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<!DOCTYPE html>
<html lang="en">
<head>
<meta charset="utf-8">
<meta http-equiv="X-UA-Compatible" content="IE=edge">
<meta name="viewport" content="width=device-width, initial-scale=1">
<meta name="description" content="Visualize and simulate Turing machines as animated state diagrams.
Create and share your own machines using a simple format. Examples and exercises are included.">
<meta name="keywords" content="Turing machine, finite automata, simulator, visualization, state diagram">
<meta name="robots" content="NOODP">
<title>Turing machine visualization</title>
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</head>
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<nav class="navbar navbar-default">
<div class="container-fluid">
<div class="navbar-header">
<span class="navbar-brand">
<span class="kern">T</span>uring <span class="kern-Ma">M</span>achine <span class="kern-Vi">V</span>isualization
</span>
</div>
<div class="navbar-form navbar-left">
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<button type="button" class="close" data-dismiss="modal" aria-label="Close"><span aria-hidden="true">×</span>
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<h4 class="modal-title" id="renameDialogHeader">Document Name</h4>
</div>
<div class="modal-body">
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<div class="modal-content">
<div class="modal-header">
<button type="button" class="close" data-dismiss="modal" aria-label="Close"><span aria-hidden="true">×</span>
</button>
<h4 class="modal-title" id="deleteDialogHeader">Delete this document?</h4>
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This will erase the document and remove it from the menu.
You can’t undo this action.
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</div>
</div>
</div>
<!-- Dialog for Resetting an Example -->
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<div class="modal-dialog" role="document">
<div class="modal-content">
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<button type="button" class="close" data-dismiss="modal" aria-label="Close"><span aria-hidden="true">×</span>
</button>
<h4 class="modal-title" id="resetExampleDialogHeader">Save a copy of your changes?</h4>
</div>
<div class="modal-body">
<p>Resetting this example will erase any changes made to it.</p>
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You can’t undo this action.
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<h4 class="modal-title" id="importPanelHeader">Import</h4>
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<div class="modal-body">
<div class="panel panel-default">
<div class="panel-heading">
<h5 class="panel-title">From GitHub gist</h5>
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<span class="input-group-addon">https://gist.github.com/</span>
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<h5 class="panel-title">From files</h5>
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<h3>Try it out</h3>
<p>
<strong>Run</strong> the machine to see it in action.
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or <strong>reset</strong> to start over.
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<section>
<h4>What’s going on?</h4>
<p>The colored circles are <em>states</em>. The squares underneath are <em>tape cells</em>.</p>
<p>The current state and tape cell are highlighted.</p>
<p>At each step, a Turing machine reads its current state and tape symbol,
and looks them up in its <em>transition table</em> for an instruction.
Each <em>instruction</em> does 3 things:</p>
<ol>
<li><strong>write</strong> a symbol to the current tape cell</li>
<li><strong>move</strong> to the left or right by one cell</li>
<li><strong>set</strong> the new state</li>
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<p>A <dfn>Turing machine</dfn> is an abstract device to model computation as rote symbol
manipulation.</p>
<p>Each machine has a finite number of states, and a finite number of possible symbols.
These are fixed before the machine starts, and do not change as the machine
runs.</p>
<p>There are an <em>infinite</em> number of tape cells, however, extending endlessly to
the left and right.
Each tape cell bears a symbol. Any cell not part of the input or not yet written to
bears the blank symbol by default.
Notice that at any step, only finitely many cells bear a non-blank symbol.</p>
<p>(As a mathematical model, a Turing machine has infinite memory (an infinite tape) so
as to not artificially restrict its power.
In practice, many machines of interest take finite memory, and can be fully
simulated for manageable input sizes.
Even machines that use infinite memory—and hence never halt—use at most one new tape
cell per step, and so can be simulated to an extent.)</p>
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<p>The formal definition of a Turing machine has slight variations but essentially is a
tuple (ordered list) comprising</p>
<ul>
<li>states Q</li>
<li>start state q₀ ∈ Q</li>
<li>input alphabet Σ</li>
<li>tape alphabet Γ, where Σ ⊆ Γ</li>
<li>blank symbol b ∈ Γ</li>
<li>transition function δ that has the type Q × Γ →
Γ × {L, R} × Q
</li>
</ul>
<p>where Q, Σ, and Γ are finite nonempty sets.
Some definitions also require that the blank symbol not be part of the input (b
∉ Σ).</p>
<p>As an example, a formal description for the “binary increment” machine is as
follows:</p>
<p>
Q = { right, carry, done }
<br>q₀ = right
<br>Σ = { 1, 0 }
<br>Γ = { 1, 0, ' ' }
<br>b = ' '
</p>
<p>
δ(right, 1) = (1, R, right)
<br>δ(right, 0) = (0, R, right)
<br>δ(right, ' ') = (' ', L, carry)
<br>δ(carry, 1) = (0, L, carry)
<br>δ(carry, 0) = (1, L, done)
<br>δ(carry, ' ') = (1, L, done)
</p>
<p>Note that for simplicity, the simulator limits each symbol to one character.
