turing machines - stanford universityweb.stanford.edu/class/archive/cs/cs103/cs103.1198...the coming...
TRANSCRIPT
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Turing MachinesPart One
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What problems can we solve with a computer?
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RegularLanguages
CFLs
All Languages
Languages recognizable by
any feasible computing machine
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All Languages
That same drawing, to scale.
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The Problem
● Finite automata accept precisely the regular languages.
● We may need unbounded memory to recognize context-free languages.
● e.g. { anbn | n ∈ ℕ } requires unbounded counting.
● How do we build an automaton with fnitely many states but unbounded memory?
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A Brief History Lesson
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Technology has solved all of mankind’s problems! No more wars or sad ever!
<ahem>
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Hilbert’s Vision
● 1900: International Congress of Mathematicians meeting in Paris● Proposes 23 unsolved problems as the agenda for the coming years● An important theme is not simply proving more theorems, but achieving automation of theorem-proving, even theorem generation. ● Humanity lives in leisure while all Truth fows efortlessly into our hands on a ticker tape!
“No one shall expel us from the Paradise that Cantor has created!” -David Hilbert
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Hi there, Reality!
● Hilbert’s agenda is both a spectacular success and a spectacular failure● Inspires some of the most impactful theoretical work in mathematical history, human history● Brings us heroes like Alan Turing and Kurt Gödel! ● These incredible results consist of utterly demolishing all the pillars of Hilbert’s vision of automated knowledge creation, within just a few years
“No one shall expel us from the Paradise that Cantor has created!” -David Hilbert
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A Simple Turing Machine
q0
qacc
qrej
q1
start
a → , R☐
a → , R☐
→ ☐ ☐, R
→ ☐ ☐, R
qacc
qrej
This is the Turing machine’s�nite��tite�nntont. It issues
commands that drive the operation of the machine.
This is the Turing machine’s�nite��tite�nntont. It issues
commands that drive the operation of the machine.
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A Simple Turing Machine
q0
qacc
qrej
q1
start
a → , R☐
a → , R☐
→ ☐ ☐, R
→ ☐ ☐, R
qacc
qrej
This is the TM’s in�nite�tipe. Each tape cell holds a tipe�
�ymbnt. Initially, all (infnitely many) tape symbols are blank.
This is the TM’s in�nite�tipe. Each tape cell holds a tipe�
�ymbnt. Initially, all (infnitely many) tape symbols are blank.
… …
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A Simple Turing Machine
q0
qacc
qrej
q1
start
a → , R☐
a → , R☐
→ ☐ ☐, R
→ ☐ ☐, R
qacc
qrej
The machine is started with the input��toing written somewhere
on the tape. The tipe�heid initially points to the frst symbol
of the input string.
The machine is started with the input��toing written somewhere
on the tape. The tipe�heid initially points to the frst symbol
of the input string.
a a a a… …
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A Simple Turing Machine
q0
qacc
qrej
q1
start
a → , R☐
a → , R☐
→ ☐ ☐, R
→ ☐ ☐, R
qacc
qrej
a a a a… …
q0
Like DFAs and NFAs, TMs begin execution in their
�tiot��tite.
Like DFAs and NFAs, TMs begin execution in their
�tiot��tite.
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a a a… …
A Simple Turing Machine
q0
qacc
qrej
q1
start
a → , R☐
a → , R☐
→ ☐ ☐, R
→ ☐ ☐, R
qacc
qrej
a
q0
At each step, the TM only looks at
the symbol immediately under
the tipe�heid.
At each step, the TM only looks at
the symbol immediately under
the tipe�heid.
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a a a… …
A Simple Turing Machine
q0
qacc
qrej
q1
start
a → , R☐ → ☐ ☐, R
qacc
qrej
a
a → , R☐ → ☐ ☐, R
q0
These two transitions originate at the current
state. We’re going to choose one of them to
follow.
These two transitions originate at the current
state. We’re going to choose one of them to
follow.
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a a a… …
A Simple Turing Machine
q0
qacc
qrej
q1
start
a → , R☐ → ☐ ☐, R
qacc
qrej
a
a → , R☐ → ☐ ☐, R
q0
Each transition has the form
oeid�→�woite,�dio
and means “if symbol oeid is under the tape head, replace it with woite and move the tape head in direction dio (L or R). The symbol denotes a blank cell.☐
Each transition has the form
oeid�→�woite,�dio
and means “if symbol oeid is under the tape head, replace it with woite and move the tape head in direction dio (L or R). The symbol denotes a blank cell.☐
![Page 17: Turing Machines - Stanford Universityweb.stanford.edu/class/archive/cs/cs103/cs103.1198...the coming years An important theme is not simply proving more theorems, but achieving automation](https://reader034.vdocuments.us/reader034/viewer/2022050502/5f9440a3ae52d04590363756/html5/thumbnails/17.jpg)
→ ☐ ☐, R
a a a… …
A Simple Turing Machine
q0
qacc
qrej
q1
start
a → , R☐ → ☐ ☐, R
qacc
qrej
a
a → , R☐
q0
Each transition has the form
oeid�→�woite,�dio
and means “if symbol oeid is under the tape head, replace it with woite and move the tape head in direction dio (L or R). The symbol denotes a blank cell.☐
Each transition has the form
oeid�→�woite,�dio
and means “if symbol oeid is under the tape head, replace it with woite and move the tape head in direction dio (L or R). The symbol denotes a blank cell.☐
![Page 18: Turing Machines - Stanford Universityweb.stanford.edu/class/archive/cs/cs103/cs103.1198...the coming years An important theme is not simply proving more theorems, but achieving automation](https://reader034.vdocuments.us/reader034/viewer/2022050502/5f9440a3ae52d04590363756/html5/thumbnails/18.jpg)
→ ☐ ☐, R
a a a… …
A Simple Turing Machine
q0
qacc
qrej
q1
start
a → , R☐ → ☐ ☐, R
qacc
qrej
a → , R☐
q0
Each transition has the form
oeid�→�woite,�dio
and means “if symbol oeid is under the tape head, replace it with woite and move the tape head in direction dio (L or R). The symbol denotes a blank cell.☐
Each transition has the form
oeid�→�woite,�dio
and means “if symbol oeid is under the tape head, replace it with woite and move the tape head in direction dio (L or R). The symbol denotes a blank cell.☐
![Page 19: Turing Machines - Stanford Universityweb.stanford.edu/class/archive/cs/cs103/cs103.1198...the coming years An important theme is not simply proving more theorems, but achieving automation](https://reader034.vdocuments.us/reader034/viewer/2022050502/5f9440a3ae52d04590363756/html5/thumbnails/19.jpg)
→ ☐ ☐, R
a a a… …
A Simple Turing Machine
q0
qacc
qrej
q1
start
a → , R☐ → ☐ ☐, R
qacc
qrej
a → , R☐
q0
Each transition has the form
oeid�→�woite,�dio
and means “if symbol oeid is under the tape head, replace it with woite and move the tape head in direction dio (L or R). The symbol denotes a blank cell.☐
Each transition has the form
oeid�→�woite,�dio
and means “if symbol oeid is under the tape head, replace it with woite and move the tape head in direction dio (L or R). The symbol denotes a blank cell.☐
![Page 20: Turing Machines - Stanford Universityweb.stanford.edu/class/archive/cs/cs103/cs103.1198...the coming years An important theme is not simply proving more theorems, but achieving automation](https://reader034.vdocuments.us/reader034/viewer/2022050502/5f9440a3ae52d04590363756/html5/thumbnails/20.jpg)
→ ☐ ☐, R
a a a… …
A Simple Turing Machine
q0
qacc
qrej
q1
start
a → , R☐ → ☐ ☐, R
qacc
qrej
a → , R☐
q1
Each transition has the form
oeid�→�woite,�dio
and means “if symbol oeid is under the tape head, replace it with woite and move the tape head in direction dio (L or R). The symbol denotes a blank cell.☐
Each transition has the form
oeid�→�woite,�dio
and means “if symbol oeid is under the tape head, replace it with woite and move the tape head in direction dio (L or R). The symbol denotes a blank cell.☐
![Page 21: Turing Machines - Stanford Universityweb.stanford.edu/class/archive/cs/cs103/cs103.1198...the coming years An important theme is not simply proving more theorems, but achieving automation](https://reader034.vdocuments.us/reader034/viewer/2022050502/5f9440a3ae52d04590363756/html5/thumbnails/21.jpg)
→ ☐ ☐, R
a a a… …
A Simple Turing Machine
q0
qacc
qrej
q1
start
a → , R☐ → ☐ ☐, R
qacc
qrej
a → , R☐
q1
![Page 22: Turing Machines - Stanford Universityweb.stanford.edu/class/archive/cs/cs103/cs103.1198...the coming years An important theme is not simply proving more theorems, but achieving automation](https://reader034.vdocuments.us/reader034/viewer/2022050502/5f9440a3ae52d04590363756/html5/thumbnails/22.jpg)
a a… …
→ ☐ ☐, R
a
A Simple Turing Machine
q0
qacc
qrej
q1
start
a → , R☐ → ☐ ☐, R
qacc
qrej
a → , R☐
q1
![Page 23: Turing Machines - Stanford Universityweb.stanford.edu/class/archive/cs/cs103/cs103.1198...the coming years An important theme is not simply proving more theorems, but achieving automation](https://reader034.vdocuments.us/reader034/viewer/2022050502/5f9440a3ae52d04590363756/html5/thumbnails/23.jpg)
a a… …
→ ☐ ☐, R
a
A Simple Turing Machine
q0
qacc
qrej
q1
start
qacc
qrej
a → , R☐
a → , R☐ → ☐ ☐, R
q1
![Page 24: Turing Machines - Stanford Universityweb.stanford.edu/class/archive/cs/cs103/cs103.1198...the coming years An important theme is not simply proving more theorems, but achieving automation](https://reader034.vdocuments.us/reader034/viewer/2022050502/5f9440a3ae52d04590363756/html5/thumbnails/24.jpg)
a a… …
→ ☐ ☐, R
a
A Simple Turing Machine
q0
qacc
qrej
q1
start
qacc
qrej
a → , R☐
→ ☐ ☐, Ra → , R☐
q1
![Page 25: Turing Machines - Stanford Universityweb.stanford.edu/class/archive/cs/cs103/cs103.1198...the coming years An important theme is not simply proving more theorems, but achieving automation](https://reader034.vdocuments.us/reader034/viewer/2022050502/5f9440a3ae52d04590363756/html5/thumbnails/25.jpg)
a a… …
→ ☐ ☐, R
A Simple Turing Machine
q0
qacc
qrej
q1
start
qacc
qrej
a → , R☐
→ ☐ ☐, Ra → , R☐
q1
![Page 26: Turing Machines - Stanford Universityweb.stanford.edu/class/archive/cs/cs103/cs103.1198...the coming years An important theme is not simply proving more theorems, but achieving automation](https://reader034.vdocuments.us/reader034/viewer/2022050502/5f9440a3ae52d04590363756/html5/thumbnails/26.jpg)
a a… …
→ ☐ ☐, R
A Simple Turing Machine
q0
qacc
qrej
q1
start
qacc
qrej
a → , R☐
→ ☐ ☐, Ra → , R☐
q1
![Page 27: Turing Machines - Stanford Universityweb.stanford.edu/class/archive/cs/cs103/cs103.1198...the coming years An important theme is not simply proving more theorems, but achieving automation](https://reader034.vdocuments.us/reader034/viewer/2022050502/5f9440a3ae52d04590363756/html5/thumbnails/27.jpg)
a a… …
→ ☐ ☐, R
A Simple Turing Machine
q0
qacc
qrej
q1
start
qacc
qrej
a → , R☐
→ ☐ ☐, Ra → , R☐
q0
![Page 28: Turing Machines - Stanford Universityweb.stanford.edu/class/archive/cs/cs103/cs103.1198...the coming years An important theme is not simply proving more theorems, but achieving automation](https://reader034.vdocuments.us/reader034/viewer/2022050502/5f9440a3ae52d04590363756/html5/thumbnails/28.jpg)
a a… …
→ ☐ ☐, R
A Simple Turing Machine
q0
qacc
qrej
q1
start
qacc
qrej
a → , R☐
→ ☐ ☐, Ra → , R☐
q0
![Page 29: Turing Machines - Stanford Universityweb.stanford.edu/class/archive/cs/cs103/cs103.1198...the coming years An important theme is not simply proving more theorems, but achieving automation](https://reader034.vdocuments.us/reader034/viewer/2022050502/5f9440a3ae52d04590363756/html5/thumbnails/29.jpg)
a a… …
→ ☐ ☐, R
A Simple Turing Machine
q0
qacc
qrej
q1
start
qacc
qrej
a → , R☐
→ ☐ ☐, Ra → , R☐
q0
![Page 30: Turing Machines - Stanford Universityweb.stanford.edu/class/archive/cs/cs103/cs103.1198...the coming years An important theme is not simply proving more theorems, but achieving automation](https://reader034.vdocuments.us/reader034/viewer/2022050502/5f9440a3ae52d04590363756/html5/thumbnails/30.jpg)
a… …
→ ☐ ☐, R
A Simple Turing Machine
q0
qacc
qrej
q1
start
qacc
qrej
a → , R☐
→ ☐ ☐, Ra → , R☐
q1
![Page 31: Turing Machines - Stanford Universityweb.stanford.edu/class/archive/cs/cs103/cs103.1198...the coming years An important theme is not simply proving more theorems, but achieving automation](https://reader034.vdocuments.us/reader034/viewer/2022050502/5f9440a3ae52d04590363756/html5/thumbnails/31.jpg)
a… …
→ ☐ ☐, R
A Simple Turing Machine
q0
qacc
qrej
q1
start
qacc
qrej
a → , R☐
→ ☐ ☐, Ra → , R☐
q1
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… …
→ ☐ ☐, R
A Simple Turing Machine
q0
qacc
qrej
q1
start
qacc
qrej
a → , R☐
→ ☐ ☐, Ra → , R☐
q0
Unlike a DFA or NFA, a TM doesn’t stop after reading all the input characters. We keep
running until the machine explicitly says to stop.
Unlike a DFA or NFA, a TM doesn’t stop after reading all the input characters. We keep
running until the machine explicitly says to stop.
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… …
→ ☐ ☐, R
A Simple Turing Machine
q0
qacc
qrej
q1
start
qacc
qrej
a → , R☐
→ ☐ ☐, Ra → , R☐
q0
![Page 34: Turing Machines - Stanford Universityweb.stanford.edu/class/archive/cs/cs103/cs103.1198...the coming years An important theme is not simply proving more theorems, but achieving automation](https://reader034.vdocuments.us/reader034/viewer/2022050502/5f9440a3ae52d04590363756/html5/thumbnails/34.jpg)
… …
→ ☐ ☐, R
A Simple Turing Machine
q0
qacc
qrej
q1
start
qacc
qrej
a → , R☐
→ ☐ ☐, Ra → , R☐
This special state is an iepting��tite. When a TM enters an accepting state, it immediately stops running and accepts whatever the
original input string was (in this case, aaaa).
