Writing Proofs | CS 139: Theory of Computation

<h1 id="writing-proofs">Writing Proofs</h1>

<hr />

<h2 id="administrivia">Administrivia</h2>
<ul>
  <li>My office hours will stay the same
    <ul>
      <li><strong>MW:</strong> 9:15am to 10:45am</li>
      <li><strong>MW:</strong> 2:45pm to 4:15pm</li>
    </ul>
  </li>
  <li>I am willing to meet outside of these office hours if needed (especially those of you with conflicts)</li>
  <li class="fragment">I will be busy from 3:00–3:30pm tomorrow due to a unmovable meeting</li>
</ul>

<h2 id="administrivia-1">Administrivia</h2>
<ul>
  <li>Remember to do reading journals!
    <ul>
      <li>They are due at <strong>8:00am</strong> before the class period they are associated with</li>
    </ul>
  </li>
  <li class="fragment">I will only respond to your reading journal if it <strong>does not</strong> qualify for credit or there is some problem</li>
</ul>

<h1 id="assignment-2">Assignment 2</h1>
<p>Due <strong>Tuesday, February 11th</strong> (before class)</p>

<h2 id="strategy-for-writing-proofs">Strategy for Writing Proofs</h2>
<ol>
  <li>Understand the problem statement
    <ul>
      <li class="fragment">Play with it</li>
      <li class="fragment">Use concrete examples</li>
      <li class="fragment">Do you think it is true? Why?</li>
    </ul>
  </li>
  <li class="fragment">Identify the parts of the statement
    <ul>
      <li class="fragment">If and only if (<code class="language-plaintext highlighter-rouge">$P \iff Q$</code>) has two parts</li>
      <li class="fragment">Induction has two parts</li>
    </ul>
  </li>
</ol>

<h2 id="strategy-for-writing-proofs-1">Strategy for Writing Proofs</h2>

<ol start="3">
  <li>Consider looking at an easier version of the problem or a special case
    <ul>
      <li class="fragment">E.g., a <code class="language-plaintext highlighter-rouge">$4\times 4$</code> grid of the <code class="language-plaintext highlighter-rouge">$2^n\times 2^n$</code> Trinomial game</li>
    </ul>
  </li>
  <li class="fragment">Identify which type of proof method should be applied</li>
  <li class="fragment">Start working out the details of the proof</li>
  <li class="fragment">If you’re stuck, take a break, talk to a classmate, or reach out to me for help
    <ul>
      <li class="fragment">Consider trying alternative approaches</li>
    </ul>
  </li>
  <li class="fragment">After finishing it, proofread it and edit it as necessary.
    <ul>
      <li class="fragment">You wouldn’t normally submit a first draft of an essay!</li>
    </ul>
  </li>
</ol>

<h2 id="types-of-proofs">Types of Proofs</h2>
<ol>
  <li><strong>Direct Proof</strong>
    <ul>
      <li>No tricks, just directly show assumptions/definitions imply the statement is true</li>
    </ul>
  </li>
  <li class="fragment"><strong>Proof by Construction</strong> (Proof by Counterexample)
    <ul>
      <li>Statements of the form “there exists a…” can be proven by just giving an example of the thing</li>
      <li class="fragment">I call these “Behold!” proofs</li>
    </ul>
  </li>
  <li class="fragment"><strong>Proof by Contraposition</strong>
    <ul>
      <li>A statement <code class="language-plaintext highlighter-rouge">$p\implies q$</code> can be proven using its contrapositive form <code class="language-plaintext highlighter-rouge">$\lnot q\implies\lnot p$</code></li>
    </ul>
  </li>
</ol>

<h2 id="types-of-proofs-1">Types of Proofs</h2>

<ol start="4">
  <li><strong>Proof by Contradiction</strong>
    <ul>
      <li>Assume the statement is false, and then show that it leads to a logical contradiction</li>
      <li class="fragment">Especially useful if your statement is of the form “for every …” since the negation becomes “there exists …”</li>
    </ul>
  </li>
  <li class="fragment"><strong>Proof by Induction</strong>
    <ul>
      <li>Create a domino effect to prove a statement of the form “for every …”</li>
      <li class="fragment">Has one or more <strong>base cases</strong> and <strong>induction cases</strong></li>
    </ul>
  </li>
</ol>

<h2 id="exercise-1">Exercise 1</h2>
<p>Let <code class="language-plaintext highlighter-rouge">$\sim:\mathbb{R}\times\mathbb{R}\rightarrow\{\text{true}, \text{false}\}$</code> be the binary relation:
<code class="language-plaintext highlighter-rouge">$$a\sim b \iff a-b \in \mathbb{Z}.$$</code></p>

<hr />

<p>Is <code class="language-plaintext highlighter-rouge">$\sim$</code>  an <strong>equivalence relation</strong>? Why or why not?</p>

<h2 id="exercise-2">Exercise 2</h2>
<p><strong>Claim:</strong> For all sets <code class="language-plaintext highlighter-rouge">$A$</code> and <code class="language-plaintext highlighter-rouge">$B$</code>, the following holds:
<code class="language-plaintext highlighter-rouge">$$A\setminus B \subseteq A\cap B$$</code></p>