Furthermore, input is not checked for conformance with an input alphabet, in
exchange for not having to define input and tape alphabets.</p>
<p>(The behavior of a Turing machine can also be described in mathematical terms if
desired.
Without going into too much detail, this involves defining how a
<em>configuration</em>
of a machine—its state, tape contents, and tape head position (current cell)—leads
to the next configuration, based on the transition function.
To start with, the tape’s contents may be defined as a function from integers to
symbols,
with the head position as an integer, and each move {L, R} adding -1 or +1
respectively to the head position.)</p>
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<p>Some aspects of the definition vary from author to author, but the differences come
down to preference and do not affect computational power.
That is, machines on one model can be simulated or converted to run on another
model.</p>
<p>Some of the variations you may come across:</p>
<ol>
<li>The tape only extends infinitely to the right. The left end is where the tape
head (current cell) begins.
Moving left at the left end keeps the tape head on the same cell.
</li>
<li>In addition to “move left” (L) or “move right” (R), a tape head movement can be
“no move” (N).
</li>
<li>Instead of moving the tape head, a movement shifts the tape itself, so that the
directions of L & R are swapped.
(Shifting the tape left is the same as moving the head right.)
</li>
<li>The machine can only halt in one of two states: a state designated the <em>accept</em>
state, or another state designated the <em>reject</em> state.
These states have no exiting transitions. All the other states must have
transitions defined for every symbol.
</li>
</ol>
<p>The simulator was designed with these and other considerations in mind.</p>
<ol>
<li>Limiting the tape was inconvenient, often requiring marking the left end with a
symbol, and writing numbers in reverse.
On the other hand, machines created for a right-infinite tape can run on a
doubly-infinite tape with little to no modification.
</li>
<li>Compared to L and R, “no move” (N) seems to be seldom used. For now it has been
omitted for conceptual simplicity.
</li>
<li>It's more intuitive to set the current cell directly by moving the tape head.
Turing also follows this convention.
(The visualization however stays centered on the head to avoid issues of the
head moving out of view.)
</li>
<li>The accept/reject convention still works; it's just not required.
In fact the simulator supports a convenient shorthand, demonstrated in some of
the examples:
omit the reject state and all transitions to it, and treat halting on a
non-accept state as an implicit rejecting transition.
This reduces tedious boilerplate code and makes for a cleaner diagram.
</li>
</ol>
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</section>
<p>
For a very readable introduction to Turing machines, including their significance and interesting
implications,
see the excellent <a href="http://plato.stanford.edu/entries/turing-machine" target="_blank">entry in
the Stanford Encyclopedia of Philosophy</a>.
</p>
</section>
<section class="col-md-4 col-sm-6">
<h3>Make your own</h3>
<p>Make spinoffs from the examples or your own creations by using
<i>Edit</i> > <i>Duplicate document</i>.
You can also start from scratch with a new blank document.</p>
<p>All it takes to describe a Turing machine is a start state, blank symbol, and transition table.</p>
<h4>Example</h4>
<pre>
# Adds 1 to a binary number.
input: '1011'
blank: ' '
start state: right
table:
# scan to the rightmost digit
right:
1: R
0: R
' ': {L: carry}
# then carry the 1
carry:
1: {write: 0, L}
0: {write: 1, L: done}
' ': {write: 1, L: done}
done:
</pre>
<p>
Here the states are <code>right</code>, <code>carry</code>, and <code>done</code>.<br>
The symbols are '<code>1</code>', '<code>0</code>', and '<code> </code>'.
</p>
<p>
We designate one state the <i>start state</i>,
and one symbol the <i>blank</i> symbol (found on unmarked tape cells).
</p>
<p>
A state and a symbol together map to an instruction.
In the <code>carry</code> state, for instance, the symbol <code>1</code> maps to the instruction <code>{write:
0, L}</code>.
When no instruction is defined, as in the <code>done</code> state, the machine halts.
</p>
<h4>Instruction format</h4>
The general form of an instruction has three parts:
<pre>{write: <var>symbol</var>, <var>move</var>: <var>state</var>}</pre>
<ul>
<li><code>{write: 1, L: done}</code> writes the symbol <code>1</code>,
moves the tape head left (<code>L</code>), and goes to the <code>done</code> state.
</li>
</ul>
<p>For brevity, you can omit the symbol and state if they stay the same.</p>
<ul>
<li><code>{L: carry}</code> writes the same symbol that was read,
moves the tape head left, and goes to the <code>carry</code> state.
</li>
<li><code>R</code> (shorthand for {R}) simply moves the tape head right.
It writes back the same symbol and stays in the same state.
</li>
</ul>
</section>
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<h3>Tips</h3>
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