This special state is an iepting��tite. When a TM enters an accepting state, it immediately stops running and accepts whatever the
original input string was (in this case, aaaa).
![Page 35: Turing Machines - Stanford Universityweb.stanford.edu/class/archive/cs/cs103/cs103.1198...the coming years An important theme is not simply proving more theorems, but achieving automation](https://reader034.vdocuments.us/reader034/viewer/2022050502/5f9440a3ae52d04590363756/html5/thumbnails/35.jpg)
… …
→ ☐ ☐, R
A Simple Turing Machine
q0
qrej
q1
start
qrej
a → , R☐
→ ☐ ☐, Ra → , R☐
This special state is an iepting��tite. When a TM enters an accepting state, it immediately stops running and accepts whatever the
original input string was (in this case, aaaa).
This special state is an iepting��tite. When a TM enters an accepting state, it immediately stops running and accepts whatever the
original input string was (in this case, aaaa).
qacc
![Page 36: Turing Machines - Stanford Universityweb.stanford.edu/class/archive/cs/cs103/cs103.1198...the coming years An important theme is not simply proving more theorems, but achieving automation](https://reader034.vdocuments.us/reader034/viewer/2022050502/5f9440a3ae52d04590363756/html5/thumbnails/36.jpg)
a a a… …a
→ ☐ ☐, R
a
A Simple Turing Machine
q0
qrej
q1
start
qrej
a → , R☐
→ ☐ ☐, Ra → , R☐
qacc
q0
![Page 37: Turing Machines - Stanford Universityweb.stanford.edu/class/archive/cs/cs103/cs103.1198...the coming years An important theme is not simply proving more theorems, but achieving automation](https://reader034.vdocuments.us/reader034/viewer/2022050502/5f9440a3ae52d04590363756/html5/thumbnails/37.jpg)
a a a… …
→ ☐ ☐, R
a
A Simple Turing Machine
q0
qrej
q1
start
qrej
a → , R☐
→ ☐ ☐, Ra → , R☐
qacc
q1
![Page 38: Turing Machines - Stanford Universityweb.stanford.edu/class/archive/cs/cs103/cs103.1198...the coming years An important theme is not simply proving more theorems, but achieving automation](https://reader034.vdocuments.us/reader034/viewer/2022050502/5f9440a3ae52d04590363756/html5/thumbnails/38.jpg)
a a a… …
→ ☐ ☐, R
A Simple Turing Machine
q0
qrej
q1
start
qrej
a → , R☐
→ ☐ ☐, Ra → , R☐
qacc
q0
![Page 39: Turing Machines - Stanford Universityweb.stanford.edu/class/archive/cs/cs103/cs103.1198...the coming years An important theme is not simply proving more theorems, but achieving automation](https://reader034.vdocuments.us/reader034/viewer/2022050502/5f9440a3ae52d04590363756/html5/thumbnails/39.jpg)
a a… …
→ ☐ ☐, R
A Simple Turing Machine
q0
qrej
q1
start
qrej
a → , R☐
→ ☐ ☐, Ra → , R☐
qacc
q1
![Page 40: Turing Machines - Stanford Universityweb.stanford.edu/class/archive/cs/cs103/cs103.1198...the coming years An important theme is not simply proving more theorems, but achieving automation](https://reader034.vdocuments.us/reader034/viewer/2022050502/5f9440a3ae52d04590363756/html5/thumbnails/40.jpg)
a… …
→ ☐ ☐, R
A Simple Turing Machine
q0
qrej
q1
start
qrej
a → , R☐
→ ☐ ☐, Ra → , R☐
qacc
q0
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… …
→ ☐ ☐, R
A Simple Turing Machine
q0
qrej
q1
start
qrej
a → , R☐
→ ☐ ☐, Ra → , R☐
qacc
q1
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… …
→ ☐ ☐, R
A Simple Turing Machine
q0
qrej
q1
start
qrej
a → , R☐
→ ☐ ☐, Ra → , R☐
qacc
This special state is a�oejeting��tite. When a TM
enters a rejecting state, it immediately stops running and rejects whatever the
original input string was (in this case, aaaaa).
This special state is a�oejeting��tite. When a TM
enters a rejecting state, it immediately stops running and rejects whatever the
original input string was (in this case, aaaaa).
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… …
→ ☐ ☐, R
A Simple Turing Machine
q0
q1
start
a → , R☐
→ ☐ ☐, Ra → , R☐
qacc
This special state is a�oejeting��tite. When a TM
enters a rejecting state, it immediately stops running and rejects whatever the
original input string was (in this case, aaaaa).
This special state is a�oejeting��tite. When a TM
enters a rejecting state, it immediately stops running and rejects whatever the
original input string was (in this case, aaaaa).
qrej
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… …
→ ☐ ☐, R
A Simple Turing Machine
q0
q1
start
a → , R☐
→ ☐ ☐, Ra → , R☐
qacc
qrej
q0
If the TM is started on the empty string ε, the entire tape is blank and the tape head is positioned at some arbitrary
location on the tape.
If the TM is started on the empty string ε, the entire tape is blank and the tape head is positioned at some arbitrary
location on the tape.
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… …
→ ☐ ☐, R
A Simple Turing Machine
q0
q1
start
a → , R☐
→ ☐ ☐, Ra → , R☐
qacc
qrej
If the TM is started on the empty string ε, the entire tape is blank and the tape head is positioned at some arbitrary
location on the tape.
If the TM is started on the empty string ε, the entire tape is blank and the tape head is positioned at some arbitrary
location on the tape.
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… …
→ ☐ ☐, R
A Simple Turing Machine
q0
q1
start
a → , R☐
→ ☐ ☐, Ra → , R☐
qacc
qrej
If the TM is started on the empty string ε, the entire tape is blank and the tape head is positioned at some arbitrary
location on the tape.
If the TM is started on the empty string ε, the entire tape is blank and the tape head is positioned at some arbitrary
location on the tape.
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The Turing Machine
● A Turing machine consists of three parts:
● A �nite-�tite�nntont that issues commands,
● an in�nite�tipe for input and scratch space, and
● a tipe�heid that can read and write a single tape cell.
● At each step, the Turing machine
● writes a symbol to the tape cell under the tape head,
● changes state, and
● moves the tape head to the left or to the right.
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Input and Tape Alphabets
● A Turing machine has two alphabets:
● An input�itphibet Σ. All input strings are written in the input alphabet.
● A tipe�itphibet Γ, where Σ Γ. The tape alphabet ⊊
contains all symbols that can be written onto the tape.
● The tape alphabet Γ can contain any number of symbols, but always contains at least one btink��ymbnt, denoted . You are guaranteed ∉ Σ.☐ ☐
● At startup, the Turing machine begins with an infnite tape of symbols with the input written at ☐
some location. The tape head is positioned at the start of the input.
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Accepting and Rejecting States
● Unlike DFAs, Turing machines do not stop processing the input when they fnish reading it.
● Turing machines decide when (and if!) they will accept or reject their input.
● Turing machines can enter infnite loops and never accept or reject; more on that later...
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→ ☐ ☐, R
Determinism
q0
qrej
q1
start
qrej
a → , R☐
→ ☐ ☐, Ra → , R☐
qacc
● Turing machines are deteomini�ti: for every combination of a (non-accepting, non-rejecting) state q and a tape symbol a ∈ Γ, there must be exactly one transition defned for that combination of q and a.
● Any transitions that are missing implicitly go straight to a rejecting state. We’ll use this later to simplify our designs.
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→ ☐ ☐, R
Determinism
q0
q1
start
a → , R☐
a → , R☐
qacc
● Turing machines are deteomini�ti: for every combination of a (non-accepting, non-rejecting) state q and a tape symbol a ∈ Γ, there must be exactly one transition defned for that combination of q and a.
● Any transitions that are missing implicitly go straight to a rejecting state. We’ll use this later to simplify our designs.
This machine is exactly the same as the
previous one.
This machine is exactly the same as the
previous one.
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q0
q1
start → ☐ ☐, L
a → b, Rb → a, R a → a, L
b → b, L → ☐ ☐, L
qacc
qrej
Run the TM shown above on the input string bba.What will the tape look like when the TM fnishes running?
Run the TM shown above on the input string bba.What will the tape look like when the TM fnishes running?
… b b a …A.
…… a a bD.
… a a b …B.
… b b a …C.
E. None of these, or two or more of these.
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q0
q1
start → ☐ ☐, L
a → b, Rb → a, R a → a, L
b → b, L → ☐ ☐, L
qacc
qrej
b a… …b
q0
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q0
q1
start → ☐ ☐, L
a → b, Rb → a, R a → a, L
b → b, L → ☐ ☐, L
qacc
qrej
b a… …a
q0
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q0
q1
start → ☐ ☐, L
a → b, Rb → a, R a → a, L
b → b, L → ☐ ☐, L
qacc
qrej
a a… …a
q0
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q0
q1
start → ☐ ☐, L
a → b, Rb → a, R a → a, L
b → b, L → ☐ ☐, L
qacc
qrej
a b… …a
q0
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q0
q1
start → ☐ ☐, L
a → b, Rb → a, R a → a, L
b → b, L → ☐ ☐, L
qacc
qrej
a b… …a
q1
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q0
q1
start → ☐ ☐, L
a → b, Rb → a, R a → a, L
b → b, L → ☐ ☐, L
qacc
qrej
a b… …a
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q0
q1
start → ☐ ☐, L
a → b, Rb → a, R a → a, L
b → b, L → ☐ ☐, L
qacc
qrej
a b… …a
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q0
q1
start → ☐ ☐, L
a → b, Rb → a, R a → a, L
b → b, L → ☐ ☐, L
qacc
qrej
If M is a Turing machine with input alphabet Σ, then the tinguige�nf�M, denoted ℒ(M), is the set
ℒ(M) = { w ∈ Σ* | M accepts w }
If M is a Turing machine with input alphabet Σ, then the tinguige�nf�M, denoted ℒ(M), is the set
ℒ(M) = { w ∈ Σ* | M accepts w }
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q0
q1
start → ☐ ☐, L
a → b, Rb → a, R a → a, L
b → b, L → ☐ ☐, L
qacc
qrej
a b… …a
q0
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q0
q1
start → ☐ ☐, L
a → b, Rb → a, R a → a, L
b → b, L → ☐ ☐, L
qacc
qrej
a b… …b
q0
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q0
q1
start → ☐ ☐, L
a → b, Rb → a, R a → a, L
b → b, L → ☐ ☐, L
qacc
qrej
b b… …b
q0
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q0
q1
start → ☐ ☐, L
a → b, Rb → a, R a → a, L
b → b, L → ☐ ☐, L
qacc
qrej
b a… …b
q0
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q0
q1
start → ☐ ☐, L
a → b, Rb → a, R a → a, L
b → b, L → ☐ ☐, L
qacc
qrej
b a… …b
q1
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q0
q1
start → ☐ ☐, L
a → b, Rb → a, R a → a, L
b → b, L → ☐ ☐, L
qacc
qrej
b a… …b
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q0
q1
start → ☐ ☐, L
a → b, Rb → a, R a → a, L
b → b, L → ☐ ☐, L
qacc
qrej
b a… …b
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q0
q1
start → ☐ ☐, L
a → b, Rb → a, R a → a, L
b → b, L → ☐ ☐, L
qacc
qrej
b a… …b
Although the tape ends with bba written on it, the original input string was aab. This shows that the TM
accepts aab, not bba.
So (ℒ M) = { w ∈ {a, b}* | w ends in b }
Although the tape ends with bba written on it, the original input string was aab. This shows that the TM
accepts aab, not bba.
So (ℒ M) = { w ∈ {a, b}* | w ends in b }
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Designing Turing Machines
● Despite their simplicity, Turing machines are very powerful computing devices.
● Today's lecture explores how to design Turing machines for various languages.
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Designing Turing Machines
● Let Σ = {0, 1} and consider the language L = {0n1n | n ∈ ℕ }.
● We know that L is context-free.
● How might we build a Turing machine for it?
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L = {0n1n | n ∈ ℕ }
0 0 0 1 1 1… …
… …
0 1 0… …
1 1 0 0… …
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A Recursive Approach
● The string ε is in L.
● The string 0w1 is in L if w is in L.
● Any string starting with 1 is not in L.
● Any string ending with 0 is not in L.
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A Sketch of the TM
0 0 0 1 1 1… …
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A Sketch of the TM
0 0 1 1 1… …
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A Sketch of the TM
0 0 1 1 1… …
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A Sketch of the TM
0 0 1 1 1… …
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A Sketch of the TM
0 0 1 1 1… …
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A Sketch of the TM
0 0 1 1 1… …
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A Sketch of the TM
0 0 1 1 1… …
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A Sketch of the TM
0 0 1 1 1… …
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A Sketch of the TM
0 0 1 1… …
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A Sketch of the TM
0 0 1 1… …
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A Sketch of the TM
0 0 1 1… …
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A Sketch of the TM
0 0 1 1… …
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A Sketch of the TM
0 0 1 1… …
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A Sketch of the TM
0 0 1 1… …
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A Sketch of the TM
0 1 1… …
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Time-Out for Announcements!
![Page 196: Turing Machines - Stanford Universityweb.stanford.edu/class/archive/cs/cs103/cs103.1198...the coming years An important theme is not simply proving more theorems, but achieving automation](https://reader034.vdocuments.us/reader034/viewer/2022050502/5f9440a3ae52d04590363756/html5/thumbnails/196.jpg)
Problem Sets
● Problem Set Five was due at 3:00PM today.
● Problem Set Six goes out today. It’s due next Friday at 3:00PM.
● Looking ahead, the fnal assignment in this course, Problem Set Seven, will go out next Friday and be due on the last day of class.
● No late days or late submissions allowed on this one, so PS6 is the last assignment you may use your late days on.
● This last assignment will be shorter and will also serve as a review of all the topics we’ve covered in the course.
![Page 197: Turing Machines - Stanford Universityweb.stanford.edu/class/archive/cs/cs103/cs103.1198...the coming years An important theme is not simply proving more theorems, but achieving automation](https://reader034.vdocuments.us/reader034/viewer/2022050502/5f9440a3ae52d04590363756/html5/thumbnails/197.jpg)
Final Exam
● As a reminder, the fnal exam for CS103 will be on Friday August 16th from 7-10 PM.
● We will release some practice fnal exams early next week.
● We’re planning to have a sit-down practice fnal on Wednesday August 1Tth right after class from 5:30-8:30 PM. Location TBD.
● The mock exam from this practice session will also be released online if you can’t make it in person.
![Page 198: Turing Machines - Stanford Universityweb.stanford.edu/class/archive/cs/cs103/cs103.1198...the coming years An important theme is not simply proving more theorems, but achieving automation](https://reader034.vdocuments.us/reader034/viewer/2022050502/5f9440a3ae52d04590363756/html5/thumbnails/198.jpg)
Another TM Design
● We've designed a TM for {0n1n | n ∈ ℕ}.