<hr />

<p>Is this claim true? Why or why not?</p>

<h2 id="exercise-3">Exercise 3</h2>
<p><strong>Claim:</strong> For all sets <code class="language-plaintext highlighter-rouge">$A$</code> and <code class="language-plaintext highlighter-rouge">$B$</code>, the following holds:
<code class="language-plaintext highlighter-rouge">$$x\not\in B \implies x\not\in A\setminus (A\setminus B)$$</code></p>

<hr />

<p>Is this claim true? Why or why not?</p>

<h2 id="exercise-4">Exercise 4</h2>
<p><strong>Claim:</strong> Every undirected graph <code class="language-plaintext highlighter-rouge">$G = (V,E)$</code> with at least two vertices has two vertices with the same degree.</p>

<hr />

<p>Is this claim true? Why or why not?</p>

<h2 id="exercise-5">Exercise 5</h2>
<p><strong>Claim:</strong> For all <code class="language-plaintext highlighter-rouge">$n\in\mathbb{N}$</code>, the following holds:
<code class="language-plaintext highlighter-rouge">$$n^3 + 2n\text{ is divisible by 3}$$</code></p>

<hr />

<p>Is this claim true? Why or why not?</p>

<h2 id="bonus-problem">Bonus Problem</h2>
<ul>
  <li>Have a board with <code class="language-plaintext highlighter-rouge">$2^n$</code> rows and <code class="language-plaintext highlighter-rouge">$2^n$</code> columns</li>
  <li>Have a bunch of L-shaped pieces</li>
  <li>Adversary selects a single square on the grid and marks it <strong>unusable</strong></li>
  <li>We win if we can arrange the L-shaped tiles in such a way that they cover the entire grid</li>
</ul>

<hr />

<ul>
  <li>Prove that we can win the game <strong>for any</strong> <code class="language-plaintext highlighter-rouge">$n$</code> and <strong>for any unusable square</strong> that the adversary selects</li>
</ul>

<h2 id="towards-a-definition-of">Towards a Definition of</h2>
<h1 id="computable">COMPUTABLE</h1>

<h2 id="definition-for-computable">Definition for “Computable”</h2>
<ul>
  <li>Recall that we defined a function to be <strong>computable</strong> if:
    <ul>
      <li>“There exists an algorithm that computes it”</li>
      <li class="fragment">In other words, there is a program that <strong>on all valid inputs</strong> will return the <strong>correct output</strong> of the function</li>
    </ul>
  </li>
  <li class="fragment">Is <code class="language-plaintext highlighter-rouge">$f(x,y) = x\cdot y$</code> computable?</li>
  <li class="fragment">Behold! A Python program that computes it!
    <div class="language-py highlighter-rouge"><div class="highlight"><pre class="highlight"><code><span class="k">def</span> <span class="nf">mult</span><span class="p">(</span><span class="n">x</span><span class="p">,</span> <span class="n">y</span><span class="p">):</span>
  <span class="s">"""Computes the product of two numbers"""</span>
  <span class="k">return</span> <span class="n">x</span> <span class="o">*</span> <span class="n">y</span>
</code></pre></div>    </div>
  </li>
</ul>

<h2 id="definition-for-computable-1">Definition for “Computable”</h2>
<ul>
  <li>We run into a few problems if we use “Python program” as our definition of algorithm:
    <ol>
      <li class="fragment">Python is <strong>complicated</strong>—it is hard to prove things like “no algorithm can compute <code class="language-plaintext highlighter-rouge">$f$</code>”</li>
      <li class="fragment">Python is too <strong>specific</strong>. What about Java? What about molecular biology?</li>
    </ol>
  </li>
  <li class="fragment">What we what:
    <ul>
      <li>The <strong>simplest</strong> definition of algorithm that is equivalent in power to <strong>all reasonable</strong> alternative definitions</li>
    </ul>
  </li>
</ul>

<h2 id="spoiler-alert">Spoiler Alert!</h2>
<ul>
  <li>The rigorous mathematical definition of algorithm we are looking for is the <strong>Turing machine</strong> (Sipser §3)</li>
  <li class="fragment">First, we will study a weaker model called a <strong>Deterministic Finite Automaton</strong> (Sipser §1)</li>
  <li class="fragment">A <strong>DFA</strong> is a Turing machine with <strong>finite memory</strong></li>
  <li class="fragment">Functions like <code class="language-plaintext highlighter-rouge">$f(n) = n^2$</code> are <strong>NOT</strong> computable in this model—but it is still a very useful model</li>
</ul>

<h2 id="another-simplification">Another Simplification</h2>
<ul>
  <li>Most computational problems have an equivalent <strong>decision</strong> variant that encapsulates all of its complexity</li>
  <li>These are much <strong>easier</strong> to work with</li>
</ul>

