Deterministic Finite Automata | CS 139: Theory of Computation

<h1 id="deterministic-finite-automata">Deterministic Finite Automata</h1>

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<h1 id="questions">Questions</h1>
<h2 id="about-assignment-2">…about Assignment 2?</h2>
<p>(Due <strong>Tuesday, February 11th</strong> before class)</p>

<h1 id="proofs">PROOFS</h1>

<h2 id="strategy-for-writing-proofs">Strategy for Writing Proofs</h2>
<ol>
  <li><strong>Understand</strong> the problem statement</li>
  <li>Identify the <strong>parts</strong> of the statement</li>
  <li>Consider looking at an <strong>easier version</strong> of the problem or a special case</li>
  <li>Identify which type of <strong>proof method</strong> should be applied</li>
  <li>Start working out the details of the proof</li>
  <li>If you’re stuck, take a break, talk to a classmate, or reach out to me for help</li>
  <li>After finishing it, proofread it and edit it as necessary.</li>
</ol>

<h2 id="types-of-proofs">Types of Proofs</h2>
<ol>
  <li><strong>Direct Proof</strong></li>
  <li><strong>Proof by Construction</strong> (“Behold!”)</li>
  <li><strong>Proof by Contraposition</strong></li>
  <li><strong>Proof by Contradiction</strong></li>
  <li><strong>Proof by Induction</strong></li>
</ol>

<h2 id="another-proof-by-induction">Another Proof by Induction</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>

<h1 id="languages">LANGUAGES</h1>

<h3 id="review-of-definitions">Review of Definitions</h3>
<ul>
  <li>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>A <strong>string</strong> is a finite sequence of symbols over an alphabet
    <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><code class="language-plaintext highlighter-rouge">$\epsilon$</code> is the <strong>empty string</strong></li>
  <li>The set of <strong>all strings</strong> over an alphabet 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>A <strong>language</strong> over an alphabet is a subset of <code class="language-plaintext highlighter-rouge">$\Sigma^\ast$</code></li>
</ul>

<h3 id="decision-problems-as-languages">Decision Problems as <strong>Languages</strong></h3>
<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>

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<dl class="fragment">
  <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>We can encode decision problems as languages</li>
  <li>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>
</ul>

<hr />

<ul>
  <li class="fragment">We defined the 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-1">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><code class="language-plaintext highlighter-rouge">$\Sigma = \{\texttt{a}, \texttt{b}, \texttt{c}, \texttt{d}\}$</code></li>
      <li class="fragment"><code class="language-plaintext highlighter-rouge">$\texttt{a}=00$</code>, <code class="language-plaintext highlighter-rouge">$\texttt{b}=01$</code>, <code class="language-plaintext highlighter-rouge">$\texttt{c}=10$</code>, <code class="language-plaintext highlighter-rouge">$\texttt{d}=11$</code></li>
      <li class="fragment"><code class="language-plaintext highlighter-rouge">$\texttt{abba}$</code> would be <code class="language-plaintext highlighter-rouge">$00010100$</code></li>
    </ul>
  </li>
</ul>

<h1 id="deterministic-finite-automata-1">DETERMINISTIC FINITE AUTOMATA</h1>

<h2 id="deterministic-finite-automata-2">Deterministic Finite Automata</h2>
<ul>
  <li>Informally, a <strong>deterministic finite automaton</strong> (<strong>DFA</strong>) is a “<em>machine</em> with <em>finite memory</em>”</li>
  <li class="fragment">As input, it receives a <strong>string</strong></li>
  <li class="fragment">It outputs one of two things:
    <ul>
      <li class="fragment"><strong>ACCEPT</strong>: akin to “YES” or “True”</li>
      <li class="fragment"><strong>REJECT</strong>: akin to “NO” or “False”</li>
    </ul>
  </li>
</ul>