● Consider this language over Σ = {0, 1}:
L = { w ∈ Σ* | w has the same number of 0s and 1s }
● This language is also not regular, but it is context-free.
● How might we design a TM for it?
![Page 199: Turing Machines - Stanford Universityweb.stanford.edu/class/archive/cs/cs103/cs103.1198...the coming years An important theme is not simply proving more theorems, but achieving automation](https://reader034.vdocuments.us/reader034/viewer/2022050502/5f9440a3ae52d04590363756/html5/thumbnails/199.jpg)
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1 → 1, R
× → ×, R
0 → ×, L
![Page 310: Turing Machines - Stanford Universityweb.stanford.edu/class/archive/cs/cs103/cs103.1198...the coming years An important theme is not simply proving more theorems, but achieving automation](https://reader034.vdocuments.us/reader034/viewer/2022050502/5f9440a3ae52d04590363756/html5/thumbnails/310.jpg)
Find0/1
Gohome
→ ☐ ☐, R Gohome
Find0Find0
Find0/1
Find1
start
×× × × × × ×… …×
Find1
0 → ×, R
0 → 0, R
× → ×, R
1 → ×, L
0 → 0, L
1 → 1, L
× → ×, L
× → ×, R
1 → ×, R
1 → 1, R
× → ×, R
0 → ×, L
![Page 311: Turing Machines - Stanford Universityweb.stanford.edu/class/archive/cs/cs103/cs103.1198...the coming years An important theme is not simply proving more theorems, but achieving automation](https://reader034.vdocuments.us/reader034/viewer/2022050502/5f9440a3ae52d04590363756/html5/thumbnails/311.jpg)
Find0/1
Gohome
→ ☐ ☐, R Gohome
Find0Find0
Find0/1
Find1
start
×× × × × × ×… …×
Find1
0 → ×, R
0 → 0, R
× → ×, R
1 → ×, L
0 → 0, L
1 → 1, L
× → ×, L
× → ×, R
1 → ×, R
1 → 1, R
× → ×, R
0 → ×, L
![Page 312: Turing Machines - Stanford Universityweb.stanford.edu/class/archive/cs/cs103/cs103.1198...the coming years An important theme is not simply proving more theorems, but achieving automation](https://reader034.vdocuments.us/reader034/viewer/2022050502/5f9440a3ae52d04590363756/html5/thumbnails/312.jpg)
Find0/1
Gohome
→ ☐ ☐, R Gohome
Find0Find0
Find0/1
Find1
start
×× × × × × ×… …×
Find1
0 → ×, R
0 → 0, R
× → ×, R
1 → ×, L
0 → 0, L
1 → 1, L
× → ×, L
× → ×, R
1 → ×, R
1 → 1, R
× → ×, R
0 → ×, L
![Page 313: Turing Machines - Stanford Universityweb.stanford.edu/class/archive/cs/cs103/cs103.1198...the coming years An important theme is not simply proving more theorems, but achieving automation](https://reader034.vdocuments.us/reader034/viewer/2022050502/5f9440a3ae52d04590363756/html5/thumbnails/313.jpg)
Find0/1
Gohome
→ ☐ ☐, R Gohome
Find0Find0
Find0/1
Find1
start
×× × × × × ×… …×
Find1
0 → ×, R
0 → 0, R
× → ×, R
1 → ×, L
0 → 0, L
1 → 1, L
× → ×, L
× → ×, R
1 → ×, R
1 → 1, R
× → ×, R
0 → ×, L
Accept!
→ � �, R
![Page 314: Turing Machines - Stanford Universityweb.stanford.edu/class/archive/cs/cs103/cs103.1198...the coming years An important theme is not simply proving more theorems, but achieving automation](https://reader034.vdocuments.us/reader034/viewer/2022050502/5f9440a3ae52d04590363756/html5/thumbnails/314.jpg)
Find0/1Find0/1
Gohome
→ ☐ ☐, R Gohome
Find0Find0
Find1
start
×× × × × × ×… …×
Find1
0 → ×, R
0 → 0, R
× → ×, R
1 → ×, L
0 → 0, L
1 → 1, L
× → ×, L
× → ×, R
1 → ×, R
1 → 1, R
× → ×, R
0 → ×, L
Accept!
→ � �, R
![Page 315: Turing Machines - Stanford Universityweb.stanford.edu/class/archive/cs/cs103/cs103.1198...the coming years An important theme is not simply proving more theorems, but achieving automation](https://reader034.vdocuments.us/reader034/viewer/2022050502/5f9440a3ae52d04590363756/html5/thumbnails/315.jpg)
Find0/1Find0/1
Gohome
→ ☐ ☐, R Gohome
Find0Find0
Find1
start
×× × × × × ×… …×
Find1
0 → ×, R
0 → 0, R
× → ×, R
1 → ×, L
0 → 0, L
1 → 1, L
× → ×, L
× → ×, R
1 → ×, R
1 → 1, R
× → ×, R
0 → ×, L
Accept!
→ � �, R
![Page 316: Turing Machines - Stanford Universityweb.stanford.edu/class/archive/cs/cs103/cs103.1198...the coming years An important theme is not simply proving more theorems, but achieving automation](https://reader034.vdocuments.us/reader034/viewer/2022050502/5f9440a3ae52d04590363756/html5/thumbnails/316.jpg)
Find0/1Find0/1
Gohome
→ ☐ ☐, R Gohome
Find0Find0
Find1
start
Find1
0 → ×, R
0 → 0, R
× → ×, R
1 → ×, L
0 → 0, L
1 → 1, L
× → ×, L
× → ×, R
1 → ×, R
1 → 1, R
× → ×, R
0 → ×, L
Accept!
→ � �, R
![Page 317: Turing Machines - Stanford Universityweb.stanford.edu/class/archive/cs/cs103/cs103.1198...the coming years An important theme is not simply proving more theorems, but achieving automation](https://reader034.vdocuments.us/reader034/viewer/2022050502/5f9440a3ae52d04590363756/html5/thumbnails/317.jpg)
Find0/1Find0/1
Gohome
→ ☐ ☐, R Gohome
Find0Find0
Find1
start
Find1
0 → ×, R
0 → 0, R
× → ×, R
1 → ×, L
0 → 0, L
1 → 1, L
× → ×, L
× → ×, R
1 → ×, R
1 → 1, R
× → ×, R
0 → ×, L
Accept!
→ � �, R
Remember that all missing transitions
implicitly reject.
Remember that all missing transitions
implicitly reject.
![Page 318: Turing Machines - Stanford Universityweb.stanford.edu/class/archive/cs/cs103/cs103.1198...the coming years An important theme is not simply proving more theorems, but achieving automation](https://reader034.vdocuments.us/reader034/viewer/2022050502/5f9440a3ae52d04590363756/html5/thumbnails/318.jpg)
Find0/1Find0/1
Gohome
→ ☐ ☐, R Gohome
Find0Find0
Find1
start
Find1
0 → ×, R
0 → 0, R
× → ×, R
1 → ×, L
0 → 0, L
1 → 1, L
× → ×, L
× → ×, R
1 → ×, R
1 → 1, R
× → ×, R
0 → ×, L
Accept!
→ � �, R
![Page 319: Turing Machines - Stanford Universityweb.stanford.edu/class/archive/cs/cs103/cs103.1198...the coming years An important theme is not simply proving more theorems, but achieving automation](https://reader034.vdocuments.us/reader034/viewer/2022050502/5f9440a3ae52d04590363756/html5/thumbnails/319.jpg)
Constant Storage
● Sometimes, a TM needs to remember some additional information that can't be put on the tape.
● In this case, you can use similar techniques from DFAs and introduce extra states into the TM's fnite-state control.
● The fnite-state control can only remember one of fnitely many things, but that might be all that you need!
![Page 320: Turing Machines - Stanford Universityweb.stanford.edu/class/archive/cs/cs103/cs103.1198...the coming years An important theme is not simply proving more theorems, but achieving automation](https://reader034.vdocuments.us/reader034/viewer/2022050502/5f9440a3ae52d04590363756/html5/thumbnails/320.jpg)
Another TM Design
● We just designed a TM for this language over Σ = {0, 1}:
L = { w ∈ Σ* | w has the same number of 0s and 1s }
● Let's do a quick review of how it worked.
![Page 321: Turing Machines - Stanford Universityweb.stanford.edu/class/archive/cs/cs103/cs103.1198...the coming years An important theme is not simply proving more theorems, but achieving automation](https://reader034.vdocuments.us/reader034/viewer/2022050502/5f9440a3ae52d04590363756/html5/thumbnails/321.jpg)
0 0 0 1 1 1 0… …1
A Leap of Faith
![Page 322: Turing Machines - Stanford Universityweb.stanford.edu/class/archive/cs/cs103/cs103.1198...the coming years An important theme is not simply proving more theorems, but achieving automation](https://reader034.vdocuments.us/reader034/viewer/2022050502/5f9440a3ae52d04590363756/html5/thumbnails/322.jpg)
0 0 1 1 1 0… …1
A Leap of Faith
![Page 323: Turing Machines - Stanford Universityweb.stanford.edu/class/archive/cs/cs103/cs103.1198...the coming years An important theme is not simply proving more theorems, but achieving automation](https://reader034.vdocuments.us/reader034/viewer/2022050502/5f9440a3ae52d04590363756/html5/thumbnails/323.jpg)
0 0 1 1 1 0… …1
A Leap of Faith
![Page 324: Turing Machines - Stanford Universityweb.stanford.edu/class/archive/cs/cs103/cs103.1198...the coming years An important theme is not simply proving more theorems, but achieving automation](https://reader034.vdocuments.us/reader034/viewer/2022050502/5f9440a3ae52d04590363756/html5/thumbnails/324.jpg)
0 0 1 1 1 0… …1
A Leap of Faith
![Page 325: Turing Machines - Stanford Universityweb.stanford.edu/class/archive/cs/cs103/cs103.1198...the coming years An important theme is not simply proving more theorems, but achieving automation](https://reader034.vdocuments.us/reader034/viewer/2022050502/5f9440a3ae52d04590363756/html5/thumbnails/325.jpg)
0 0 1 1 0… …1
A Leap of Faith
![Page 326: Turing Machines - Stanford Universityweb.stanford.edu/class/archive/cs/cs103/cs103.1198...the coming years An important theme is not simply proving more theorems, but achieving automation](https://reader034.vdocuments.us/reader034/viewer/2022050502/5f9440a3ae52d04590363756/html5/thumbnails/326.jpg)
0 0 1 1 0… …1
A Leap of Faith
![Page 327: Turing Machines - Stanford Universityweb.stanford.edu/class/archive/cs/cs103/cs103.1198...the coming years An important theme is not simply proving more theorems, but achieving automation](https://reader034.vdocuments.us/reader034/viewer/2022050502/5f9440a3ae52d04590363756/html5/thumbnails/327.jpg)
0 0 1 1 0… …1
A Leap of Faith
![Page 328: Turing Machines - Stanford Universityweb.stanford.edu/class/archive/cs/cs103/cs103.1198...the coming years An important theme is not simply proving more theorems, but achieving automation](https://reader034.vdocuments.us/reader034/viewer/2022050502/5f9440a3ae52d04590363756/html5/thumbnails/328.jpg)
0 0 1 1 0… …1
A Leap of Faith
![Page 329: Turing Machines - Stanford Universityweb.stanford.edu/class/archive/cs/cs103/cs103.1198...the coming years An important theme is not simply proving more theorems, but achieving automation](https://reader034.vdocuments.us/reader034/viewer/2022050502/5f9440a3ae52d04590363756/html5/thumbnails/329.jpg)
0 1 1 0… …1
A Leap of Faith
![Page 330: Turing Machines - Stanford Universityweb.stanford.edu/class/archive/cs/cs103/cs103.1198...the coming years An important theme is not simply proving more theorems, but achieving automation](https://reader034.vdocuments.us/reader034/viewer/2022050502/5f9440a3ae52d04590363756/html5/thumbnails/330.jpg)
0 1 1 0… …1
A Leap of Faith
![Page 331: Turing Machines - Stanford Universityweb.stanford.edu/class/archive/cs/cs103/cs103.1198...the coming years An important theme is not simply proving more theorems, but achieving automation](https://reader034.vdocuments.us/reader034/viewer/2022050502/5f9440a3ae52d04590363756/html5/thumbnails/331.jpg)
0 1 1 0… …1
A Leap of Faith
How do we know that this blank isn't one of
the infnitely many blanks after our input
string?
How do we know that this blank isn't one of
the infnitely many blanks after our input
string?
