Low-Level Description
B R q1 reject
1 R q2 reject accept
x R q3 x R q
2 B L q4
1 R q2 x R q
3 reject
1 L q5 rejectx L q
4
1 L q6 x L q
5 accept
x x 1 x x 1
1 L q6 x L q
6 B R q2
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Bxx1
q6
q5
1
q4
q5
q2
B
q4
q2
q3
x
q3
q6
q1
q0
q1
q0 reject
A Low-Level Description
B R q1 reject
1 R q2 reject accept
x R q3 x R q
2 B L q4
1 R q2 x R q
3 reject
1 L q5 rejectx L q
4
1 L q6 x L q
5 accept
x x 1 x x 1
1 L q6 x L q
6 B R q2
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B
q2
q6
Bxx1
q6
q5
1
q4
q5
q2
q4
q3
x
q3
q1
q0
q1
q0 reject
A Low-Level Description
B R q1 reject
1 R q2 reject accept
x R q3 x R q
2 B L q4
1 R q2 x R q
3 reject
1 L q5 rejectx L q
4
1 L q6 x L q
5 accept
x x 1 x x 1
1 L q6 x L q
6 B R q2
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x B
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Bx1
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q5
1
q4
q5
q2
q4
q3
x
q3
q1
q0
q1
q0 reject
A Low-Level Description
B R q1 reject
1 R q2 reject accept
x R q3 x R q
2 B L q4
1 R q2 x R q
3 reject
1 L q5 rejectx L q
4
1 L q6 x L q
5 accept
x x 1 x x 1
1 L q6 x L q
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B R q1 reject
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x R q3 x R q
2 B L q4
1 R q2 x R q
3 reject
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5 accept
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A Low-Level Description
B R q1 reject
1 R q2 reject accept
x R q3 x R q
2 B L q4
1 R q2 x R q
3 reject
1 L q5 rejectx L q
4
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5 accept
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A Low-Level Description
B R q1 reject
1 R q2 reject accept
x R q3 x R q
2 B L q4
1 R q2 x R q
3 reject
1 L q5 rejectx L q
4
1 L q6 x L q
5 accept
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A Low-Level Description
B R q1 reject
1 R q2 reject accept
x R q3 x R q
2 B L q4
1 R q2 x R q
3 reject
1 L q5 rejectx L q
4
1 L q6 x L q
5 accept
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A Low-Level Description
B R q1 reject
1 R q2 reject accept
x R q3 x R q
2 B L q4
1 R q2 x R q
3 reject
1 L q5 rejectx L q
4
1 L q6 x L q
5 accept
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A Low-Level Description
B R q1 reject
1 R q2 reject accept
x R q3 x R q
2 B L q4
1 R q2 x R q
3 reject
1 L q5 rejectx L q
4
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5 accept
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A Low-Level Description
B R q1 reject
1 R q2 reject accept
x R q3 x R q
2 B L q4
1 R q2 x R q
3 reject
1 L q5 rejectx L q
4
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5 accept
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A Low-Level Description
B R q1 reject
1 R q2 reject accept
x R q3 x R q
2 B L q4
1 R q2 x R q
3 reject
1 L q5 rejectx L q
4
1 L q6 x L q
5 accept
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A Low-Level Description
B R q1 reject
1 R q2 reject accept
x R q3 x R q
2 B L q4
1 R q2 x R q
3 reject
1 L q5 rejectx L q
4
1 L q6 x L q
5 accept
x x x x x 1
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A Low-Level Description
B R q1 reject
1 R q2 reject accept
x R q3 x R q
2 B L q4
1 R q2 x R q
3 reject
1 L q5 rejectx L q
4
1 L q6 x L q
5 accept
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A Low-Level Description
B R q1 reject
1 R q2 reject accept
x R q3 x R q
2 B L q4
1 R q2 x R q
3 reject
1 L q5 rejectx L q
4
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5 accept
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A Low-Level Description
B R q1 reject
1 R q2 reject accept
x R q3 x R q
2 B L q4
1 R q2 x R q
3 reject
1 L q5 rejectx L q
4
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5 accept
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A Low-Level Description
B R q1 reject
1 R q2 reject accept
x R q3 x R q
2 B L q4
1 R q2 x R q
3 reject
1 L q5 rejectx L q
4
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5 accept
x x x x x 1
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A Low-Level Description
B R q1 reject
1 R q2 reject accept
x R q3 x R q
2 B L q4
1 R q2 x R q
3 reject
1 L q5 rejectx L q
4
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5 accept
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A Low-Level Description
B R q1 reject
