Pause For Breath | CS 139: Theory of Computation

<h1 id="pause-for-breath">Pause For Breath</h1>

<hr />

<h2 id="administrivia">Administrivia</h2>
<ul>
  <li>Assignment 7 due today at 11:59 PM</li>
  <li>Any questions about it?</li>
  <li class="fragment">I will post solutions tomorrow morning</li>
</ul>

<h2 id="administrivia-1">Administrivia</h2>
<ul>
  <li><strong>Exam 2</strong> is on Thursday
    <ul>
      <li><strong>Open</strong> notes, <strong>open</strong> text, 75 minute exam</li>
      <li>Covers Sipser chapters 1, 3, 4, and parts of 5</li>
    </ul>
  </li>
  <li class="fragment">Exam procedures:
    <ul>
      <li class="fragment">I will release the exam at 11:00 AM CDT on Gradescope</li>
      <li class="fragment">From that time, you will have <strong>120 minutes</strong> to complete the exam and upload your solutions to gradescope (extra time for scanning and uploading)</li>
      <li class="fragment">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="administrivia-2">Administrivia</h2>
<ul>
  <li>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>
</ul>

<h1 id="exercises">Exercises</h1>

<h2 id="exercise-1">Exercise 1</h2>
<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>

<hr />

<ul>
  <li class="fragment">Turing recognizable means that there is a TM that <strong>accepts</strong> all <code class="language-plaintext highlighter-rouge">$w\in L$</code> and <strong>does not accept</strong> all <code class="language-plaintext highlighter-rouge">$w\not\in L$</code></li>
  <li class="fragment"><code class="language-plaintext highlighter-rouge">$A_\text{TM} = \{\langle M, w\rangle \mid \text{ $M$ is a TM that accepts $w$}\}$</code></li>
  <li class="fragment"><code class="language-plaintext highlighter-rouge">$L \le_m A_\text{TM}$</code> means there is a computable function <code class="language-plaintext highlighter-rouge">$f$</code> such that <code class="language-plaintext highlighter-rouge">$w\in L$</code> <strong>if and only if</strong> <code class="language-plaintext highlighter-rouge">$f(w) \in A_\text{TM}$</code></li>
</ul>

<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><strong>Claim:</strong> the following language is decidable
<code class="language-plaintext highlighter-rouge">$$\text{FINITE}_\text{DFA}
    = \{ \langle M\rangle \mid \text{$M$ is a DFA and $L(M)$ is finite} \}$$</code></p>

<hr />

<ul>
  <li class="fragment"><strong>3-2-1-GO:</strong> Do you think this is true or false?</li>
</ul>

<h2 id="scratch-paper">Scratch Paper</h2>
<ul>
  <li>Does the following DFA recognize a finite language?</li>
  <li class="fragment">Why or why not?</li>
</ul>

<hr />

</div>

<h2 id="scratch-paper-1">Scratch Paper</h2>
<ul>
  <li>What about this DFA?</li>
</ul>

<hr />

</div>

<hr />

<ul>
  <li class="fragment">In general, how can we determine whether a DFA will recognize an infinite language?</li>
  <li class="fragment"><strong>Observation</strong>: A DFA recognizes an infinite language if there is a <strong>cycle</strong> on an accepting path</li>
</ul>

<h2 id="exercise-3-from-kozen-1997">Exercise 3 (from Kozen 1997)</h2>
<p>Which of these languages about TMs are decidable?</p>