![Page 332: Turing Machines - Stanford Universityweb.stanford.edu/class/archive/cs/cs103/cs103.1198...the coming years An important theme is not simply proving more theorems, but achieving automation](https://reader034.vdocuments.us/reader034/viewer/2022050502/5f9440a3ae52d04590363756/html5/thumbnails/332.jpg)
The Solution
0 0 0 1 1 1 0… …1
![Page 333: Turing Machines - Stanford Universityweb.stanford.edu/class/archive/cs/cs103/cs103.1198...the coming years An important theme is not simply proving more theorems, but achieving automation](https://reader034.vdocuments.us/reader034/viewer/2022050502/5f9440a3ae52d04590363756/html5/thumbnails/333.jpg)
The Solution
× 0 0 1 1 1 0… …1
![Page 334: Turing Machines - Stanford Universityweb.stanford.edu/class/archive/cs/cs103/cs103.1198...the coming years An important theme is not simply proving more theorems, but achieving automation](https://reader034.vdocuments.us/reader034/viewer/2022050502/5f9440a3ae52d04590363756/html5/thumbnails/334.jpg)
The Solution
× 0 0 1 1 1 0… …1
![Page 335: Turing Machines - Stanford Universityweb.stanford.edu/class/archive/cs/cs103/cs103.1198...the coming years An important theme is not simply proving more theorems, but achieving automation](https://reader034.vdocuments.us/reader034/viewer/2022050502/5f9440a3ae52d04590363756/html5/thumbnails/335.jpg)
The Solution
× 0 0 1 1 1 0… …1
![Page 336: Turing Machines - Stanford Universityweb.stanford.edu/class/archive/cs/cs103/cs103.1198...the coming years An important theme is not simply proving more theorems, but achieving automation](https://reader034.vdocuments.us/reader034/viewer/2022050502/5f9440a3ae52d04590363756/html5/thumbnails/336.jpg)
The Solution
× 0 0 × 1 1 0… …1
![Page 337: Turing Machines - Stanford Universityweb.stanford.edu/class/archive/cs/cs103/cs103.1198...the coming years An important theme is not simply proving more theorems, but achieving automation](https://reader034.vdocuments.us/reader034/viewer/2022050502/5f9440a3ae52d04590363756/html5/thumbnails/337.jpg)
The Solution
× 0 0 × 1 1 0… …1
![Page 338: Turing Machines - Stanford Universityweb.stanford.edu/class/archive/cs/cs103/cs103.1198...the coming years An important theme is not simply proving more theorems, but achieving automation](https://reader034.vdocuments.us/reader034/viewer/2022050502/5f9440a3ae52d04590363756/html5/thumbnails/338.jpg)
The Solution
× 0 0 × 1 1 0… …1
![Page 339: Turing Machines - Stanford Universityweb.stanford.edu/class/archive/cs/cs103/cs103.1198...the coming years An important theme is not simply proving more theorems, but achieving automation](https://reader034.vdocuments.us/reader034/viewer/2022050502/5f9440a3ae52d04590363756/html5/thumbnails/339.jpg)
The Solution
× 0 0 × 1 1 0… …1
![Page 340: Turing Machines - Stanford Universityweb.stanford.edu/class/archive/cs/cs103/cs103.1198...the coming years An important theme is not simply proving more theorems, but achieving automation](https://reader034.vdocuments.us/reader034/viewer/2022050502/5f9440a3ae52d04590363756/html5/thumbnails/340.jpg)
The Solution
× 0 0 × 1 1 0… …1
![Page 341: Turing Machines - Stanford Universityweb.stanford.edu/class/archive/cs/cs103/cs103.1198...the coming years An important theme is not simply proving more theorems, but achieving automation](https://reader034.vdocuments.us/reader034/viewer/2022050502/5f9440a3ae52d04590363756/html5/thumbnails/341.jpg)
The Solution
× 0 0 × 1 1 0… …1
![Page 342: Turing Machines - Stanford Universityweb.stanford.edu/class/archive/cs/cs103/cs103.1198...the coming years An important theme is not simply proving more theorems, but achieving automation](https://reader034.vdocuments.us/reader034/viewer/2022050502/5f9440a3ae52d04590363756/html5/thumbnails/342.jpg)
The Solution
× × 0 × 1 1 0… …1
![Page 343: Turing Machines - Stanford Universityweb.stanford.edu/class/archive/cs/cs103/cs103.1198...the coming years An important theme is not simply proving more theorems, but achieving automation](https://reader034.vdocuments.us/reader034/viewer/2022050502/5f9440a3ae52d04590363756/html5/thumbnails/343.jpg)
The Solution
× × 0 × 1 1 0… …1
![Page 344: Turing Machines - Stanford Universityweb.stanford.edu/class/archive/cs/cs103/cs103.1198...the coming years An important theme is not simply proving more theorems, but achieving automation](https://reader034.vdocuments.us/reader034/viewer/2022050502/5f9440a3ae52d04590363756/html5/thumbnails/344.jpg)
The Solution
× × 0 × 1 1 0… …1
![Page 345: Turing Machines - Stanford Universityweb.stanford.edu/class/archive/cs/cs103/cs103.1198...the coming years An important theme is not simply proving more theorems, but achieving automation](https://reader034.vdocuments.us/reader034/viewer/2022050502/5f9440a3ae52d04590363756/html5/thumbnails/345.jpg)
The Solution
× × 0 × × 1 0… …1
![Page 346: Turing Machines - Stanford Universityweb.stanford.edu/class/archive/cs/cs103/cs103.1198...the coming years An important theme is not simply proving more theorems, but achieving automation](https://reader034.vdocuments.us/reader034/viewer/2022050502/5f9440a3ae52d04590363756/html5/thumbnails/346.jpg)
The Solution
× × 0 × × 1 0… …1
![Page 347: Turing Machines - Stanford Universityweb.stanford.edu/class/archive/cs/cs103/cs103.1198...the coming years An important theme is not simply proving more theorems, but achieving automation](https://reader034.vdocuments.us/reader034/viewer/2022050502/5f9440a3ae52d04590363756/html5/thumbnails/347.jpg)
The Solution
× × 0 × × 1 0… …1
![Page 348: Turing Machines - Stanford Universityweb.stanford.edu/class/archive/cs/cs103/cs103.1198...the coming years An important theme is not simply proving more theorems, but achieving automation](https://reader034.vdocuments.us/reader034/viewer/2022050502/5f9440a3ae52d04590363756/html5/thumbnails/348.jpg)
The Solution
× × 0 × × 1 0… …1
![Page 349: Turing Machines - Stanford Universityweb.stanford.edu/class/archive/cs/cs103/cs103.1198...the coming years An important theme is not simply proving more theorems, but achieving automation](https://reader034.vdocuments.us/reader034/viewer/2022050502/5f9440a3ae52d04590363756/html5/thumbnails/349.jpg)
The Solution
× × 0 × × 1 0… …1
![Page 350: Turing Machines - Stanford Universityweb.stanford.edu/class/archive/cs/cs103/cs103.1198...the coming years An important theme is not simply proving more theorems, but achieving automation](https://reader034.vdocuments.us/reader034/viewer/2022050502/5f9440a3ae52d04590363756/html5/thumbnails/350.jpg)
The Solution
× × 0 × × 1 0… …1
![Page 351: Turing Machines - Stanford Universityweb.stanford.edu/class/archive/cs/cs103/cs103.1198...the coming years An important theme is not simply proving more theorems, but achieving automation](https://reader034.vdocuments.us/reader034/viewer/2022050502/5f9440a3ae52d04590363756/html5/thumbnails/351.jpg)
A Diferent Idea
![Page 352: Turing Machines - Stanford Universityweb.stanford.edu/class/archive/cs/cs103/cs103.1198...the coming years An important theme is not simply proving more theorems, but achieving automation](https://reader034.vdocuments.us/reader034/viewer/2022050502/5f9440a3ae52d04590363756/html5/thumbnails/352.jpg)
0 0 0 1 1 1 0… …1
A Diferent Strategy
![Page 353: Turing Machines - Stanford Universityweb.stanford.edu/class/archive/cs/cs103/cs103.1198...the coming years An important theme is not simply proving more theorems, but achieving automation](https://reader034.vdocuments.us/reader034/viewer/2022050502/5f9440a3ae52d04590363756/html5/thumbnails/353.jpg)
0 0 0 1 1 1 0… …1
A Diferent Strategy
Could we sort
the characters
of this string?
Could we sort
the characters
of this string?
![Page 354: Turing Machines - Stanford Universityweb.stanford.edu/class/archive/cs/cs103/cs103.1198...the coming years An important theme is not simply proving more theorems, but achieving automation](https://reader034.vdocuments.us/reader034/viewer/2022050502/5f9440a3ae52d04590363756/html5/thumbnails/354.jpg)
0 0 0 1 1 1 0… …1
A Diferent Strategy
Observation 1: A string
of 0s and 1s is sorted
if it matches the
regex 0*1*.
Observation 1: A string
of 0s and 1s is sorted
if it matches the
regex 0*1*.
![Page 355: Turing Machines - Stanford Universityweb.stanford.edu/class/archive/cs/cs103/cs103.1198...the coming years An important theme is not simply proving more theorems, but achieving automation](https://reader034.vdocuments.us/reader034/viewer/2022050502/5f9440a3ae52d04590363756/html5/thumbnails/355.jpg)
0 0 0 1 1 1 0… …1
A Diferent Strategy
Observation 1: A string
of 0s and 1s is sorted
if it matches the
regex 0*1*.
Observation 1: A string
of 0s and 1s is sorted
if it matches the
regex 0*1*.
![Page 356: Turing Machines - Stanford Universityweb.stanford.edu/class/archive/cs/cs103/cs103.1198...the coming years An important theme is not simply proving more theorems, but achieving automation](https://reader034.vdocuments.us/reader034/viewer/2022050502/5f9440a3ae52d04590363756/html5/thumbnails/356.jpg)
0 0 0 1 1 1 0… …1
A Diferent Strategy
Observation 1: A string
of 0s and 1s is sorted
if it matches the
regex 0*1*.
Observation 1: A string
of 0s and 1s is sorted
if it matches the
regex 0*1*.
![Page 357: Turing Machines - Stanford Universityweb.stanford.edu/class/archive/cs/cs103/cs103.1198...the coming years An important theme is not simply proving more theorems, but achieving automation](https://reader034.vdocuments.us/reader034/viewer/2022050502/5f9440a3ae52d04590363756/html5/thumbnails/357.jpg)
0 0 0 1 1 1 0… …1
A Diferent Strategy
Observation 1: A string
of 0s and 1s is sorted
if it matches the
regex 0*1*.
Observation 1: A string
of 0s and 1s is sorted
if it matches the
regex 0*1*.
![Page 358: Turing Machines - Stanford Universityweb.stanford.edu/class/archive/cs/cs103/cs103.1198...the coming years An important theme is not simply proving more theorems, but achieving automation](https://reader034.vdocuments.us/reader034/viewer/2022050502/5f9440a3ae52d04590363756/html5/thumbnails/358.jpg)
0 0 0 1 1 1 0… …1
A Diferent Strategy
Observation 1: A string
of 0s and 1s is sorted
if it matches the
regex 0*1*.
Observation 1: A string
of 0s and 1s is sorted
if it matches the
regex 0*1*.
![Page 359: Turing Machines - Stanford Universityweb.stanford.edu/class/archive/cs/cs103/cs103.1198...the coming years An important theme is not simply proving more theorems, but achieving automation](https://reader034.vdocuments.us/reader034/viewer/2022050502/5f9440a3ae52d04590363756/html5/thumbnails/359.jpg)
0 0 0 1 1 1 0… …1
A Diferent Strategy
Observation 1: A string
of 0s and 1s is sorted
if it matches the
regex 0*1*.
Observation 1: A string
of 0s and 1s is sorted
if it matches the
regex 0*1*.
![Page 360: Turing Machines - Stanford Universityweb.stanford.edu/class/archive/cs/cs103/cs103.1198...the coming years An important theme is not simply proving more theorems, but achieving automation](https://reader034.vdocuments.us/reader034/viewer/2022050502/5f9440a3ae52d04590363756/html5/thumbnails/360.jpg)
0 0 0 1 1 1 0… …1
A Diferent Strategy
Observation 1: A string
of 0s and 1s is sorted
if it matches the
regex 0*1*.
Observation 1: A string
of 0s and 1s is sorted
if it matches the
regex 0*1*.
![Page 361: Turing Machines - Stanford Universityweb.stanford.edu/class/archive/cs/cs103/cs103.1198...the coming years An important theme is not simply proving more theorems, but achieving automation](https://reader034.vdocuments.us/reader034/viewer/2022050502/5f9440a3ae52d04590363756/html5/thumbnails/361.jpg)
0 0 0 1 1 1 0… …1
A Diferent Strategy
Observation 1: A string
of 0s and 1s is sorted
if it matches the
regex 0*1*.
Observation 1: A string
of 0s and 1s is sorted
if it matches the
regex 0*1*.
![Page 362: Turing Machines - Stanford Universityweb.stanford.edu/class/archive/cs/cs103/cs103.1198...the coming years An important theme is not simply proving more theorems, but achieving automation](https://reader034.vdocuments.us/reader034/viewer/2022050502/5f9440a3ae52d04590363756/html5/thumbnails/362.jpg)
…0 0 0 1 1 1… 01
A Diferent Strategy
Observation 1: A string
of 0s and 1s is sorted
if it matches the
regex 0*1*.
Observation 1: A string
of 0s and 1s is sorted
if it matches the
regex 0*1*.
![Page 363: Turing Machines - Stanford Universityweb.stanford.edu/class/archive/cs/cs103/cs103.1198...the coming years An important theme is not simply proving more theorems, but achieving automation](https://reader034.vdocuments.us/reader034/viewer/2022050502/5f9440a3ae52d04590363756/html5/thumbnails/363.jpg)
…0 0 0 1 1 1… 01
A Diferent Strategy
Observation 2: A
string of 0s and 1s is
not sorted if it
contains 10 as a
substring.
Observation 2: A
string of 0s and 1s is
not sorted if it
contains 10 as a
substring.
![Page 364: Turing Machines - Stanford Universityweb.stanford.edu/class/archive/cs/cs103/cs103.1198...the coming years An important theme is not simply proving more theorems, but achieving automation](https://reader034.vdocuments.us/reader034/viewer/2022050502/5f9440a3ae52d04590363756/html5/thumbnails/364.jpg)
A Diferent Strategy
…0 0 0 1 1 1… 11
Observation 2: A
string of 0s and 1s is
not sorted if it
contains 10 as a
substring.
Observation 2: A
string of 0s and 1s is
not sorted if it
contains 10 as a
substring.
![Page 365: Turing Machines - Stanford Universityweb.stanford.edu/class/archive/cs/cs103/cs103.1198...the coming years An important theme is not simply proving more theorems, but achieving automation](https://reader034.vdocuments.us/reader034/viewer/2022050502/5f9440a3ae52d04590363756/html5/thumbnails/365.jpg)
A Diferent Strategy
…0 0 0 1 1 1… 10
Observation 2: A
string of 0s and 1s is
not sorted if it
contains 10 as a
substring.
Observation 2: A
string of 0s and 1s is
not sorted if it
contains 10 as a
substring.
![Page 366: Turing Machines - Stanford Universityweb.stanford.edu/class/archive/cs/cs103/cs103.1198...the coming years An important theme is not simply proving more theorems, but achieving automation](https://reader034.vdocuments.us/reader034/viewer/2022050502/5f9440a3ae52d04590363756/html5/thumbnails/366.jpg)
0 0 0 1 1 1 1… …0
A Diferent Strategy
Observation 2: A
string of 0s and 1s is
not sorted if it
contains 10 as a
substring.
Observation 2: A
string of 0s and 1s is
not sorted if it
contains 10 as a
substring.
![Page 367: Turing Machines - Stanford Universityweb.stanford.edu/class/archive/cs/cs103/cs103.1198...the coming years An important theme is not simply proving more theorems, but achieving automation](https://reader034.vdocuments.us/reader034/viewer/2022050502/5f9440a3ae52d04590363756/html5/thumbnails/367.jpg)
0 0 0 1 1 1 1… …0
A Diferent Strategy
Idea: Repeatedly fnd
a copy of 10 and
replace it with 01.
Idea: Repeatedly fnd
a copy of 10 and
replace it with 01.
![Page 368: Turing Machines - Stanford Universityweb.stanford.edu/class/archive/cs/cs103/cs103.1198...the coming years An important theme is not simply proving more theorems, but achieving automation](https://reader034.vdocuments.us/reader034/viewer/2022050502/5f9440a3ae52d04590363756/html5/thumbnails/368.jpg)
0 0 0 1 1 1 1… …0
A Diferent Strategy
Idea: Repeatedly fnd
a copy of 10 and
replace it with 01.
Idea: Repeatedly fnd
a copy of 10 and
replace it with 01.
![Page 369: Turing Machines - Stanford Universityweb.stanford.edu/class/archive/cs/cs103/cs103.1198...the coming years An important theme is not simply proving more theorems, but achieving automation](https://reader034.vdocuments.us/reader034/viewer/2022050502/5f9440a3ae52d04590363756/html5/thumbnails/369.jpg)
0 0 0 1 1 1 1… …0
A Diferent Strategy
Idea: Repeatedly fnd
a copy of 10 and
replace it with 01.
