Review For Exam | CS 139: Theory of Computation

<h1 id="review">Review</h1>

<hr />

<h2 id="assignment-9">Assignment 9</h2>
<ul>
  <li><strong>Due:</strong> today at 11:59 PM</li>
  <li>Send me a link to your video when you are done</li>
</ul>

<h2 id="extra-credit-opportunity">Extra Credit Opportunity</h2>
<ul>
  <li>I will post each group’s presentation on Blackboard soon after receiving all of them</li>
  <li>For each presentation, I will add +0.33% to your final grade if you watch the presentation and fill out a short rubric about it</li>
  <li class="fragment">If you do this for all of the presentations, it will be a total of 4% added to your final grade</li>
  <li class="fragment">Your feedback will be shared <strong>anonymously</strong> with the group and will not affect their grade</li>
</ul>

<h2 id="final-exam">Final Exam</h2>
<ul>
  <li><strong>Final Exam</strong> is on Wednesday, May 13th
    <ul>
      <li><strong>Open</strong> notes, <strong>open</strong> text, 110 minute exam</li>
      <li>Covers Sipser chapters 1, 3, 4, 5, and 7</li>
    </ul>
  </li>
  <li>Exam procedures:
    <ul>
      <li>Administered via a Gradescope feature that allows timed assessments <strong>without</strong> printing or scanning</li>
      <li>You may take the exam at any time from
        <ul>
          <li>12:00 AM CDT to 11:59 PM CDT</li>
        </ul>
      </li>
      <li>Once you start, you will have 110 minutes to complete the exam</li>
      <li>You may <strong>not</strong> use Google or receive help from another other source other than your notes and the text</li>
    </ul>
  </li>
</ul>

<h2 id="final-exam-1">Final Exam</h2>
<ul>
  <li>Old Practice problems from Sipser:
    <ul>
      <li><strong>Chapter 1</strong>: 2, 6, 7, 11, 12, 14, 18, 20, 22, 29, 30, 31, 38, 40, 43, 46</li>
      <li><strong>Chapter 3</strong>: 7, 8, 15, 16</li>
      <li><strong>Chapter 4</strong>: 2, 3, 4, 5, 10, 12, 13, 21</li>
      <li><strong>Chapter 5</strong>: 4, 5, 6, 7</li>
    </ul>
  </li>
  <li>New Practice problems from Sipser:
    <ul>
      <li><strong>Chapter 5</strong>: 9, 10, 14, 20, 22</li>
      <li><strong>Chapter 7</strong>: 1, 5, 6, 7, 9, 10, 16, 18, 20</li>
    </ul>
  </li>
</ul>

<h1 id="reflections-on-the-course">Reflections on the Course</h1>

<h2 id="course-topics">Course Topics</h2>
<ol>
  <li>Complexity Theory
    <ul>
      <li>The study of the <strong>difficulty</strong> of computational problems</li>
    </ul>
  </li>
  <li>Computability Theory
    <ul>
      <li>The study of the <strong>limits</strong> of computation</li>
    </ul>
  </li>
  <li>Automata Theory
    <ul>
      <li>The study of abstract <strong>machines</strong></li>
    </ul>
  </li>
</ol>

<h2 id="1-complexity-theory">1. Complexity Theory</h2>
<ul>
  <li>On the first day of class we looked at:</li>
</ul>

<hr />

<ol>
  <li>Given two integers <code class="language-plaintext highlighter-rouge">$a,b\in\mathbb{Z}$</code>, compute <code class="language-plaintext highlighter-rouge">$a\cdot b$</code>.</li>
  <li class="fragment">Given an integer <code class="language-plaintext highlighter-rouge">$n\in\mathbb{Z}$</code>, find two factors <code class="language-plaintext highlighter-rouge">$a,b\in\mathbb{Z}$</code> such that <code class="language-plaintext highlighter-rouge">$n = a\cdot b$</code>.</li>
  <li class="fragment">Given a natural number <code class="language-plaintext highlighter-rouge">$n\in\mathbb{N}$</code>, output <strong>True</strong> if <code class="language-plaintext highlighter-rouge">$n$</code> is prime and <strong>False</strong> otherwise</li>
</ol>

<h2 id="1-complexity-theory-1">1. Complexity Theory</h2>
<ul>
  <li>Other interesting things regarding complexity theory
    <ul>
      <li class="fragment">Some problems such as the <strong>shortest path problem</strong> are in <code class="language-plaintext highlighter-rouge">$\text{P}$</code> and are trivial to compute</li>
      <li class="fragment">Others such as the <strong>longest path problem</strong> are <code class="language-plaintext highlighter-rouge">$\text{NP}$</code> complete and are computationally intractable</li>
    </ul>
  </li>
</ul>

