Special Topic | CS 139: Theory of Computation

<h1 id="special-topic">Special Topic</h1>

<hr />

<h2 id="administrivia">Administrivia</h2>
<ul>
  <li>Classes will be moved to be <strong>online</strong> for the two weeks after spring break</li>
  <li>More information will be released over spring break on how this will look in this course</li>
</ul>

<h2 id="nondeterministic-tms">Nondeterministic TMs</h2>
<h1 id="revisited">Revisited</h1>

<h2 id="nondeterministic-tms-1">Nondeterministic TMs</h2>
<ul>
  <li>Similar to NFAs being equivalent in power to DFAs, <strong>nondeterministic Turing machines</strong> are equivalent to the normal deterministic kind</li>
</ul>

<h2 id="nondeterministic-tms-2">Nondeterministic TMs</h2>

<ul>
  <li>Suppose I have the following Turing nondeterministic machine
    <div class="language-plaintext highlighter-rouge"><div class="highlight"><pre class="highlight"><code>q0 _ 0 r q1
q0 _ 1 r q1
q1 _ 0 r q2
q1 _ 1 r q2
q2 _ 0 r q3
q2 _ 1 r q3
</code></pre></div>    </div>
  </li>
  <li>If we initialize this Turing machine with an <strong>empty tape</strong>, what will it do?</li>
</ul>

<h2 id="nondeterministic-tms-3">Nondeterministic TMs</h2>

</div>

<hr />
<p><code class="language-plaintext highlighter-rouge">$q_0\sqcup\sqcup\cdots$</code></p>

<hr />

<h2 id="nondeterministic-tms-4">Nondeterministic TMs</h2>
<div class="jekyll-diagrams diagrams graphviz">
  

<svg width="250pt" height="98pt" viewBox="0.00 0.00 250.39 98.00" xmlns="http://www.w3.org/2000/svg" xmlns:xlink="http://www.w3.org/1999/xlink">
<g id="graph0" class="graph" transform="scale(1 1) rotate(0) translate(4 94)">
<title>dfa</title>
<polygon fill="white" stroke="transparent" points="-4,4 -4,-94 246.39,-94 246.39,4 -4,4" />
<g id="node1" class="node">
<title></title>
</g>

<g id="node2" class="node">
<title>q0</title>
<ellipse fill="none" stroke="black" cx="117" cy="-45" rx="27" ry="18" />
<text text-anchor="middle" x="117" y="-41.3" font-family="Times,serif" font-size="14.00">q0</text>
</g>

<g id="edge1" class="edge">
<title>->q0</title>
<path fill="none" stroke="black" d="M54.4,-45C62.39,-45 71.31,-45 79.82,-45" />
<polygon fill="black" stroke="black" points="79.92,-48.5 89.92,-45 79.92,-41.5 79.92,-48.5" />
</g>

<g id="node3" class="node">
<title>a q1</title>
<ellipse fill="gray" stroke="black" cx="211.2" cy="-72" rx="30.59" ry="18" />
<text text-anchor="middle" x="211.2" y="-68.3" font-family="Times,serif" font-size="14.00">a q1</text>
</g>

<g id="edge2" class="edge">
<title>q0->a q1</title>
<path fill="none" stroke="black" d="M142.22,-52.09C151.85,-54.91 163.12,-58.21 173.7,-61.31" />
<polygon fill="black" stroke="black" points="172.96,-64.74 183.54,-64.19 174.93,-58.02 172.96,-64.74" />
</g>

<g id="node4" class="node">
<title>b q1</title>
<ellipse fill="gray" stroke="black" cx="211.2" cy="-18" rx="31.4" ry="18" />
<text text-anchor="middle" x="211.2" y="-14.3" font-family="Times,serif" font-size="14.00">b q1</text>
</g>

<g id="edge3" class="edge">
<title>q0->b q1</title>
<path fill="none" stroke="black" d="M142.22,-37.91C151.68,-35.14 162.75,-31.9 173.16,-28.85" />
<polygon fill="black" stroke="black" points="174.27,-32.17 182.88,-26 172.3,-25.45 174.27,-32.17" />
</g>
</g>
</svg>

</div>

<hr />
<p><code class="language-plaintext highlighter-rouge">$\texttt{a}q_1\#\texttt{b}q_1\sqcup\sqcup\cdots$</code></p>

<hr />

<h2 id="nondeterministic-tms-5">Nondeterministic TMs</h2>
<div class="jekyll-diagrams diagrams graphviz">
  

<svg width="360pt" height="206pt" viewBox="0.00 0.00 360.49 206.00" xmlns="http://www.w3.org/2000/svg" xmlns:xlink="http://www.w3.org/1999/xlink">
<g id="graph0" class="graph" transform="scale(1 1) rotate(0) translate(4 202)">
<title>dfa</title>
<polygon fill="white" stroke="transparent" points="-4,4 -4,-202 356.49,-202 356.49,4 -4,4" />
<g id="node1" class="node">
<title></title>
</g>