Idea: Repeatedly fnd
a copy of 10 and
replace it with 01.
![Page 370: Turing Machines - Stanford Universityweb.stanford.edu/class/archive/cs/cs103/cs103.1198...the coming years An important theme is not simply proving more theorems, but achieving automation](https://reader034.vdocuments.us/reader034/viewer/2022050502/5f9440a3ae52d04590363756/html5/thumbnails/370.jpg)
0 0 0 1 1 1 1… …0
A Diferent Strategy
Idea: Repeatedly fnd
a copy of 10 and
replace it with 01.
Idea: Repeatedly fnd
a copy of 10 and
replace it with 01.
![Page 371: Turing Machines - Stanford Universityweb.stanford.edu/class/archive/cs/cs103/cs103.1198...the coming years An important theme is not simply proving more theorems, but achieving automation](https://reader034.vdocuments.us/reader034/viewer/2022050502/5f9440a3ae52d04590363756/html5/thumbnails/371.jpg)
0 0 0 1 1 1 1… …0
A Diferent Strategy
Idea: Repeatedly fnd
a copy of 10 and
replace it with 01.
Idea: Repeatedly fnd
a copy of 10 and
replace it with 01.
![Page 372: Turing Machines - Stanford Universityweb.stanford.edu/class/archive/cs/cs103/cs103.1198...the coming years An important theme is not simply proving more theorems, but achieving automation](https://reader034.vdocuments.us/reader034/viewer/2022050502/5f9440a3ae52d04590363756/html5/thumbnails/372.jpg)
0 0 0 1 1 1 1… …0
A Diferent Strategy
Idea: Repeatedly fnd
a copy of 10 and
replace it with 01.
Idea: Repeatedly fnd
a copy of 10 and
replace it with 01.
![Page 373: Turing Machines - Stanford Universityweb.stanford.edu/class/archive/cs/cs103/cs103.1198...the coming years An important theme is not simply proving more theorems, but achieving automation](https://reader034.vdocuments.us/reader034/viewer/2022050502/5f9440a3ae52d04590363756/html5/thumbnails/373.jpg)
0 0 0 1 1 1 1… …0
A Diferent Strategy
Idea: Repeatedly fnd
a copy of 10 and
replace it with 01.
Idea: Repeatedly fnd
a copy of 10 and
replace it with 01.
![Page 374: Turing Machines - Stanford Universityweb.stanford.edu/class/archive/cs/cs103/cs103.1198...the coming years An important theme is not simply proving more theorems, but achieving automation](https://reader034.vdocuments.us/reader034/viewer/2022050502/5f9440a3ae52d04590363756/html5/thumbnails/374.jpg)
0 0 0 1 1 1 1… …0
A Diferent Strategy
Idea: Repeatedly fnd
a copy of 10 and
replace it with 01.
Idea: Repeatedly fnd
a copy of 10 and
replace it with 01.
![Page 375: Turing Machines - Stanford Universityweb.stanford.edu/class/archive/cs/cs103/cs103.1198...the coming years An important theme is not simply proving more theorems, but achieving automation](https://reader034.vdocuments.us/reader034/viewer/2022050502/5f9440a3ae52d04590363756/html5/thumbnails/375.jpg)
0 0 0 1 1 1 1… …0
A Diferent Strategy
Idea: Repeatedly fnd
a copy of 10 and
replace it with 01.
Idea: Repeatedly fnd
a copy of 10 and
replace it with 01.
![Page 376: Turing Machines - Stanford Universityweb.stanford.edu/class/archive/cs/cs103/cs103.1198...the coming years An important theme is not simply proving more theorems, but achieving automation](https://reader034.vdocuments.us/reader034/viewer/2022050502/5f9440a3ae52d04590363756/html5/thumbnails/376.jpg)
0 0 0 1 1 1 1… …0
A Diferent Strategy
Idea: Repeatedly fnd
a copy of 10 and
replace it with 01.
Idea: Repeatedly fnd
a copy of 10 and
replace it with 01.
![Page 377: Turing Machines - Stanford Universityweb.stanford.edu/class/archive/cs/cs103/cs103.1198...the coming years An important theme is not simply proving more theorems, but achieving automation](https://reader034.vdocuments.us/reader034/viewer/2022050502/5f9440a3ae52d04590363756/html5/thumbnails/377.jpg)
0 0 0 1 1 1 1… …0
A Diferent Strategy
Idea: Repeatedly fnd
a copy of 10 and
replace it with 01.
Idea: Repeatedly fnd
a copy of 10 and
replace it with 01.
![Page 378: Turing Machines - Stanford Universityweb.stanford.edu/class/archive/cs/cs103/cs103.1198...the coming years An important theme is not simply proving more theorems, but achieving automation](https://reader034.vdocuments.us/reader034/viewer/2022050502/5f9440a3ae52d04590363756/html5/thumbnails/378.jpg)
0 0 0 1 1 1 1… …0
A Diferent Strategy
Idea: Repeatedly fnd
a copy of 10 and
replace it with 01.
Idea: Repeatedly fnd
a copy of 10 and
replace it with 01.
![Page 379: Turing Machines - Stanford Universityweb.stanford.edu/class/archive/cs/cs103/cs103.1198...the coming years An important theme is not simply proving more theorems, but achieving automation](https://reader034.vdocuments.us/reader034/viewer/2022050502/5f9440a3ae52d04590363756/html5/thumbnails/379.jpg)
0 0 0 1 1 1 1… …0
A Diferent Strategy
Idea: Repeatedly fnd
a copy of 10 and
replace it with 01.
Idea: Repeatedly fnd
a copy of 10 and
replace it with 01.
![Page 380: Turing Machines - Stanford Universityweb.stanford.edu/class/archive/cs/cs103/cs103.1198...the coming years An important theme is not simply proving more theorems, but achieving automation](https://reader034.vdocuments.us/reader034/viewer/2022050502/5f9440a3ae52d04590363756/html5/thumbnails/380.jpg)
0 0 0 1 1 1 1… …0
A Diferent Strategy
Idea: Repeatedly fnd
a copy of 10 and
replace it with 01.
Idea: Repeatedly fnd
a copy of 10 and
replace it with 01.
![Page 381: Turing Machines - Stanford Universityweb.stanford.edu/class/archive/cs/cs103/cs103.1198...the coming years An important theme is not simply proving more theorems, but achieving automation](https://reader034.vdocuments.us/reader034/viewer/2022050502/5f9440a3ae52d04590363756/html5/thumbnails/381.jpg)
0 0 0 1 1 1 1… …0
A Diferent Strategy
Idea: Repeatedly fnd
a copy of 10 and
replace it with 01.
Idea: Repeatedly fnd
a copy of 10 and
replace it with 01.
![Page 382: Turing Machines - Stanford Universityweb.stanford.edu/class/archive/cs/cs103/cs103.1198...the coming years An important theme is not simply proving more theorems, but achieving automation](https://reader034.vdocuments.us/reader034/viewer/2022050502/5f9440a3ae52d04590363756/html5/thumbnails/382.jpg)
0 0 0 1 1 1 1… …1
A Diferent Strategy
Idea: Repeatedly fnd
a copy of 10 and
replace it with 01.
Idea: Repeatedly fnd
a copy of 10 and
replace it with 01.
![Page 383: Turing Machines - Stanford Universityweb.stanford.edu/class/archive/cs/cs103/cs103.1198...the coming years An important theme is not simply proving more theorems, but achieving automation](https://reader034.vdocuments.us/reader034/viewer/2022050502/5f9440a3ae52d04590363756/html5/thumbnails/383.jpg)
0 0 0 1 1 0 1… …1
A Diferent Strategy
Idea: Repeatedly fnd
a copy of 10 and
replace it with 01.
Idea: Repeatedly fnd
a copy of 10 and
replace it with 01.
![Page 384: Turing Machines - Stanford Universityweb.stanford.edu/class/archive/cs/cs103/cs103.1198...the coming years An important theme is not simply proving more theorems, but achieving automation](https://reader034.vdocuments.us/reader034/viewer/2022050502/5f9440a3ae52d04590363756/html5/thumbnails/384.jpg)
0 0 0 1 1 0 1… …1
A Diferent Strategy
Idea: Repeatedly fnd
a copy of 10 and
replace it with 01.
Idea: Repeatedly fnd
a copy of 10 and
replace it with 01.
![Page 385: Turing Machines - Stanford Universityweb.stanford.edu/class/archive/cs/cs103/cs103.1198...the coming years An important theme is not simply proving more theorems, but achieving automation](https://reader034.vdocuments.us/reader034/viewer/2022050502/5f9440a3ae52d04590363756/html5/thumbnails/385.jpg)
0 0 0 1 1 0 1… …1
A Diferent Strategy
Idea: Repeatedly fnd
a copy of 10 and
replace it with 01.
Idea: Repeatedly fnd
a copy of 10 and
replace it with 01.
![Page 386: Turing Machines - Stanford Universityweb.stanford.edu/class/archive/cs/cs103/cs103.1198...the coming years An important theme is not simply proving more theorems, but achieving automation](https://reader034.vdocuments.us/reader034/viewer/2022050502/5f9440a3ae52d04590363756/html5/thumbnails/386.jpg)
Let's Build It!
![Page 387: Turing Machines - Stanford Universityweb.stanford.edu/class/archive/cs/cs103/cs103.1198...the coming years An important theme is not simply proving more theorems, but achieving automation](https://reader034.vdocuments.us/reader034/viewer/2022050502/5f9440a3ae52d04590363756/html5/thumbnails/387.jpg)
0 0 1 1 0 0… …
![Page 388: Turing Machines - Stanford Universityweb.stanford.edu/class/archive/cs/cs103/cs103.1198...the coming years An important theme is not simply proving more theorems, but achieving automation](https://reader034.vdocuments.us/reader034/viewer/2022050502/5f9440a3ae52d04590363756/html5/thumbnails/388.jpg)
0 0 1 1 0 0… …
start
![Page 389: Turing Machines - Stanford Universityweb.stanford.edu/class/archive/cs/cs103/cs103.1198...the coming years An important theme is not simply proving more theorems, but achieving automation](https://reader034.vdocuments.us/reader034/viewer/2022050502/5f9440a3ae52d04590363756/html5/thumbnails/389.jpg)
0*
0 0 1 1 0 0… …
start
![Page 390: Turing Machines - Stanford Universityweb.stanford.edu/class/archive/cs/cs103/cs103.1198...the coming years An important theme is not simply proving more theorems, but achieving automation](https://reader034.vdocuments.us/reader034/viewer/2022050502/5f9440a3ae52d04590363756/html5/thumbnails/390.jpg)
0*
0 0 1 1 0 0… …
start
0*
![Page 391: Turing Machines - Stanford Universityweb.stanford.edu/class/archive/cs/cs103/cs103.1198...the coming years An important theme is not simply proving more theorems, but achieving automation](https://reader034.vdocuments.us/reader034/viewer/2022050502/5f9440a3ae52d04590363756/html5/thumbnails/391.jpg)
0*
0 0 1 1 0 0… …
start
0*0 → 0, R
![Page 392: Turing Machines - Stanford Universityweb.stanford.edu/class/archive/cs/cs103/cs103.1198...the coming years An important theme is not simply proving more theorems, but achieving automation](https://reader034.vdocuments.us/reader034/viewer/2022050502/5f9440a3ae52d04590363756/html5/thumbnails/392.jpg)
0*
0 0 1 1 0 0… …
start
0*0 → 0, R
![Page 393: Turing Machines - Stanford Universityweb.stanford.edu/class/archive/cs/cs103/cs103.1198...the coming years An important theme is not simply proving more theorems, but achieving automation](https://reader034.vdocuments.us/reader034/viewer/2022050502/5f9440a3ae52d04590363756/html5/thumbnails/393.jpg)
0*
0 0 1 1 0 0… …
start
0*0 → 0, R
![Page 394: Turing Machines - Stanford Universityweb.stanford.edu/class/archive/cs/cs103/cs103.1198...the coming years An important theme is not simply proving more theorems, but achieving automation](https://reader034.vdocuments.us/reader034/viewer/2022050502/5f9440a3ae52d04590363756/html5/thumbnails/394.jpg)
0*
0 0 1 1 0 0… …
start
0*0 → 0, R 1 → 1, R
![Page 395: Turing Machines - Stanford Universityweb.stanford.edu/class/archive/cs/cs103/cs103.1198...the coming years An important theme is not simply proving more theorems, but achieving automation](https://reader034.vdocuments.us/reader034/viewer/2022050502/5f9440a3ae52d04590363756/html5/thumbnails/395.jpg)
0*0*
0 0 1 1 0 0… …
start
0 → 0, R 1 → 1, R
![Page 396: Turing Machines - Stanford Universityweb.stanford.edu/class/archive/cs/cs103/cs103.1198...the coming years An important theme is not simply proving more theorems, but achieving automation](https://reader034.vdocuments.us/reader034/viewer/2022050502/5f9440a3ae52d04590363756/html5/thumbnails/396.jpg)
0*0*
0 0 1 1 0 0… …
start
0 → 0, R 0*1*1 → 1, R
![Page 397: Turing Machines - Stanford Universityweb.stanford.edu/class/archive/cs/cs103/cs103.1198...the coming years An important theme is not simply proving more theorems, but achieving automation](https://reader034.vdocuments.us/reader034/viewer/2022050502/5f9440a3ae52d04590363756/html5/thumbnails/397.jpg)
0*0*
0 0 1 1 0 0… …
start
0 → 0, R 0*1*1 → 1, R
1 → 1, R
![Page 398: Turing Machines - Stanford Universityweb.stanford.edu/class/archive/cs/cs103/cs103.1198...the coming years An important theme is not simply proving more theorems, but achieving automation](https://reader034.vdocuments.us/reader034/viewer/2022050502/5f9440a3ae52d04590363756/html5/thumbnails/398.jpg)
0*0*
0 0 1 1 0 0… …
start
0 → 0, R 0*1*1 → 1, R
1 → 1, R
![Page 399: Turing Machines - Stanford Universityweb.stanford.edu/class/archive/cs/cs103/cs103.1198...the coming years An important theme is not simply proving more theorems, but achieving automation](https://reader034.vdocuments.us/reader034/viewer/2022050502/5f9440a3ae52d04590363756/html5/thumbnails/399.jpg)
0*0*
0 0 1 1 0 0… …
start
0 → 0, R 0*1*1 → 1, R
1 → 1, R
0 → ?, ?
![Page 400: Turing Machines - Stanford Universityweb.stanford.edu/class/archive/cs/cs103/cs103.1198...the coming years An important theme is not simply proving more theorems, but achieving automation](https://reader034.vdocuments.us/reader034/viewer/2022050502/5f9440a3ae52d04590363756/html5/thumbnails/400.jpg)
0*0*
0 0 1 1 0 0… …
start
0 → 0, R 0*1*1 → 1, R
1 → 1, R
0 → ?, ?