1 R q2 reject accept
x R q3 x R q
2 B L q4
1 R q2 x R q
3 reject
1 L q5 rejectx L q
4
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5 accept
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A Low-Level Description
B R q1 reject
1 R q2 reject accept
x R q3 x R q
2 B L q4
1 R q2 x R q
3 reject
1 L q5 rejectx L q
4
1 L q6 x L q
5 accept
x x x x x 1
1 L q6 x L q
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A Low-Level Description
B R q1 reject
1 R q2 reject accept
x R q3 x R q
2 B L q4
1 R q2 x R q
3 reject
1 L q5 rejectx L q
4
1 L q6 x L q
5 accept
x x x x x 1
1 L q6 x L q
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A Low-Level Description
B R q1 reject
1 R q2 reject accept
x R q3 x R q
2 B L q4
1 R q2 x R q
3 reject
1 L q5 rejectx L q
4
1 L q6 x L q
5 accept
x x x x x 1
1 L q6 x L q
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A Low-Level Description
B R q1 reject
1 R q2 reject accept
x R q3 x R q
2 B L q4
1 R q2 x R q
3 reject
1 L q5 rejectx L q
4
1 L q6 x L q
5 accept
x x x x x 1
1 L q6 x L q
6 B R q2
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1
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A Low-Level Description
B R q1 reject
1 R q2 reject accept
x R q3 x R q
2 B L q4
1 R q2 x R q
3 reject
1 L q5 rejectx L q
4
1 L q6 x L q
5 accept
x x x x x 1
1 L q6 x L q
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A Low-Level Description
B R q1 reject
1 R q2 reject accept
x R q3 x R q
2 B L q4
1 R q2 x R q
3 reject
1 L q5 rejectx L q
4
1 L q6 x L q
5 accept
x x x x x 1
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A Low-Level Description
B R q1 reject
1 R q2 reject accept
x R q3 x R q
2 B L q4
1 R q2 x R q
3 reject
1 L q5 rejectx L q
4
1 L q6 x L q
5 accept
x x x x x 1
1 L q6 x L q
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1
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x
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A Low-Level Description
B R q1 reject
1 R q2 reject accept
x R q3 x R q
2 B L q4
1 R q2 x R q
3 reject
1 L q5 rejectx L q
4
1 L q6 x L q
5 accept
x x x x x 1
1 L q6 x L q
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1
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A Low-Level Description
B R q1 reject
1 R q2 reject accept
x R q3 x R q
2 B L q4
1 R q2 x R q
3 reject
1 L q5 rejectx L q
4
1 L q6 x L q
5 accept
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1 L q6 x L q
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1
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q1
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A Low-Level Description
B R q1 reject
1 R q2 reject accept
x R q3 x R q
2 B L q4
1 R q2 x R q
3 reject
1 L q5 rejectx L q
4
1 L q6 x L q
5 accept
x x x x x 1
1 L q6 x L q
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1
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q1
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A Low-Level Description
B R q1 reject
1 R q2 reject accept
x R q3 x R q
2 B L q4
1 R q2 x R q
3 reject
1 L q5 rejectx L q
4
1 L q6 x L q
5 accept
x x x x x 1
1 L q6 x L q
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A Low-Level Description
B R q1 reject
1 R q2 reject accept
x R q3 x R q
2 B L q4
1 R q2 x R q
3 reject
1 L q5 rejectx L q
4
1 L q6 x L q
5 accept
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1 L q6 x L q
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A Low-Level Description
B R q1 reject
1 R q2 reject accept
x R q3 x R q
2 B L q4
1 R q2 x R q
3 reject
1 L q5 rejectx L q
4
1 L q6 x L q
5 accept
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1 L q6 x L q
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A Low-Level Description
B R q1 reject
1 R q2 reject accept
x R q3 x R q
2 B L q4
1 R q2 x R q
3 reject
1 L q5 rejectx L q
4
1 L q6 x L q
5 accept
x x x x x 1
1 L q6 x L q
6 B R q2
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Bxx1
q4
q5
1
q2
B
q4
q2
q3
q3
q6
q6
q5
x
q1
q0
q1
q0 reject
A Low-Level Description
B R q1 reject
1 R q2 reject accept
x R q3 x R q
2 B L q4
1 R q2 x R q
3 reject
1 L q5 rejectx L q
4
1 L q6 x L q
5 accept
x x x x x 1
1 L q6 x L q
6 B R q2
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Why This Matters
● From this point forward, we are not going to concern ourselves (that much) with low-level descriptions of Turing Machines.