<g id="node2" class="node">
<title>q0</title>
<ellipse fill="none" stroke="black" cx="117" cy="-99" rx="27" ry="18" />
<text text-anchor="middle" x="117" y="-95.3" font-family="Times,serif" font-size="14.00">q0</text>
</g>

<g id="edge1" class="edge">
<title>->q0</title>
<path fill="none" stroke="black" d="M54.4,-99C62.39,-99 71.31,-99 79.82,-99" />
<polygon fill="black" stroke="black" points="79.92,-102.5 89.92,-99 79.92,-95.5 79.92,-102.5" />
</g>

<g id="node3" class="node">
<title>a q1</title>
<ellipse fill="none" stroke="black" cx="211.2" cy="-126" rx="30.59" ry="18" />
<text text-anchor="middle" x="211.2" y="-122.3" font-family="Times,serif" font-size="14.00">a q1</text>
</g>

<g id="edge2" class="edge">
<title>q0->a q1</title>
<path fill="none" stroke="black" d="M142.22,-106.09C151.85,-108.91 163.12,-112.21 173.7,-115.31" />
<polygon fill="black" stroke="black" points="172.96,-118.74 183.54,-118.19 174.93,-112.02 172.96,-118.74" />
</g>

<g id="node4" class="node">
<title>b q1</title>
<ellipse fill="none" stroke="black" cx="211.2" cy="-72" rx="31.4" ry="18" />
<text text-anchor="middle" x="211.2" y="-68.3" font-family="Times,serif" font-size="14.00">b q1</text>
</g>

<g id="edge3" class="edge">
<title>q0->b q1</title>
<path fill="none" stroke="black" d="M142.22,-91.91C151.68,-89.14 162.75,-85.9 173.16,-82.85" />
<polygon fill="black" stroke="black" points="174.27,-86.17 182.88,-80 172.3,-79.45 174.27,-86.17" />
</g>

<g id="node5" class="node">
<title>aa q2</title>
<ellipse fill="gray" stroke="black" cx="315.44" cy="-180" rx="36.29" ry="18" />
<text text-anchor="middle" x="315.44" y="-176.3" font-family="Times,serif" font-size="14.00">aa q2</text>
</g>

<g id="edge4" class="edge">
<title>a q1->aa q2</title>
<path fill="none" stroke="black" d="M234.64,-137.85C248.17,-145 265.67,-154.24 280.83,-162.25" />
<polygon fill="black" stroke="black" points="279.43,-165.47 289.91,-167.04 282.7,-159.28 279.43,-165.47" />
</g>

<g id="node6" class="node">
<title>ab q2</title>
<ellipse fill="gray" stroke="black" cx="315.44" cy="-126" rx="36.29" ry="18" />
<text text-anchor="middle" x="315.44" y="-122.3" font-family="Times,serif" font-size="14.00">ab q2</text>
</g>

<g id="edge5" class="edge">
<title>a q1->ab q2</title>
<path fill="none" stroke="black" d="M241.77,-126C250.17,-126 259.52,-126 268.64,-126" />
<polygon fill="black" stroke="black" points="268.78,-129.5 278.78,-126 268.78,-122.5 268.78,-129.5" />
</g>

<g id="node7" class="node">
<title>ba q2</title>
<ellipse fill="gray" stroke="black" cx="315.44" cy="-72" rx="36.29" ry="18" />
<text text-anchor="middle" x="315.44" y="-68.3" font-family="Times,serif" font-size="14.00">ba q2</text>
</g>

<g id="edge6" class="edge">
<title>b q1->ba q2</title>
<path fill="none" stroke="black" d="M242.6,-72C250.8,-72 259.87,-72 268.72,-72" />
<polygon fill="black" stroke="black" points="268.94,-75.5 278.94,-72 268.94,-68.5 268.94,-75.5" />
</g>

<g id="node8" class="node">
<title>bb q2</title>
<ellipse fill="gray" stroke="black" cx="315.44" cy="-18" rx="37.09" ry="18" />
<text text-anchor="middle" x="315.44" y="-14.3" font-family="Times,serif" font-size="14.00">bb q2</text>
</g>

<g id="edge7" class="edge">
<title>b q1->bb q2</title>
<path fill="none" stroke="black" d="M234.89,-60.02C248.34,-52.91 265.63,-43.78 280.66,-35.84" />
<polygon fill="black" stroke="black" points="282.46,-38.85 289.67,-31.08 279.19,-32.66 282.46,-38.85" />
</g>
</g>
</svg>

</div>

<hr />
<p><code class="language-plaintext highlighter-rouge">$\texttt{aa}q_2\#\texttt{ba}q_2\#\texttt{ab}q_2\#\texttt{bb}q_2\sqcup\sqcup\cdots$</code></p>