Based on what we’re trying to do, what should this transition say?
Based on what we’re trying to do, what should this transition say?
![Page 401: Turing Machines - Stanford Universityweb.stanford.edu/class/archive/cs/cs103/cs103.1198...the coming years An important theme is not simply proving more theorems, but achieving automation](https://reader034.vdocuments.us/reader034/viewer/2022050502/5f9440a3ae52d04590363756/html5/thumbnails/401.jpg)
0*0*
0 0 1 1 0… …
start
0 → 0, R 0*1*1 → 1, R
1 → 1, R
0 → ?, ?
0
![Page 402: Turing Machines - Stanford Universityweb.stanford.edu/class/archive/cs/cs103/cs103.1198...the coming years An important theme is not simply proving more theorems, but achieving automation](https://reader034.vdocuments.us/reader034/viewer/2022050502/5f9440a3ae52d04590363756/html5/thumbnails/402.jpg)
0*0*
0 0 1 1 0… …
start
0 → 0, R 0*1*1 → 1, R
1 → 1, R
0 → 1, L
0
![Page 403: Turing Machines - Stanford Universityweb.stanford.edu/class/archive/cs/cs103/cs103.1198...the coming years An important theme is not simply proving more theorems, but achieving automation](https://reader034.vdocuments.us/reader034/viewer/2022050502/5f9440a3ae52d04590363756/html5/thumbnails/403.jpg)
0*0*
0 0 1 1 0… …
start
0 → 0, R 0*1*1 → 1, R
1 → 1, R
0 → 1, L
1
0*1*
![Page 404: Turing Machines - Stanford Universityweb.stanford.edu/class/archive/cs/cs103/cs103.1198...the coming years An important theme is not simply proving more theorems, but achieving automation](https://reader034.vdocuments.us/reader034/viewer/2022050502/5f9440a3ae52d04590363756/html5/thumbnails/404.jpg)
0*0*
0 0 1 1 0… …
start
0 → 0, R 0*1*1 → 1, R
1 → 1, R
Fix
01
0 → 1, L
1
0*1*
![Page 405: Turing Machines - Stanford Universityweb.stanford.edu/class/archive/cs/cs103/cs103.1198...the coming years An important theme is not simply proving more theorems, but achieving automation](https://reader034.vdocuments.us/reader034/viewer/2022050502/5f9440a3ae52d04590363756/html5/thumbnails/405.jpg)
0*0*
0 0 1 1 0… …
start
0 → 0, R 0*1*1 → 1, R
1 → 1, R
Fix
01
0 → 1, L
1
0*1*
1 → 0, L
![Page 406: Turing Machines - Stanford Universityweb.stanford.edu/class/archive/cs/cs103/cs103.1198...the coming years An important theme is not simply proving more theorems, but achieving automation](https://reader034.vdocuments.us/reader034/viewer/2022050502/5f9440a3ae52d04590363756/html5/thumbnails/406.jpg)
0*0*
0 0 1 0 0… …
start
0 → 0, R 0*1*1 → 1, R
1 → 1, R
Fix
01
0 → 1, L
1
0*1*
1 → 0, LFix
01
![Page 407: Turing Machines - Stanford Universityweb.stanford.edu/class/archive/cs/cs103/cs103.1198...the coming years An important theme is not simply proving more theorems, but achieving automation](https://reader034.vdocuments.us/reader034/viewer/2022050502/5f9440a3ae52d04590363756/html5/thumbnails/407.jpg)
0*0*
0 0 1 0 0… …
start
0 → 0, R 0*1*1 → 1, R
1 → 1, R
Fix
01
0 → 1, L
1
0*1*
1 → 0, LFix
01
![Page 408: Turing Machines - Stanford Universityweb.stanford.edu/class/archive/cs/cs103/cs103.1198...the coming years An important theme is not simply proving more theorems, but achieving automation](https://reader034.vdocuments.us/reader034/viewer/2022050502/5f9440a3ae52d04590363756/html5/thumbnails/408.jpg)
0*0*
0 0 1 0 0… …
start
0 → 0, R 0*1*1 → 1, R
1 → 1, R
Fix
01
0 → 1, L
1
0*1*
Go
Home
1 → 0, LFix
01
![Page 409: Turing Machines - Stanford Universityweb.stanford.edu/class/archive/cs/cs103/cs103.1198...the coming years An important theme is not simply proving more theorems, but achieving automation](https://reader034.vdocuments.us/reader034/viewer/2022050502/5f9440a3ae52d04590363756/html5/thumbnails/409.jpg)
0*0*
0 0 1 0 0… …
start
0 → 0, R 0*1*1 → 1, R
1 → 1, R
Fix
01
0 → 1, L
1
0*1*
Go
Home
1 → 0, LFix
01
0 → 0, L
1 → 1, L
![Page 410: Turing Machines - Stanford Universityweb.stanford.edu/class/archive/cs/cs103/cs103.1198...the coming years An important theme is not simply proving more theorems, but achieving automation](https://reader034.vdocuments.us/reader034/viewer/2022050502/5f9440a3ae52d04590363756/html5/thumbnails/410.jpg)
0*0*
0 0 1 0 0… …
start
0 → 0, R 0*1*1 → 1, R
1 → 1, R
Fix
01
0 → 1, L
1
0*1*
Go
Home
1 → 0, LFix
01
0 → 0, L
1 → 1, L
![Page 411: Turing Machines - Stanford Universityweb.stanford.edu/class/archive/cs/cs103/cs103.1198...the coming years An important theme is not simply proving more theorems, but achieving automation](https://reader034.vdocuments.us/reader034/viewer/2022050502/5f9440a3ae52d04590363756/html5/thumbnails/411.jpg)
0*0*
0 0 1 0 0… …
start
0 → 0, R 0*1*1 → 1, R
1 → 1, R
Fix
01
0 → 1, L
1
0*1*
Go
Home
1 → 0, LFix
01
0 → 0, L
1 → 1, L
![Page 412: Turing Machines - Stanford Universityweb.stanford.edu/class/archive/cs/cs103/cs103.1198...the coming years An important theme is not simply proving more theorems, but achieving automation](https://reader034.vdocuments.us/reader034/viewer/2022050502/5f9440a3ae52d04590363756/html5/thumbnails/412.jpg)
0*0*
0 0 1 0 0… …
start
0 → 0, R 0*1*1 → 1, R
1 → 1, R
Fix
01
0 → 1, L
1
0*1*
Go
Home
1 → 0, LFix
01
0 → 0, L
1 → 1, L
![Page 413: Turing Machines - Stanford Universityweb.stanford.edu/class/archive/cs/cs103/cs103.1198...the coming years An important theme is not simply proving more theorems, but achieving automation](https://reader034.vdocuments.us/reader034/viewer/2022050502/5f9440a3ae52d04590363756/html5/thumbnails/413.jpg)
0*0*
0 0 1 0 0… …
start
0 → 0, R 0*1*1 → 1, R
1 → 1, R
Fix
01
0 → 1, L
1
0*1*
Go
Home
1 → 0, LFix
01
0 → 0, L
1 → 1, L
→ � �, R
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1 → 1, L
→ � �, R
![Page 453: Turing Machines - Stanford Universityweb.stanford.edu/class/archive/cs/cs103/cs103.1198...the coming years An important theme is not simply proving more theorems, but achieving automation](https://reader034.vdocuments.us/reader034/viewer/2022050502/5f9440a3ae52d04590363756/html5/thumbnails/453.jpg)
0* 0*1*0*
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Home
Fix
01
Go
Home
0*1*
Fix
01
1 → 1, R
0 0 0 0 1… …
start
0 → 0, R 1 → 1, R
0 → 1, L
1
1 → 0, L0 → 0, L
1 → 1, L
→ � �, R
Our ultimate goal here was to sort
everything so we could hand it of to
the machine to check for 0n1n. Let's
rewind the tape head back to the start.
Our ultimate goal here was to sort
everything so we could hand it of to
the machine to check for 0n1n. Let's
rewind the tape head back to the start.
![Page 454: Turing Machines - Stanford Universityweb.stanford.edu/class/archive/cs/cs103/cs103.1198...the coming years An important theme is not simply proving more theorems, but achieving automation](https://reader034.vdocuments.us/reader034/viewer/2022050502/5f9440a3ae52d04590363756/html5/thumbnails/454.jpg)
0* 0*1*0*
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Home
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01
Go
Home
0*1*
Fix
01
1 → 1, R
0 0 0 0 1… …
start
0 → 0, R 1 → 1, R
0 → 1, L
1
1 → 0, L0 → 0, L
1 → 1, L
→ � �, R
To
Start
→ � �, L
![Page 455: Turing Machines - Stanford Universityweb.stanford.edu/class/archive/cs/cs103/cs103.1198...the coming years An important theme is not simply proving more theorems, but achieving automation](https://reader034.vdocuments.us/reader034/viewer/2022050502/5f9440a3ae52d04590363756/html5/thumbnails/455.jpg)
0*1*0* 0*1*0*
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01
Go
Home
Fix
01
1 → 1, R
0 0 0 0 1… …
start
0 → 0, R 1 → 1, R
0 → 1, L
1
1 → 0, L0 → 0, L
1 → 1, L
→ � �, R
To
Start
→ � �, L
![Page 456: Turing Machines - Stanford Universityweb.stanford.edu/class/archive/cs/cs103/cs103.1198...the coming years An important theme is not simply proving more theorems, but achieving automation](https://reader034.vdocuments.us/reader034/viewer/2022050502/5f9440a3ae52d04590363756/html5/thumbnails/456.jpg)
0*1*0* 0*1*0*
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Home
Fix
01
Go
Home
Fix
01
1 → 1, R
0 0 0 0 1… …
start
0 → 0, R 1 → 1, R
0 → 1, L
1
1 → 0, L0 → 0, L
1 → 1, L
→ � �, R
To
Start
→ � �, L
0 → 0, L
1 → 1, L
![Page 457: Turing Machines - Stanford Universityweb.stanford.edu/class/archive/cs/cs103/cs103.1198...the coming years An important theme is not simply proving more theorems, but achieving automation](https://reader034.vdocuments.us/reader034/viewer/2022050502/5f9440a3ae52d04590363756/html5/thumbnails/457.jpg)
0*1*0* 0*1*0*
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01
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Home
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01
1 → 1, R
0 0 0 0 1… …
start
0 → 0, R 1 → 1, R
0 → 1, L
1
1 → 0, L0 → 0, L
1 → 1, L
→ � �, R
To
Start
→ � �, L
0 → 0, L
1 → 1, L
![Page 458: Turing Machines - Stanford Universityweb.stanford.edu/class/archive/cs/cs103/cs103.1198...the coming years An important theme is not simply proving more theorems, but achieving automation](https://reader034.vdocuments.us/reader034/viewer/2022050502/5f9440a3ae52d04590363756/html5/thumbnails/458.jpg)
0*1*0* 0*1*0*
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01
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Home
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01
1 → 1, R
0 0 0 0 1… …
start
0 → 0, R 1 → 1, R
0 → 1, L
1
1 → 0, L0 → 0, L
1 → 1, L
→ � �, R
To
Start
→ � �, L
0 → 0, L
1 → 1, L
![Page 459: Turing Machines - Stanford Universityweb.stanford.edu/class/archive/cs/cs103/cs103.1198...the coming years An important theme is not simply proving more theorems, but achieving automation](https://reader034.vdocuments.us/reader034/viewer/2022050502/5f9440a3ae52d04590363756/html5/thumbnails/459.jpg)
0*1*0* 0*1*0*
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Home
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01
Go
Home
Fix
01
1 → 1, R
0 0 0 0 1… …
start
0 → 0, R 1 → 1, R
0 → 1, L
1
1 → 0, L0 → 0, L
1 → 1, L
→ � �, R
To
Start
→ � �, L
0 → 0, L
1 → 1, L
![Page 460: Turing Machines - Stanford Universityweb.stanford.edu/class/archive/cs/cs103/cs103.1198...the coming years An important theme is not simply proving more theorems, but achieving automation](https://reader034.vdocuments.us/reader034/viewer/2022050502/5f9440a3ae52d04590363756/html5/thumbnails/460.jpg)
0*1*0* 0*1*0*
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01
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Home
Fix
01
1 → 1, R
0 0 0 0 1… …
start
0 → 0, R 1 → 1, R
0 → 1, L
1
1 → 0, L0 → 0, L
1 → 1, L
→ � �, R
To
Start
→ � �, L
0 → 0, L
1 → 1, L
![Page 461: Turing Machines - Stanford Universityweb.stanford.edu/class/archive/cs/cs103/cs103.1198...the coming years An important theme is not simply proving more theorems, but achieving automation](https://reader034.vdocuments.us/reader034/viewer/2022050502/5f9440a3ae52d04590363756/html5/thumbnails/461.jpg)
0*1*0* 0*1*0*
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Home
Fix
01
Go
Home
Fix
01
1 → 1, R
0 0 0 0 1… …
start
0 → 0, R 1 → 1, R
0 → 1, L
1
1 → 0, L0 → 0, L
1 → 1, L
→ � �, R
To
Start
→ � �, L
0 → 0, L
1 → 1, L
![Page 462: Turing Machines - Stanford Universityweb.stanford.edu/class/archive/cs/cs103/cs103.1198...the coming years An important theme is not simply proving more theorems, but achieving automation](https://reader034.vdocuments.us/reader034/viewer/2022050502/5f9440a3ae52d04590363756/html5/thumbnails/462.jpg)
0*1*0* 0*1*0*
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Home
Fix
01
Go
Home
Fix
01
1 → 1, R
0 0 0 0 1… …
start
0 → 0, R 1 → 1, R
0 → 1, L
1
1 → 0, L0 → 0, L
1 → 1, L
→ � �, R
To
Start
→ � �, L
0 → 0, L
1 → 1, L
![Page 463: Turing Machines - Stanford Universityweb.stanford.edu/class/archive/cs/cs103/cs103.1198...the coming years An important theme is not simply proving more theorems, but achieving automation](https://reader034.vdocuments.us/reader034/viewer/2022050502/5f9440a3ae52d04590363756/html5/thumbnails/463.jpg)
0*1*0* 0*1*0*
Go
Home
Fix
01
Go
Home
Fix
01
1 → 1, R
0 0 0 0 1… …
start
0 → 0, R 1 → 1, R
0 → 1, L
1
1 → 0, L0 → 0, L
1 → 1, L
→ � �, R
To
Start
→ � �, L
0 → 0, L
1 → 1, L Start
0n1n
→ � �, R
![Page 464: Turing Machines - Stanford Universityweb.stanford.edu/class/archive/cs/cs103/cs103.1198...the coming years An important theme is not simply proving more theorems, but achieving automation](https://reader034.vdocuments.us/reader034/viewer/2022050502/5f9440a3ae52d04590363756/html5/thumbnails/464.jpg)
0*1*0* 0*1*0*
Go
Home
Fix
01
Go
Home
Fix
01
1 → 1, R
0 0 0 0 1… …
start
0 → 0, R 1 → 1, R
0 → 1, L
1
1 → 0, L0 → 0, L
1 → 1, L
→ � �, R
To
Start
→ � �, L
0 → 0, L
1 → 1, L Start
0n1n
→ � �, R
![Page 465: Turing Machines - Stanford Universityweb.stanford.edu/class/archive/cs/cs103/cs103.1198...the coming years An important theme is not simply proving more theorems, but achieving automation](https://reader034.vdocuments.us/reader034/viewer/2022050502/5f9440a3ae52d04590363756/html5/thumbnails/465.jpg)
0*1*0* 0*1*0*
Go
Home
Fix
01
Go
Home
Fix
01
1 → 1, R
start
0 → 0, R 1 → 1, R
0 → 1, L
1 → 0, L0 → 0, L
1 → 1, L
→ � �, R
To
Start
→ � �, L
0 → 0, L
1 → 1, L Start
0n1n
→ � �, R
![Page 466: Turing Machines - Stanford Universityweb.stanford.edu/class/archive/cs/cs103/cs103.1198...the coming years An important theme is not simply proving more theorems, but achieving automation](https://reader034.vdocuments.us/reader034/viewer/2022050502/5f9440a3ae52d04590363756/html5/thumbnails/466.jpg)
0*1*0* 0*1*0*
Go
Home
Fix
01
Go
Home
Fix
01
1 → 1, R
start
0 → 0, R 1 → 1, R
0 → 1, L
1 → 0, L0 → 0, L
1 → 1, L
→ � �, R
To
Start
→ � �, L
0 → 0, L
1 → 1, L Start
0n1n
→ � �, R
This is just a
placeholder. Imagine
snapping in the entire
TM for 0n1n into this
diagram, putting the
start state in the
dashed area.