● If you are asked to explicitly construct a Turing machine, you should provide a transition table or state-transition diagram.
● If you are asked to prove that you could solve a problem with a Turing machine (for example, to prove that a language is recursively enumerable), just provide a high-level description.
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The Church-Turing Thesis
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Just how powerful are Turing machines?
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Tweaking Models of Computation
● If we add a stack to a finite automaton, we get the (more powerful) pushdown automaton.
● If we replace a PDA's stack with an infinite tape, we get the (more powerful) Turing machine.
● What happens if we try making changes to the Turing machine?
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The Multitrack Turing Machine
0 0 1
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The Multitrack Turing Machine
0 0 1
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The Multitrack Turing Machine
0 0 1
+
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0
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The Multitrack Turing Machine
0 0 1
+ +
? ?
0 0
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The Multitrack Turing Machine
0 0 1
+ + -
? ? !
0 0 1
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The Multitrack Turing Machine
0 0 1
+ + -
? ! !
0 0 1
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The Multitrack Turing Machine
0 0 1
+ + -
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The Multitrack Turing Machine
● A multitrack Turing machine is a Turing machine in which the tape is split into a finite number of parallel tracks.
● Each cell in each track can hold a single symbol.
● The finite-state control can base its transition on the symbols stored in each of the tracks.
● The finite-state control can write a new symbol back to each of the tracks.
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The Multitrack Turing Machine
● Clearly, the multitrack Turing machine is at least as powerful as the regular Turing machine.● Just have one track.
● Is the multitrack TM strictly more powerful than the regular TM?
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Simulation Revisited
● Two models of computation are said to be equivalent if they can accept precisely the same languages.
● Key technique: If each model of computation can simulate the other, they must be equally powerful.● If we can simulate one machine with another, then any
machine of one type can be turned into a machine of the other type.
● We last saw this with NFAs and DFAs.● Can we prove that a TM can simulate a multitrack
TM?
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Simulating a Multitrack TM
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Simulating a Multitrack TM
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Simulating a Multitrack TM
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Simulating a Multitrack TM
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Simulating a Multitrack TM
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Simulating a Multitrack TM
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Simulating a Multitrack TM
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Simulating a Multitrack TM
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Simulating a Multitrack TM
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Simulating a Multitrack TM
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Simulating a Multitrack TM
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Simulating a Multitrack TM
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Simulating a Multitrack TM
0 ? -
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Simulating a Multitrack TM
0 ? -
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Simulating a Multitrack TM
0 ? -
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(0, ?, -, B)
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Simulating a Multitrack TM
0 ? -
0 ? -
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(0, ?, -, *)
(0, ?, -, B)
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Simulating a Multitrack TM
● Given a multitrack TM M, we can build a single-track TM M' as follows:● Use a stack alphabet Γ consisting of tuples of
symbols rather than individual symbols.● On startup, replace each input symbol with a tuple
where each extra track begins blank.● Then, use the same finite-state control as M.
● Turing machines and multitrack Turing machines have the same computational power.
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The Multitape Turing Machine
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The Multitape Turing Machine
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The Multitape Turing Machine
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The Multitape Turing Machine
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The Multitape Turing Machine
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The Multitape Turing Machine
● A multitape Turing machine (MTTM) is a Turing machine with finitely many additional tapes and additional tape heads.
● Each tape head can move independently of the others.
● Each tape head can write symbols independently of the others.
● The finite-state control issues commands to each tape head at the same time and can base its decisions on all of the other tapes.
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The Multitape Turing Machine
● Clearly, the multitape Turing machine is at least as powerful as the regular Turing machine (just use zero extra tapes).
● However, the MTTM is no more powerful than a regular Turing machine.