<h2 id="2-computability-theory">2. Computability Theory</h2>
<ul>
  <li>We also defined a computational model that correctly captures our sense of “computable”—the <strong>Turing machine</strong></li>
  <li class="fragment">Many problems are literally impossible for computers to solve:
    <ul>
      <li class="fragment">Checking if two programs are equivalent (<code class="language-plaintext highlighter-rouge">$\text{EQ}_\text{TM}$</code>)</li>
      <li class="fragment">Checking if a program halts (<code class="language-plaintext highlighter-rouge">$\text{HALT}$</code>)</li>
      <li class="fragment">Knowing if a set of self-assembling tiles will build the shape you’re interested in</li>
    </ul>
  </li>
</ul>

<h2 id="3-automata-theory">3. Automata Theory</h2>
<ul>
  <li>We also explored <strong>finite state machines</strong> as an alternative computational model
    <ul>
      <li class="fragment">Is a much weaker notion of computation</li>
      <li class="fragment">Still has lots of applications in string searching, compilers, and real-time processing of enormous datasets</li>
    </ul>
  </li>
</ul>

<h1 id="np-completeness">NP Completeness</h1>

<h2 id="undirected-hamiltonian-path">Undirected Hamiltonian Path</h2>
<ul>
  <li>The <strong>Undirected Hamiltonian Path</strong> problem is:</li>
</ul>

<p><code class="language-plaintext highlighter-rouge">$$\text{UHAMPATH} = \{\langle G, s, t\rangle \mid \text{$G$ has a simple path from $s$ to $t$}\\\text{that visits every vertex exactly once}\}$$</code></p>

<ul>
  <li class="fragment">The only difference here is that <code class="language-plaintext highlighter-rouge">$G$</code> is an <strong>undirected</strong> graph instead of a directed one</li>
</ul>

<hr />

<ul>
  <li class="fragment"><strong>Claim:</strong> <code class="language-plaintext highlighter-rouge">$\text{UHAMPATH}$</code> is NP complete</li>
</ul>

<h2 id="undirected-hamiltonian-path-1">Undirected Hamiltonian Path</h2>
<ul>
  <li>Obviously <code class="language-plaintext highlighter-rouge">$\text{UHAMPATH}$</code> is in NP—verifier would be nearly identical to <code class="language-plaintext highlighter-rouge">$\text{HAMPATH}$</code>’s</li>
  <li class="fragment">Can we show <code class="language-plaintext highlighter-rouge">$\text{HAMPATH}\le_p\text{UHAMPATH}$</code>?</li>
</ul>

<h2 id="double-sat">Double SAT</h2>
<ul>
  <li>Show that the following language is NP complete
<code class="language-plaintext highlighter-rouge">$$\text{DOUBLE-SAT} = \{\langle \Phi\rangle \mid \text{$\Phi$ has at least two}\\\text{unique satisfying assignments }\}$$</code></li>
</ul>

<h2 id="double-sat-1">Double SAT</h2>

<p><code class="language-plaintext highlighter-rouge">$$\text{DOUBLE-SAT} = \{\langle \Phi\rangle \mid \text{$\Phi$ has at least two}\\\text{unique satisfying assignments }\}$$</code></p>

<hr />

<ul>
  <li><strong>Exercise:</strong> Show that it is in NP
    <ul>
      <li class="fragment"><strong>Idea:</strong> Have a machine nondeterministically select two assignments, make sure they’re different, and check if they both satisfy the formula</li>
    </ul>
  </li>
  <li class="fragment"><strong>Exercise:</strong> Show that it is NP hard
    <ul>
      <li class="fragment">Show that <code class="language-plaintext highlighter-rouge">$\text{3SAT} \le_p \text{DOUBLE-SAT}$</code></li>
    </ul>
  </li>
</ul>

<h1 id="chapter-5">Chapter 5</h1>

<h2 id="exercise-1">Exercise 1</h2>
<p>Prove or disprove the following statement:</p>

<p><strong>Claim</strong>: <code class="language-plaintext highlighter-rouge">$L$</code> is Turing recognizable if and only if <code class="language-plaintext highlighter-rouge">$L\le_m A_\text{TM}$</code>.</p>

<h2 id="exercise-2">Exercise 2</h2>
<p>Prove or disprove the following statement:</p>

<p><strong>Claim</strong>: If <code class="language-plaintext highlighter-rouge">$L$</code> is Turing recognizable, then so is <code class="language-plaintext highlighter-rouge">$\overline{L}$</code></p>

<h2 id="exercise-3">Exercise 3</h2>
<ul>
  <li>Give a low-level Turing machine description that computes the function  <code class="language-plaintext highlighter-rouge">$f:\{0\}^\ast\rightarrow\{1\}^\ast$</code> defined as
<code class="language-plaintext highlighter-rouge">$$f(0^k) = 1^{2k}$$</code></li>
</ul>