This is just a
placeholder. Imagine
snapping in the entire
TM for 0n1n into this
diagram, putting the
start state in the
dashed area.
![Page 467: Turing Machines - Stanford Universityweb.stanford.edu/class/archive/cs/cs103/cs103.1198...the coming years An important theme is not simply proving more theorems, but achieving automation](https://reader034.vdocuments.us/reader034/viewer/2022050502/5f9440a3ae52d04590363756/html5/thumbnails/467.jpg)
0*1*0* 0*1*0*
Go
Home
Fix
01
Go
Home
Fix
01
1 → 1, R
0 0 0 0… …
start
0 → 0, R 1 → 1, R
0 → 1, L
1 → 0, L0 → 0, L
1 → 1, L
→ � �, R
To
Start
→ � �, L
0 → 0, L
1 → 1, L Start
0n1n
→ � �, R
![Page 468: Turing Machines - Stanford Universityweb.stanford.edu/class/archive/cs/cs103/cs103.1198...the coming years An important theme is not simply proving more theorems, but achieving automation](https://reader034.vdocuments.us/reader034/viewer/2022050502/5f9440a3ae52d04590363756/html5/thumbnails/468.jpg)
0* 0*1*0* 0*1*
Go
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Fix
01
Go
Home
Fix
01
1 → 1, R
0 0 0 0… …
start
0 → 0, R 1 → 1, R
0 → 1, L
1 → 0, L0 → 0, L
1 → 1, L
→ � �, R
To
Start
→ � �, L
0 → 0, L
1 → 1, L Start
0n1n
→ � �, R
![Page 469: Turing Machines - Stanford Universityweb.stanford.edu/class/archive/cs/cs103/cs103.1198...the coming years An important theme is not simply proving more theorems, but achieving automation](https://reader034.vdocuments.us/reader034/viewer/2022050502/5f9440a3ae52d04590363756/html5/thumbnails/469.jpg)
0* 0*1*0* 0*1*
Go
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Fix
01
Go
Home
Fix
01
1 → 1, R
0 0 0 0… …
start
0 → 0, R 1 → 1, R
0 → 1, L
1 → 0, L0 → 0, L
1 → 1, L
→ � �, R
To
Start
→ � �, L
0 → 0, L
1 → 1, L Start
0n1n
→ � �, R
![Page 470: Turing Machines - Stanford Universityweb.stanford.edu/class/archive/cs/cs103/cs103.1198...the coming years An important theme is not simply proving more theorems, but achieving automation](https://reader034.vdocuments.us/reader034/viewer/2022050502/5f9440a3ae52d04590363756/html5/thumbnails/470.jpg)
0* 0*1*0* 0*1*
Go
Home
Fix
01
Go
Home
Fix
01
1 → 1, R
0 0 0 0… …
start
0 → 0, R 1 → 1, R
0 → 1, L
1 → 0, L0 → 0, L
1 → 1, L
→ � �, R
To
Start
→ � �, L
0 → 0, L
1 → 1, L Start
0n1n
→ � �, R
![Page 471: Turing Machines - Stanford Universityweb.stanford.edu/class/archive/cs/cs103/cs103.1198...the coming years An important theme is not simply proving more theorems, but achieving automation](https://reader034.vdocuments.us/reader034/viewer/2022050502/5f9440a3ae52d04590363756/html5/thumbnails/471.jpg)
0* 0*1*0* 0*1*
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Fix
01
Go
Home
Fix
01
1 → 1, R
0 0 0 0… …
start
0 → 0, R 1 → 1, R
0 → 1, L
1 → 0, L0 → 0, L
1 → 1, L
→ � �, R
To
Start
→ � �, L
0 → 0, L
1 → 1, L Start
0n1n
→ � �, R
![Page 472: Turing Machines - Stanford Universityweb.stanford.edu/class/archive/cs/cs103/cs103.1198...the coming years An important theme is not simply proving more theorems, but achieving automation](https://reader034.vdocuments.us/reader034/viewer/2022050502/5f9440a3ae52d04590363756/html5/thumbnails/472.jpg)
0* 0*1*0* 0*1*
Go
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Fix
01
Go
Home
Fix
01
1 → 1, R
0 0 0 0… …
start
0 → 0, R 1 → 1, R
0 → 1, L
1 → 0, L0 → 0, L
1 → 1, L
→ � �, R
To
Start
→ � �, L
0 → 0, L
1 → 1, L Start
0n1n
→ � �, R
![Page 473: Turing Machines - Stanford Universityweb.stanford.edu/class/archive/cs/cs103/cs103.1198...the coming years An important theme is not simply proving more theorems, but achieving automation](https://reader034.vdocuments.us/reader034/viewer/2022050502/5f9440a3ae52d04590363756/html5/thumbnails/473.jpg)
0* 0*1*0* 0*1*
Go
Home
Fix
01
Go
Home
Fix
01
1 → 1, R
0 0 0 0… …
start
0 → 0, R 1 → 1, R
0 → 1, L
1 → 0, L0 → 0, L
1 → 1, L
→ � �, R
To
Start
→ � �, L
0 → 0, L
1 → 1, L Start
0n1n
→ � �, R
→ � �, L
![Page 474: Turing Machines - Stanford Universityweb.stanford.edu/class/archive/cs/cs103/cs103.1198...the coming years An important theme is not simply proving more theorems, but achieving automation](https://reader034.vdocuments.us/reader034/viewer/2022050502/5f9440a3ae52d04590363756/html5/thumbnails/474.jpg)
0*0* 0*1*0*1*
Go
Home
Fix
01
Go
Home
Fix
01
1 → 1, R
0 0 0 0… …
start
0 → 0, R 1 → 1, R
0 → 1, L
1 → 0, L0 → 0, L
1 → 1, L
→ � �, R
To
Start
→ � �, L
0 → 0, L
1 → 1, L Start
0n1n
→ � �, R
→ � �, L
![Page 475: Turing Machines - Stanford Universityweb.stanford.edu/class/archive/cs/cs103/cs103.1198...the coming years An important theme is not simply proving more theorems, but achieving automation](https://reader034.vdocuments.us/reader034/viewer/2022050502/5f9440a3ae52d04590363756/html5/thumbnails/475.jpg)
0*0* 0*1*0*1*
Go
Home
Fix
01
Go
Home
Fix
01
1 → 1, R
0 0 0 0… …
start
0 → 0, R 1 → 1, R
0 → 1, L
1 → 0, L0 → 0, L
1 → 1, L
→ � �, R
To
Start
→ � �, L
0 → 0, L
1 → 1, L Start
0n1n
→ � �, R
→ � �, L
![Page 476: Turing Machines - Stanford Universityweb.stanford.edu/class/archive/cs/cs103/cs103.1198...the coming years An important theme is not simply proving more theorems, but achieving automation](https://reader034.vdocuments.us/reader034/viewer/2022050502/5f9440a3ae52d04590363756/html5/thumbnails/476.jpg)
0*0* 0*1*0*1*
Go
Home
Fix
01
Go
Home
Fix
01
1 → 1, R
0 0 0 0… …
start
0 → 0, R 1 → 1, R
0 → 1, L
1 → 0, L0 → 0, L
1 → 1, L
→ � �, R
To
Start
→ � �, L
0 → 0, L
1 → 1, L Start
0n1n
→ � �, R
→ � �, L
![Page 477: Turing Machines - Stanford Universityweb.stanford.edu/class/archive/cs/cs103/cs103.1198...the coming years An important theme is not simply proving more theorems, but achieving automation](https://reader034.vdocuments.us/reader034/viewer/2022050502/5f9440a3ae52d04590363756/html5/thumbnails/477.jpg)
0*0* 0*1*0*1*
Go
Home
Fix
01
Go
Home
Fix
01
1 → 1, R
0 0 0 0… …
start
0 → 0, R 1 → 1, R
0 → 1, L
1 → 0, L0 → 0, L
1 → 1, L
→ � �, R
To
Start
→ � �, L
0 → 0, L
1 → 1, L Start
0n1n
→ � �, R
→ � �, L
![Page 478: Turing Machines - Stanford Universityweb.stanford.edu/class/archive/cs/cs103/cs103.1198...the coming years An important theme is not simply proving more theorems, but achieving automation](https://reader034.vdocuments.us/reader034/viewer/2022050502/5f9440a3ae52d04590363756/html5/thumbnails/478.jpg)
0*0* 0*1*0*1*
Go
Home
Fix
01
Go
Home
Fix
01
1 → 1, R
0 0 0 0… …
start
0 → 0, R 1 → 1, R
0 → 1, L
1 → 0, L0 → 0, L
1 → 1, L
→ � �, R
To
Start
→ � �, L
0 → 0, L
1 → 1, L Start
0n1n
→ � �, R
→ � �, L
![Page 479: Turing Machines - Stanford Universityweb.stanford.edu/class/archive/cs/cs103/cs103.1198...the coming years An important theme is not simply proving more theorems, but achieving automation](https://reader034.vdocuments.us/reader034/viewer/2022050502/5f9440a3ae52d04590363756/html5/thumbnails/479.jpg)
0*0* 0*1*0*1*
Go
Home
Fix
01
Go
Home
Fix
01
1 → 1, R
0 0 0 0… …
start
0 → 0, R 1 → 1, R
0 → 1, L
1 → 0, L0 → 0, L
1 → 1, L
→ � �, R
To
Start
→ � �, L
0 → 0, L
1 → 1, L Start
0n1n
→ � �, R
→ � �, L
![Page 480: Turing Machines - Stanford Universityweb.stanford.edu/class/archive/cs/cs103/cs103.1198...the coming years An important theme is not simply proving more theorems, but achieving automation](https://reader034.vdocuments.us/reader034/viewer/2022050502/5f9440a3ae52d04590363756/html5/thumbnails/480.jpg)
0* 0*1*
Fix
01
Go
Home
To
Start
Start
0n1n
start
1 → 1, R 1 → 1, R0 → 0, R
0 → 1, L
1 → 0, L0 → 0, L
1 → 1, L
→ � �, R
→ � �, L → � �, L
0 → 0, L
1 → 1, L
→ � �, R
![Page 481: Turing Machines - Stanford Universityweb.stanford.edu/class/archive/cs/cs103/cs103.1198...the coming years An important theme is not simply proving more theorems, but achieving automation](https://reader034.vdocuments.us/reader034/viewer/2022050502/5f9440a3ae52d04590363756/html5/thumbnails/481.jpg)
0* 0*1*
Fix
01
Go
Home
To
Start
Start
0n1n
start
1 → 1, R 1 → 1, R0 → 0, R
0 → 1, L
1 → 0, L0 → 0, L
1 → 1, L
→ � �, R
→ � �, L → � �, L
0 → 0, L
1 → 1, L
→ � �, R
This TM will sort any sequence of 0s and
1s, but it might take a while.
Fun problem: design a TM that sorts a
string of 0s and 1s, but does so while
taking way fewer steps than this machine.
This TM will sort any sequence of 0s and
1s, but it might take a while.
Fun problem: design a TM that sorts a
string of 0s and 1s, but does so while
taking way fewer steps than this machine.
![Page 482: Turing Machines - Stanford Universityweb.stanford.edu/class/archive/cs/cs103/cs103.1198...the coming years An important theme is not simply proving more theorems, but achieving automation](https://reader034.vdocuments.us/reader034/viewer/2022050502/5f9440a3ae52d04590363756/html5/thumbnails/482.jpg)
TM Subroutines
● A TM subroutine is a Turing machine that, instead of accepting or rejecting an input, does some sort of processing job.
● TM subroutines let us compose larger TMs out of smaller TMs, just as you'd write a larger program using lots of smaller helper functions.
● Here, we saw a TM subroutine that sorts a sequence of 0s and 1s into ascending order.
![Page 483: Turing Machines - Stanford Universityweb.stanford.edu/class/archive/cs/cs103/cs103.1198...the coming years An important theme is not simply proving more theorems, but achieving automation](https://reader034.vdocuments.us/reader034/viewer/2022050502/5f9440a3ae52d04590363756/html5/thumbnails/483.jpg)
TM Subroutines
● Typically, when a subroutine is done running, you have it enter a state marked “done” with a dashed line around it.
● When we're composing multiple subroutines together – which we'll do in a bit – the idea is that we'll snap in some real state for the “done” state.
![Page 484: Turing Machines - Stanford Universityweb.stanford.edu/class/archive/cs/cs103/cs103.1198...the coming years An important theme is not simply proving more theorems, but achieving automation](https://reader034.vdocuments.us/reader034/viewer/2022050502/5f9440a3ae52d04590363756/html5/thumbnails/484.jpg)
Where We Stand
● What have we seen TMs do so far?
● Operate on numbers.
● Sort sequences of values.
● Break tasks down into smaller pieces.
● Here are a few other tasks TMs can do:
● Work with base-10 numbers.
● Increment and decrement numbers.
● Add numbers.
● Aren’t these, youknow, the thingscomputers do?
If you’re curious to see how this is
done, check the appendix for this
lecture. You aren’t required to do
this, though.
If you’re curious to see how this is
done, check the appendix for this
lecture. You aren’t required to do
this, though.