<ul>
  <li class="fragment"><code class="language-plaintext highlighter-rouge">$L_1 = \{\langle M \rangle \mid \text{$M$ has $\ge$ 481 states}\}$</code></li>
  <li class="fragment"><code class="language-plaintext highlighter-rouge">$L_2 = \{\langle M \rangle \mid \text{$M$ takes $\ge$ 481 steps on input $\epsilon$}\}$</code></li>
  <li class="fragment"><code class="language-plaintext highlighter-rouge">$L_3 = \{\langle M \rangle \mid \text{$M$ takes $\ge$ 481 steps on some input}\}$</code></li>
  <li class="fragment"><code class="language-plaintext highlighter-rouge">$L_4 = \{\langle M \rangle \mid \text{$M$ takes $\ge$ 481 steps on all inputs}\}$</code></li>
  <li class="fragment"><code class="language-plaintext highlighter-rouge">$L_5 = \{\langle M \rangle \mid \text{$M$ moves its head to the 481th cell on $\epsilon$}\}$</code></li>
  <li class="fragment"><code class="language-plaintext highlighter-rouge">$L_6 = \{\langle M \rangle \mid \text{$M$ accepts $\epsilon$}\}$</code></li>
  <li class="fragment"><code class="language-plaintext highlighter-rouge">$L_7 = \{\langle M \rangle \mid \text{$M$ accepts some string}\}$</code></li>
  <li class="fragment"><code class="language-plaintext highlighter-rouge">$L_8 = \{\langle M \rangle \mid \text{$M$ accepts every string}\}$</code></li>
  <li class="fragment"><code class="language-plaintext highlighter-rouge">$L_9 = \{\langle M \rangle \mid \text{$M$ accepts a finite set}\}$</code></li>
</ul>

<ul>
  <li><code class="language-plaintext highlighter-rouge">$L_1 = \{\langle M \rangle \mid \text{$M$ has $\ge$ 481 states}\}$</code>
    <ul>
      <li class="fragment"><strong>Decidable:</strong> (count the number of states)</li>
    </ul>
  </li>
  <li class="fragment"><code class="language-plaintext highlighter-rouge">$L_2 = \{\langle M \rangle \mid \text{$M$ takes $\ge$ 481 steps on input $\epsilon$}\}$</code>
    <ul>
      <li class="fragment"><strong>Decidable:</strong> (simulate for 481 steps on <code class="language-plaintext highlighter-rouge">$\epsilon$</code>)</li>
    </ul>
  </li>
  <li class="fragment"><code class="language-plaintext highlighter-rouge">$L_3 = \{\langle M \rangle \mid \text{$M$ takes $\ge$ 481 steps on some input}\}$</code>
    <ul>
      <li class="fragment"><strong>Decidable:</strong> (simulate 481 steps on all inputs <code class="language-plaintext highlighter-rouge">$\le$</code> length 481)</li>
    </ul>
  </li>
  <li class="fragment"><code class="language-plaintext highlighter-rouge">$L_4 = \{\langle M \rangle \mid \text{$M$ takes $\ge$ 481 steps on all inputs}\}$</code>
    <ul>
      <li class="fragment"><strong>Decidable:</strong> (simulate 481 steps on all inputs <code class="language-plaintext highlighter-rouge">$\le$</code> length 481)</li>
    </ul>
  </li>
  <li class="fragment"><code class="language-plaintext highlighter-rouge">$L_5 = \{\langle M \rangle \mid \text{$M$ moves its head to the 481th cell on $\epsilon$}\}$</code>
    <ul>
      <li class="fragment"><strong>Decidable:</strong> (finite # of configurations of 481 tape cells)</li>
    </ul>
  </li>
</ul>

<ul>
  <li><code class="language-plaintext highlighter-rouge">$L_6 = \{\langle M \rangle \mid \text{$M$ accepts $\epsilon$}\}$</code>
    <ul>
      <li class="fragment"><strong>Undecidable</strong></li>
    </ul>
  </li>
  <li class="fragment"><code class="language-plaintext highlighter-rouge">$L_7 = \{\langle M \rangle \mid \text{$M$ accepts some string}\}$</code>
    <ul>
      <li class="fragment"><strong>Undecidable</strong></li>
    </ul>
  </li>
  <li class="fragment"><code class="language-plaintext highlighter-rouge">$L_8 = \{\langle M \rangle \mid \text{$M$ accepts every string}\}$</code>
    <ul>
      <li class="fragment"><strong>Undecidable</strong></li>
    </ul>
  </li>
  <li class="fragment"><code class="language-plaintext highlighter-rouge">$L_9 = \{\langle M \rangle \mid \text{$M$ accepts a finite set}\}$</code>
    <ul>
      <li class="fragment"><strong>Undecidable</strong></li>
    </ul>
  </li>
</ul>