● Idea: Show how to simulate an multitape Turing machine using a multitrack Turing machine.● Since multitrack TMs are equally powerful as
regular TMs, this shows that standard Turing machines are at least as powerful as MTTMs.
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Simulating an MTTM
● Idea: Build a multitrack machine with one track per tape.● Add in special symbols representing “the tape head is
here, and the symbol is x.”● Repeatedly scan across the tracks, applying the
transition for each tape one at a time.● Scan across the tracks to find where the tape head is.● Apply the transition function to determine which direction the
head moves.● Replace the symbol in that direction with a marked symbol
indicating that the tape head is now there.
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Modern Computers
CPU
137 3141 42
1004
2 7 1 8 2 8 0…
ProgramCounter
Registers
0 4 8 12 16 20 N
Data Bus
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Modern Computers
CPU
137 3141 42
1004
2 7 1 8 2 8 0…
ProgramCounter
Registers
0 4 8 12 16 20 N
Data Bus
Fetches and executes
instructions
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Modern Computers
CPU
137 3141 42
1004
2 7 1 8 2 8 0…
ProgramCounter
Registers
0 4 8 12 16 20 N
Data Bus
Fetches and executes
instructions
Holds address of next
instruction
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Modern Computers
CPU
137 3141 42
1004
2 7 1 8 2 8 0…
ProgramCounter
Registers
0 4 8 12 16 20 N
Data Bus
Fetches and executes
instructions
Holds address of next
instruction
Stores values for quick access
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Modern Computers
CPU
137 3141 42
1004
2 7 1 8 2 8 0…
ProgramCounter
Registers
0 4 8 12 16 20 N
Data Bus
Fetches and executes
instructions
Holds address of next
instruction
Stores values for quick accessHolds values
for later use
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Modern Computers
● A modern computer has● A CPU for fetching, decoding, and executing
instructions,● A program counter for keeping track of the next
instruction to execute,● A finite set of fixed-size registers for doing quick
computation,● A finite random-access memory of
fixed-size integers for holding results.
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An Idealized Computer
● An ideal random-access machine has● A CPU for fetching, decoding, and executing
instructions,● A program counter for keeping track of the next
instruction to execute,● A finite set of unbounded registers for doing quick
computation,● An infinite random-access memory of
unbounded integers for holding results.
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The Idealized Computer
● The idealized computer is powerful enough to simulate a Turing machine:● Dedicate a special memory location to the state of
the TM.● Use the infinite memory to hold the contents of the
TM's infinite tape.
● Is the idealized computer any more powerful than the Turing machine?
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1370
421
02
23
14
……
7PC
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1370
421
02
23
14
……
7PC
0
:
1
,
1
:
3
7
4
2
,
RAM Tape
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1370
421
02
23
14
……
7PC
0
:
1
,
1
:
3
7
4
2
,
7
RAM Tape
Register Tape
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1370
421
02
23
14
……
7PC
0
:
1
,
1
:
3
7
4
2
,
7 1
3
7
2
=
1
+
4
7
9
RAM Tape
Register Tape
Scratch Tape
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Simulating the Idealized Computer
● Have three tapes:● A RAM tape for holding the contents of RAM.● A register tape for holding the program counter and other registers.● A scratch tape for performing extra work.
● To execute an instruction:● Fetch the instruction from memory by scanning the memory tape for
the appropriate instruction.● Copy the instruction to the scratch tape.● Decode the instruction and fetch the operands from the RAM tape.● Perform the operation on the scratch tape.● Write the result back to the RAM tape.
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All of these different models of computation are no more powerful than the basic Turing machine.
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Just how powerful are Turing machines?
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The Church-Turing Thesis states that
Any feasible model of computation is equivalent in power to a Turing machine.
This statement cannot be proven or disproven, but is widely considered true.
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All Languages
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RegularLanguages
All Languages
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RegularLanguages DCFLs
All Languages
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RegularLanguages CFLsDCFLs
All Languages
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RegularLanguages CFLsDCFLs
All Languages
Turing-Recognizable
Languages
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RegularLanguages CFLsDCFLs
All Languages
Problems Solvable by
Any Feasible Computing
Machine
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Next Time
● The Limits of Turing Machines● What problems are provably impossible to solve
with Turing machines?● In other words, what problems are beyond the
realm of what can be computed?