<hr />

<h2 id="encoding-objects-as">Encoding Objects as</h2>
<h1 id="strings">Strings</h1>

<h2 id="encoding-objects-as-strings">Encoding Objects as Strings</h2>
<ul>
  <li>There are an <strong>infinite</strong> number of ways to encode an object like a graph as a string</li>
  <li>The encoding of choice doesn’t matter that much as long as it uniquely specifies the desired object</li>
  <li>Sipser uses notation <code class="language-plaintext highlighter-rouge">$\langle G\rangle$</code> to denote an arbitrary string-encoding of a finite object like a graph</li>
</ul>

<h2 id="example">Example</h2>
<p>Show that the following language is <strong>decidable</strong></p>

<p><code class="language-plaintext highlighter-rouge">$$L = \{\langle S, n\rangle \mid S\subseteq\mathbb{N}\text{ is finite and }n\in S\}$$</code></p>

<hr />

<ul>
  <li>This is basically asking us to write a function that takes in a finite set (or a list) of numbers and check if <code class="language-plaintext highlighter-rouge">$n$</code> is an element of that list</li>
</ul>

<h2 id="example-1">Example</h2>
<p><code class="language-plaintext highlighter-rouge">$$L = \{\langle S, n\rangle \mid S\subseteq\mathbb{N}\text{ is finite and }n\in S\}$$</code></p>

<hr />

<div class="language-py highlighter-rouge"><div class="highlight"><pre class="highlight"><code><span class="k">def</span> <span class="nf">decider</span><span class="p">(</span><span class="n">S</span><span class="p">,</span> <span class="n">n</span><span class="p">):</span>
    <span class="s">"""Checks if n is an element of S<br>
    Parameters:
        S: a list of integers
        n: an integer
    """</span>
    <span class="k">for</span> <span class="n">x</span> <span class="ow">in</span> <span class="n">S</span><span class="p">:</span>
        <span class="k">if</span> <span class="n">x</span> <span class="o">==</span> <span class="n">n</span><span class="p">:</span>
            <span class="k">return</span> <span class="bp">True</span>
    <span class="k">return</span> <span class="bp">False</span>
</code></pre></div></div>

<h2 id="example-2">Example</h2>
<p><code class="language-plaintext highlighter-rouge">$$L = \{\langle S, n\rangle \mid S\subseteq\mathbb{N}\text{ is finite and }n\in S\}$$</code></p>

<hr />

<ul>
  <li>Here is a <strong>high-level</strong> Turing machine that decides <code class="language-plaintext highlighter-rouge">$L$</code></li>
</ul>

<div class="language-plaintext highlighter-rouge"><div class="highlight"><pre class="highlight"><code>M = On input &lt;S,n&gt;,
  0. Check to see if &lt;S,n&gt; is properly formatted; if not,
     immediately REJECT
  1. For each element x in S
     a. Compare x to n
     b. If they are the same, then immediately ACCEPT; otherwise
        continue with the for-loop
  2. If we checked all elements of S and still did not find n,
     then REJECT
</code></pre></div></div>

<h2 id="another-example">Another Example</h2>
<p><code class="language-plaintext highlighter-rouge">$$L = \{\langle p\rangle \mid p\text{ is a polynomial with integral roots}\}$$</code></p>

<hr />

<ul>
  <li class="fragment">Is <code class="language-plaintext highlighter-rouge">$L$</code> <strong>decidable</strong>?</li>
  <li class="fragment">Is <code class="language-plaintext highlighter-rouge">$L$</code> <strong>Turing recognizable</strong>?</li>
</ul>

<h2 id="another-example-1">Another Example</h2>

<p><code class="language-plaintext highlighter-rouge">$$L = \{\langle p\rangle \mid p\text{ is a polynomial with integral roots}\}$$</code></p>

<hr />

<div class="language-plaintext highlighter-rouge"><div class="highlight"><pre class="highlight"><code>M = On input &lt;p&gt; where p is a polynomial over variables x1,...,xk,
  1. For each possible integral assignment of x1,...,xk:
     a. Evaluate the polynomial p with the current assignment
     b. If p evaluates to zero, then immediately ACCEPT
  2. If no assignment ever evaluates to zero, then REJECT
</code></pre></div></div>

<hr />

<ul>
  <li class="fragment">Will line 2 ever be reached by this machine?
    <ul>
      <li class="fragment">No. There are an infinite number of assignments to the variables, so the for-loop is infinite</li>
    </ul>
  </li>
  <li class="fragment">Does this machine recognize <code class="language-plaintext highlighter-rouge">$L$</code>?</li>
</ul>

<h1 id="church-turing-thesis">Church-Turing Thesis</h1>

<h1 id="special-topic-1">Special Topic</h1>