![Page 485: Turing Machines - Stanford Universityweb.stanford.edu/class/archive/cs/cs103/cs103.1198...the coming years An important theme is not simply proving more theorems, but achieving automation](https://reader034.vdocuments.us/reader034/viewer/2022050502/5f9440a3ae52d04590363756/html5/thumbnails/485.jpg)
How to Turing machines compare with standard, run-of-the-mill computers?
![Page 486: Turing Machines - Stanford Universityweb.stanford.edu/class/archive/cs/cs103/cs103.1198...the coming years An important theme is not simply proving more theorems, but achieving automation](https://reader034.vdocuments.us/reader034/viewer/2022050502/5f9440a3ae52d04590363756/html5/thumbnails/486.jpg)
Real and “Ideal” Computers
● A real computer has memory limitations: you have a fnite amount of RAM, a fnite amount of disk space, etc.
● This makes them equivalent to fnite automata.
● However, as computers get more and more powerful, the amount of memory available keeps increasing.
● An idealized computer is like a regular computer, but with unlimited RAM and disk space. It functions just like a regular computer, but never runs out of memory.
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Claim 1: Idealized computers can simulate Turing machines.
“Anything that can be done with a TMcan also be done with an unbounded-
memory computer.”
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Clear a1
Clear a1
Go toend
Checkfor 0
Go toend
Checkfor 0
Go tostart
Go tostart
start
0 → , R☐ 0 → 0, R
1 → 1, R
→ ☐ ☐, L
1 → , L☐0 → 0, L
1 → 1, L
→ ☐ ☐, R
qaccqa
→ ☐ ☐, R
qaccqr
1 → , R☐
→ ☐ ☐, R
0 → 0, R
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Clear a1q₂
Go toend
Checkfor 0
q₁q₀
Go tostartq₃
start
0 → , R☐ 0 → 0, R
1 → 1, R
→ ☐ ☐, L
1 → , L☐0 → 0, L
1 → 1, L
→ ☐ ☐, R
qaccqa
→ ☐ ☐, R
qaccqr
1 → , R☐
→ ☐ ☐, R
0 → 0, R
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Clear a1q₂
Go toend
Checkfor 0
q₁q₀
Go tostartq₃
start
0 → , R☐ 0 → 0, R
1 → 1, R
→ ☐ ☐, L
1 → , L☐0 → 0, L
1 → 1, L
→ ☐ ☐, R
qaccqa
→ ☐ ☐, R
qaccqr
1 → , R☐
→ ☐ ☐, R
0 → 0, R
q₀
q₁
q₂
q₃
0
q₁ ◻ R
1
qr ◻ R
◻
qa ◻ R
q₁ 0 R q₁ 1 R q₂ ◻ L
qr 0 R q₃ ◻ L q
r ◻ R
q₃ 0 L q₃ 1 L q₀ ◻ R
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Clear a1q₂
Go toend
Checkfor 0
q₁q₀
Go tostartq₃
start
0 → , R☐ 0 → 0, R
1 → 1, R
→ ☐ ☐, L
1 → , L☐0 → 0, L
1 → 1, L
→ ☐ ☐, R
qaccqa
→ ☐ ☐, R
qaccqr
1 → , R☐
→ ☐ ☐, R
0 → 0, R
q₀
q₁
q₂
q₃
0
q₁ ◻ R
1
qr ◻ R
◻
qa ◻ R
q₁ 0 R q₁ 1 R q₂ ◻ L
qr 0 R q₃ ◻ L q
r ◻ R
q₃ 0 L q₃ 1 L q₀ ◻ R
The TM's fnite-state control can be encoded as a table, just like we did for DFAs.
The TM's fnite-state control can be encoded as a table, just like we did for DFAs.
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Simulating a TM
● To simulate a TM, the computer would need to be able to keep track of
● the fnite-state control,
● the current state,
● the position of the tape head, and
● the tape contents.
● The tape contents are infnite, but that's because there are infnitely many blanks on both sides.
● We only need to store the “interesting” part of the tape (the parts that have been read from or written to so far.)
0 0 0 1 1 1 ……
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Simulating a TM
● To simulate a TM, the computer would need to be able to keep track of
● the fnite-state control,
● the current state,
● the position of the tape head, and
● the tape contents.
● The tape contents are infnite, but that's because there are infnitely many blanks on both sides.
● We only need to store the “interesting” part of the tape (the parts that have been read from or written to so far.)
…… 0 0 0 1 1 1
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Claim 2: Turing machines can simulate idealized computers.
“Anything that can be done with an unbounded-memory computer can be done
with a TM.”
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What We've Seen
● TMs can
● implement loops (basically, every TM we've seen).
● make function calls (subroutines).
● keep track of natural numbers (written in unary or in decimal on the tape).
● perform elementary arithmetic (equality testing, multiplication, addition, increment, decrement, etc.).
● perform if/else tests (diferent transitions based on diferent cases).
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What Else Can TMs Do?
● Maintain variables.
● Have a dedicated part of the tape where the variables are stored.
● We've seen this before: you can kinda sorta think of our machine for { 0n1n | n ∈ ℕ } as checking if two variables are equal.
● Maintain arrays and linked structures.
● Divide the tape into diferent regions corresponding to memory locations.
● Represent arrays and linked structures by keeping track of the ID of one of those regions.
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A CS107 Perspective
● Internally, computers execute by using basic operations like
● simple arithmetic,
● memory reads and writes,
● branches and jumps,
● register operations,
● etc.
● Each of these are simple enough that they could be simulated by a Turing machine.
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A Leap of Faith
● It may require a leap of faith, but anything you can do with a computer (excluding randomness and user input) can be performed by a Turing machine.
● The resulting TM might be colossal, or really slow, or both, but it would still faithfully simulate the computer.
● We're going to take this as an article of faith in CS103. If you’re curious for more details, come talk to me after class.
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Wait, You’re Saying a TM Can Do…
Sure! A picture is
just a 2D array of
colors, and a color
can be represented
as a series of
numbers.
“cat pictures?”
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If you think about
it, a video is just a
series of pictures!“cat videos?”
“cat pictures?”
Wait, You’re Saying a TM Can Do…
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Yes! Write encodings of notes to
play on the TM tape. Hook up a
speaker device that reads the tape
and makes sound.
“music?”
Yes! View all networked
computers as one gigantic
machine.
“chat messages over the internet?”
Wait, You’re Saying a TM Can Do…
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Just how powerful are Turing machines?
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Efective Computation
● An efective method of computation is a form of computation with the following properties:
● The computation consists of a set of steps.
● There are fxed rules governing how one step leads to the next.
● Any computation that yields an answer does so in fnitely many steps.
● Any computation that yields an answer always yields the correct answer.
● This is not a formal defnition. Rather, it's a set of properties we expect out of a computational system.
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The Church-Turing Thesis claims that
every efective method of computation is either equivalent to or weaker than a Turing machine.
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RegularLanguages
CFLs
All Languages
Problems Solvable by
Any Feasible Computing
Machine
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RegularLanguages
CFLs
All Languages
Problems solvable by
Turing Machines
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TMs ≈ Computers
● Because Turing machines have the same computational powers as regular computers, we can (essentially) reason about Turing machines by reasoning about actual computer programs.
● Going forward, we're going to switch back and forth between TMs and computer programs based on whatever is most appropriate.
● In fact, our eventual proofs about the existence of impossible problems will involve a good amount of pseudocode. Stay tuned for details!
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What problems can we solve with a computer?
What kind
of
computer?
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What problems can we solve with a computer?
What does it
mean to
“solve” a
problem?
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The Hailstone Sequence
● Consider the following procedure, starting with some n ∈ ℕ, where n > 0:
● If n = 1, you are done.
● If n is even, set n = n / 2.
● Otherwise, set n = 3n + 1.
● Repeat.
● Question: Given a number n, does this process terminate?
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11
34
17
52
26
13
40
20
10
5
16
8
4
2
1
· If n = 1, stop.
· If n is even, set n = n / 2.
· Otherwise, set n = 3n + 1.
· Repeat.
· If n = 1, stop.
· If n is even, set n = n / 2.
· Otherwise, set n = 3n + 1.
· Repeat.
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The Hailstone Sequence
● Let Σ = {1} and consider the language
L = { 1n | n > 0 and the hailstone sequence terminates for n }.
● Could we build a TM for L?
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The Hailstone Turing Machine
… 1 1 1 1 1 …
If the input is ε, reject.
While the input is not 1:
· If the input has even length, halve the length of the string.
· If the input has odd length, triple the length of the string and append a 1.
Accept.
If the input is ε, reject.
While the input is not 1:
· If the input has even length, halve the length of the string.
· If the input has odd length, triple the length of the string and append a 1.
Accept.
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The Hailstone Turing Machine
… 1 1 1 1 1 …
If the input is ε, reject.
While the input is not 1:
· If the input has even length, halve the length of the string.
· If the input has odd length, triple the length of the string and append a 1.
Accept.
If the input is ε, reject.
While the input is not 1:
· If the input has even length, halve the length of the string.
· If the input has odd length, triple the length of the string and append a 1.
Accept.
Poof!
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The Hailstone Turing Machine
… 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 …
If the input is ε, reject.
While the input is not 1:
· If the input has even length, halve the length of the string.
· If the input has odd length, triple the length of the string and append a 1.
Accept.
If the input is ε, reject.
While the input is not 1:
· If the input has even length, halve the length of the string.
· If the input has odd length, triple the length of the string and append a 1.
Accept.
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The Hailstone Turing Machine
… 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 …
If the input is ε, reject.
While the input is not 1:
· If the input has even length, halve the length of the string.
· If the input has odd length, triple the length of the string and append a 1.
Accept.
If the input is ε, reject.
While the input is not 1:
· If the input has even length, halve the length of the string.
· If the input has odd length, triple the length of the string and append a 1.
Accept.
Poof!
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The Hailstone Turing Machine
… 1 1 1 1 1 1 1 1 …
If the input is ε, reject.
While the input is not 1:
· If the input has even length, halve the length of the string.
· If the input has odd length, triple the length of the string and append a 1.
Accept.
If the input is ε, reject.
While the input is not 1:
· If the input has even length, halve the length of the string.
· If the input has odd length, triple the length of the string and append a 1.
Accept.
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The Hailstone Turing Machine
… 1 1 1 1 1 1 1 1 …
If the input is ε, reject.
While the input is not 1:
· If the input has even length, halve the length of the string.
· If the input has odd length, triple the length of the string and append a 1.
Accept.
If the input is ε, reject.
While the input is not 1:
· If the input has even length, halve the length of the string.
· If the input has odd length, triple the length of the string and append a 1.
Accept.
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The Hailstone Turing Machine
… 1 1 1 1 1 1 1 1 …
If the input is ε, reject.
While the input is not 1:
· If the input has even length, halve the length of the string.
· If the input has odd length, triple the length of the string and append a 1.
Accept.
If the input is ε, reject.
While the input is not 1:
· If the input has even length, halve the length of the string.
· If the input has odd length, triple the length of the string and append a 1.
Accept.
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The Hailstone Turing Machine
… 1 1 1 1 …
If the input is ε, reject.
While the input is not 1:
· If the input has even length, halve the length of the string.
· If the input has odd length, triple the length of the string and append a 1.
Accept.
If the input is ε, reject.
While the input is not 1:
· If the input has even length, halve the length of the string.
· If the input has odd length, triple the length of the string and append a 1.
Accept.
![Page 521: Turing Machines - Stanford Universityweb.stanford.edu/class/archive/cs/cs103/cs103.1198...the coming years An important theme is not simply proving more theorems, but achieving automation](https://reader034.vdocuments.us/reader034/viewer/2022050502/5f9440a3ae52d04590363756/html5/thumbnails/521.jpg)
The Hailstone Turing Machine
… 1 1 …
If the input is ε, reject.
While the input is not 1:
· If the input has even length, halve the length of the string.
· If the input has odd length, triple the length of the string and append a 1.
Accept.
If the input is ε, reject.
While the input is not 1:
· If the input has even length, halve the length of the string.
· If the input has odd length, triple the length of the string and append a 1.
Accept.
![Page 522: Turing Machines - Stanford Universityweb.stanford.edu/class/archive/cs/cs103/cs103.1198...the coming years An important theme is not simply proving more theorems, but achieving automation](https://reader034.vdocuments.us/reader034/viewer/2022050502/5f9440a3ae52d04590363756/html5/thumbnails/522.jpg)
The Hailstone Turing Machine
… 1 …
If the input is ε, reject.
While the input is not 1:
· If the input has even length, halve the length of the string.
· If the input has odd length, triple the length of the string and append a 1.
Accept.
If the input is ε, reject.
While the input is not 1:
· If the input has even length, halve the length of the string.
· If the input has odd length, triple the length of the string and append a 1.
Accept.
![Page 523: Turing Machines - Stanford Universityweb.stanford.edu/class/archive/cs/cs103/cs103.1198...the coming years An important theme is not simply proving more theorems, but achieving automation](https://reader034.vdocuments.us/reader034/viewer/2022050502/5f9440a3ae52d04590363756/html5/thumbnails/523.jpg)
Does this Turing machine accept all nonempty strings?
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The Collatz Conjecture
● It is unknown whether this process will terminate for all natural numbers.
● In other words, no one knows whether the TM described in the previous slides will always stop running!
● The conjecture (unproven claim) that this always terminates is called the Collatz Conjecture.
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The Collatz Conjecture
“Mathematics may not be readyfor such problems.” - Paul Erdős
The fact that the Collatz Conjecture is unresolved is useful later on for building
intuitions. Keep this in mind!
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An Important Observation
● Unlike fnite automata, which automatically halt after all the input is read, TMs keep running until they explicitly enter an accept or reject state.
● As a result, it’s possible for a TM to run forever without accepting or rejecting.
● This leads to several important questions:
● How do we formally defne what it means to build a TM for a language?
● What implications does this have about problem-solving?
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Very Important Terminology
● Let M be a Turing machine and let w be a string.
● M accepts w if it enters an accept state when run on w.
● M rejects w if it enters a reject state when run on w.
● M loops infnitely on w (or just loops on w) if when run on w it enters neither an accept nor a reject state.
● M does not accept w if it either rejects w or loops infnitely on w.
● M does not reject w w if it either accepts w or loops on w.
● M halts on w if it accepts w or rejects w.
Accept
Loop
Rejectdoes not accept
does not reject
halts
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The Language of a TM
● The language of a Turing machine M, denoted (ℒ M), is the set of all strings that M accepts:
ℒ(M) = { w ∈ Σ* | M accepts w }
● For any w ∈ (ℒ M), M accepts w.
● For any w ∉ (ℒ M), M does not accept w.
● M might reject w, or it might loop on w.
● A language is called recognizable if it is the language of some TM.
● A TM M where (ℒ M) = L is called a recognizer for L.
● Notation: the class RE is the set of all recognizable languages.
L ∈ RE ↔ L is recognizable
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What do you think? Does thatcorrespond to what you think it
means to solve a problem?