Introduction To Theory | CS 139: Theory of Computation

<h1 id="theory-of-computation">THEORY OF COMPUTATION</h1>

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<h1 id="welcome">Welcome!</h1>
<p>My name is Titus Klinge<br />
(Please call me <strong>Titus</strong>)</p>

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**Office:** 305 Collier-Scripps<br />
**Email:** <titus.klinge@drake.edu><br />
**Phone:** (515) 271-4599

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**Office Hours:**<br />
**MW:** 9:15pm–10:45am<br /> **MW:** 2:45pm–4:15pm

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<h1 id="introductions">Introductions</h1>
<ul>
  <li>Getting to know you is important to me, so we are going to do the following:</li>
</ul>

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<ol>
  <li class="fragment">I will <strong>badly mispronounce</strong> your name</li>
  <li class="fragment">You will <strong>correct</strong> my terrible pronunciation and (if you are comfortable) share one <strong>cool thing you did over break</strong></li>
</ol>

<h2 id="course-website">Course Website</h2>

<p><a href="https://titusklinge.com/teaching/2020s/cs139/">analytics.drake.edu/teaching/2020s/cs139</a></p>

<h2 id="personal-website">Personal Website</h2>

<p><a href="http://titusklinge.com/">http://titusklinge.com</a></p>

<h1 id="course-themes">COURSE THEMES</h1>

<h2 id="1-complexity-theory">1. Complexity Theory</h2>
<blockquote>
  <p>The study of the <strong>difficulty</strong> of computational problems</p>
</blockquote>

<h2 id="1-complexity-theory-1">1. Complexity Theory</h2>
<ul>
  <li><strong>Classify</strong> each of the following computational problems as “easy” or “hard” for computers to solve</li>
  <li><strong>Discuss</strong> with your neighbors</li>
</ul>

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<ol>
  <li class="fragment">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="2-computability-theory">2. Computability Theory</h2>
<blockquote>
  <p>The study of the <strong>limits</strong> of computation</p>
</blockquote>

<h2 id="2-computability-theory-1">2. Computability Theory</h2>
<ul>
  <li>Let <code class="language-plaintext highlighter-rouge">$f:\mathbb{N}\rightarrow\mathbb{N}$</code> be a function defined by <code class="language-plaintext highlighter-rouge">$$f(n) = n^2$$</code></li>
  <li class="fragment">Is <code class="language-plaintext highlighter-rouge">$f$</code> <strong>computable</strong>?</li>
  <li class="fragment">With your neighbor, come up with your own definition of <strong>computable</strong> and write it down</li>
</ul>

<h2 id="computable-functions">Computable Functions</h2>
<p>A function <code class="language-plaintext highlighter-rouge">$f:\mathbb{N}\rightarrow\mathbb{N}$</code> is <strong>computable</strong> if…</p>

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<ul>
  <li>It can be computed in a finite number of operations</li>
  <li>There is an “algorithm” that, when given <code class="language-plaintext highlighter-rouge">$n$</code>, outputs <code class="language-plaintext highlighter-rouge">$f(n)$</code>
    <ul>
      <li>An algorithm: a series of steps / operations</li>
    </ul>
  </li>
  <li>If a “machine” can compute the function</li>
  <li>Can these machines have an infinite amount of memory?</li>
</ul>

<h2 id="computable-numbers">Computable Numbers</h2>
<ul>
  <li>Recall that <code class="language-plaintext highlighter-rouge">$\pi\in\mathbb{R}$</code> is the number <code class="language-plaintext highlighter-rouge">$$\pi = 3.1415926535... $$</code></li>
  <li class="fragment">Is <code class="language-plaintext highlighter-rouge">$\pi$</code> <strong>computable</strong>?</li>
  <li class="fragment">With your neighbor, come up with your own definition of <strong>computable</strong> and write it down</li>
</ul>

<h2 id="3-automata-theory">3. Automata Theory</h2>
<blockquote>
  <p>The study of abstract <strong>machines</strong></p>
</blockquote>

<h2 id="3-automata-theory-1">3. Automata Theory</h2>
<ul>
  <li>Definition of <strong>automaton</strong> (from Merriam-Webster)</li>
</ul>

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<ol>
  <li class="fragment">A mechanism that is relatively self-operating (i.e. <strong>robot</strong>)</li>
  <li class="fragment">A <strong>machine</strong> or control mechanism designed to follow automatically a predetermined sequence of operations or respond to encoded instructions</li>
</ol>

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<ul>
  <li class="fragment">We will be primarily focusing on <strong>finite state machines</strong></li>
</ul>

<h1 id="assignment-1">Assignment 1</h1>
<p>Due <strong>Tuesday, February 4th</strong> (before class)</p>

<ol>
  <li>Read the <strong>syllabus</strong></li>
  <li>Complete a <strong>questionnaire</strong></li>
  <li>Review of <strong>prereqs</strong></li>
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