<ul>
  <li>What is the general pattern about undecidable properties of Turing machines?</li>
</ul>

<hr />

<ul>
  <li><code class="language-plaintext highlighter-rouge">$L_1 = \{\langle M \rangle \mid \text{$M$ has $\ge$ 481 states}\}$</code></li>
  <li><code class="language-plaintext highlighter-rouge">$L_2 = \{\langle M \rangle \mid \text{$M$ takes $\ge$ 481 steps on input $\epsilon$}\}$</code></li>
  <li><code class="language-plaintext highlighter-rouge">$L_3 = \{\langle M \rangle \mid \text{$M$ takes $\ge$ 481 steps on some input}\}$</code></li>
  <li><code class="language-plaintext highlighter-rouge">$L_4 = \{\langle M \rangle \mid \text{$M$ takes $\ge$ 481 steps on all inputs}\}$</code></li>
  <li><code class="language-plaintext highlighter-rouge">$L_5 = \{\langle M \rangle \mid \text{$M$ moves its head to the 481th cell on $\epsilon$}\}$</code></li>
  <li><code class="language-plaintext highlighter-rouge">$L_6 = \{\langle M \rangle \mid \text{$M$ accepts $\epsilon$}\}$</code></li>
  <li><code class="language-plaintext highlighter-rouge">$L_7 = \{\langle M \rangle \mid \text{$M$ accepts some string}\}$</code></li>
  <li><code class="language-plaintext highlighter-rouge">$L_8 = \{\langle M \rangle \mid \text{$M$ accepts every string}\}$</code></li>
  <li><code class="language-plaintext highlighter-rouge">$L_9 = \{\langle M \rangle \mid \text{$M$ accepts a finite set}\}$</code></li>
</ul>

<h2 id="exercise-4">Exercise 4</h2>
<p><strong>Claim:</strong> the following language is decidable
<code class="language-plaintext highlighter-rouge">$$\text{FINITE}_\text{TM}
    = \{ \langle M\rangle \mid \text{$M$ is a TM and $L(M)$ is finite} \}$$</code></p>

<h2 id="exercise-5">Exercise 5</h2>
<p>Let <code class="language-plaintext highlighter-rouge">$f:\{\texttt{x}\}^\ast\rightarrow\{0, 1, 2\}$</code> be the function defined by
<code class="language-plaintext highlighter-rouge">$$f(\texttt{x}^n) = (n\;\%\;3)$$</code>
Show that <code class="language-plaintext highlighter-rouge">$f$</code> is computable by giving a <strong>low-level</strong> Turing machine that computes it</p>

<h2 id="exercise-6">Exercise 6</h2>
<ul>
  <li>Let <code class="language-plaintext highlighter-rouge">$\Sigma = \{\texttt{(}, \texttt{)}\}$</code> be an alphabet (just parentheses)</li>
  <li class="fragment"><code class="language-plaintext highlighter-rouge">$P = \{w\in\Sigma^\ast \mid \text{$w$ has perfectly matched parentheses}\}$</code>
    <ul>
      <li class="fragment"><code class="language-plaintext highlighter-rouge">$\texttt{(()())}\in P$</code></li>
      <li class="fragment"><code class="language-plaintext highlighter-rouge">$\texttt{(()))()}\not\in P$</code></li>
    </ul>
  </li>
  <li class="fragment">Prove that <code class="language-plaintext highlighter-rouge">$P$</code> is not regular using the pumping lemma</li>
</ul>