<hr />

<dl>
  <dt><strong>Optimization Problem</strong></dt>
  <dd>What is the shortest path from <code class="language-plaintext highlighter-rouge">$s$</code> to <code class="language-plaintext highlighter-rouge">$t$</code> in <code class="language-plaintext highlighter-rouge">$G = (V,E)$</code>?</dd>
  <dt><strong>Decision Problem</strong></dt>
  <dd>Does <code class="language-plaintext highlighter-rouge">$G$</code> have a path from <code class="language-plaintext highlighter-rouge">$s$</code> to <code class="language-plaintext highlighter-rouge">$t$</code> of length <code class="language-plaintext highlighter-rouge">$\le k$</code>?</dd>
</dl>

<h3 id="decision-problems-are-languages">Decision Problems are <strong>Languages</strong></h3>
<ul>
  <li class="fragment">An <strong>alphabet</strong> <code class="language-plaintext highlighter-rouge">$\Sigma$</code> is a finite set of <strong>symbols</strong>
    <ul>
      <li>e.g. <code class="language-plaintext highlighter-rouge">$\Sigma = \{0,1\}$</code> and <code class="language-plaintext highlighter-rouge">$\Sigma = \{\texttt{a},\texttt{b},\texttt{c}, \ldots, \texttt{z}\}$</code></li>
    </ul>
  </li>
  <li class="fragment">A <strong>string</strong> over an alphabet <code class="language-plaintext highlighter-rouge">$\Sigma$</code> is a finite sequence of symbols in <code class="language-plaintext highlighter-rouge">$\Sigma$</code>
    <ul>
      <li>e.g. <code class="language-plaintext highlighter-rouge">$\texttt{abba}$</code> is a string over <code class="language-plaintext highlighter-rouge">$\{\texttt{a},\texttt{b}\}$</code></li>
    </ul>
  </li>
  <li class="fragment">The <strong>empty string</strong> is denoted <code class="language-plaintext highlighter-rouge">$\epsilon$</code></li>
</ul>

<h3 id="decision-problems-are-languages-1">Decision Problems are <strong>Languages</strong></h3>
<ul>
  <li>The set of all strings over an alphabet <code class="language-plaintext highlighter-rouge">$\Sigma$</code> is denoted <code class="language-plaintext highlighter-rouge">$\Sigma^\ast$</code>
<code class="language-plaintext highlighter-rouge">$$\{0,1\}^\ast = \{\epsilon, 0, 1, 00, 01, 10, 11, 000, \ldots, \}$$</code></li>
  <li class="fragment">A <strong>language</strong> over <code class="language-plaintext highlighter-rouge">$\Sigma$</code> is a subset of <code class="language-plaintext highlighter-rouge">$\Sigma^\ast$</code></li>
</ul>

<h3 id="decision-problems-are-languages-2">Decision Problems are <strong>Languages</strong></h3>
<ul>
  <li>We can encode decision problems as languages</li>
  <li class="fragment">For example, consider the decision problem:
    <ul>
      <li>“Given <code class="language-plaintext highlighter-rouge">$x,y,z\in\mathbb{Z}_{\ge 0}$</code>, does <code class="language-plaintext highlighter-rouge">$x+y=z$</code>?”</li>
    </ul>
  </li>
  <li class="fragment">To define this as a language, we first need an alphabet:
<code class="language-plaintext highlighter-rouge">$$\Sigma = \{0, 1, 2, \ldots, 9\}\cup \{+,=\}$$</code></li>
  <li class="fragment">Then we can define the language:
<code class="language-plaintext highlighter-rouge">$$\text{ADD}_2 = \{x+y=z\mid x,y,z\in\{0,\ldots,9\}^\ast\\\text{where the sum of $x$ and $y$ is $z$}\}$$</code></li>
</ul>

<h3 id="decision-problems-are-languages-3">Decision Problems are <strong>Languages</strong></h3>
<ul>
  <li>It is also easy to encode strings in an arbitrary alphabet <code class="language-plaintext highlighter-rouge">$\Sigma$</code> in the <strong>binary alphabet</strong>: <code class="language-plaintext highlighter-rouge">$\{0,1\}$</code></li>
  <li class="fragment">For example,
    <ul>
      <li>If <code class="language-plaintext highlighter-rouge">$\Sigma = \{\texttt{a}, \texttt{b}, \texttt{c}, \texttt{d}\}$</code></li>
      <li>Let <code class="language-plaintext highlighter-rouge">$00=\texttt{a}$</code>, <code class="language-plaintext highlighter-rouge">$01=\texttt{b}$</code>, <code class="language-plaintext highlighter-rouge">$10=\texttt{c}$</code>, <code class="language-plaintext highlighter-rouge">$11=\texttt{d}$</code></li>
      <li>Thus the string <code class="language-plaintext highlighter-rouge">$\texttt{abba}$</code> would be <code class="language-plaintext highlighter-rouge">$00010100$</code> in binary</li>
    </ul>
  </li>
  <li class="fragment">Your laptop stores everything in binary, too!</li>
</ul>

<h1 id="assignment-2-1">Assignment 2</h1>
<p>Due <strong>Tuesday, February 11th</strong> (before class)</p>