<h2 id="deterministic-finite-automata-3">Deterministic Finite Automata</h2>
<ul>
  <li>A DFA processes its input string <strong>one symbol at a time</strong> from left to right (and it can’t go backwards)</li>
  <li class="fragment">A DFA <strong>accepts</strong> the string if it is in one of its <strong>accepting states</strong> after it finishes processing the input</li>
  <li class="fragment">If a DFA is in <strong>any other state</strong> when it is finished, it will <strong>reject</strong> its input</li>
  <li class="fragment">The <strong>language</strong> of a DFA <code class="language-plaintext highlighter-rouge">$M$</code>, written <code class="language-plaintext highlighter-rouge">$L(M)$</code>, is the set of strings that <code class="language-plaintext highlighter-rouge">$M$</code> accepts.</li>
</ul>

<h2 id="deterministic-finite-automata-4">Deterministic Finite Automata</h2>
<ul>
  <li>Visually, a DFA <code class="language-plaintext highlighter-rouge">$M$</code> looks like the following:</li>
</ul>

</div>

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<ul>
  <li class="fragment">What is the language <code class="language-plaintext highlighter-rouge">$L(M)$</code>?
    <ul>
      <li class="fragment">“All binary strings that start with 1”</li>
      <li class="fragment"><code class="language-plaintext highlighter-rouge">$L(M) = \{1w \mid w \in \{0,1\}^\ast\}$</code></li>
    </ul>
  </li>
</ul>

<h2 id="dfa-exercises">DFA Exercises</h2>
<ul>
  <li>Let <code class="language-plaintext highlighter-rouge">$\Sigma = \{\texttt{a}, \texttt{b}\}$</code></li>
  <li>Construct a DFA for each of the following languages</li>
</ul>

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

<p><code class="language-plaintext highlighter-rouge">$L(M) = \emptyset$</code></p>

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</div>

<h2 id="exercise-2">Exercise 2</h2>
<p><code class="language-plaintext highlighter-rouge">$L(M) = \{x\texttt{a}y\texttt{a}z \mid x,y,z\in\Sigma^\ast\}$</code></p>

<hr />

</div>

<h2 id="exercise-3">Exercise 3</h2>
<p><code class="language-plaintext highlighter-rouge">$L(M) = \{w\in\Sigma^\ast\mid w\text{ contains the substring }\texttt{abba}\}$</code></p>

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</div>

<h2 id="exercise-4">Exercise 4</h2>
<p><code class="language-plaintext highlighter-rouge">$L(M) = \{ w\in\Sigma^\ast\mid w\text{ has the same # of }\texttt{a}\text{'s and }\texttt{b}\text{'s}\}$</code></p>

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<ul>
  <li class="fragment">This is impossible! No such DFA exists</li>
  <li class="fragment">Why? Because DFAs only have <strong>finite</strong> memory and cannot count to arbitrarily large numbers</li>
</ul>

<h2 id="formal-definition-of-dfas">Formal Definition of DFAs</h2>
<p>A <strong>deterministic finite automaton</strong> (<strong>DFA</strong>) <code class="language-plaintext highlighter-rouge">$M$</code> is a 5-tuple:
<code class="language-plaintext highlighter-rouge">$$M = (Q, \Sigma, \delta, q_0, F)$$</code></p>

<hr />

<ul>
  <li class="fragment"><code class="language-plaintext highlighter-rouge">$Q$</code> is a finite set of <strong>states</strong></li>
  <li class="fragment"><code class="language-plaintext highlighter-rouge">$\Sigma$</code> is an <strong>alphabet</strong> (finite set of symbols)</li>
  <li class="fragment"><code class="language-plaintext highlighter-rouge">$\delta:Q\times\Sigma\rightarrow Q$</code> is the <strong>transition function</strong></li>
  <li class="fragment"><code class="language-plaintext highlighter-rouge">$q_0\in Q$</code> is the <strong>start state</strong></li>
  <li class="fragment"><code class="language-plaintext highlighter-rouge">$F\subseteq Q$</code> is the set of <strong>accepting states</strong></li>
</ul>

<h2 id="formal-dfa-exercise">Formal DFA Exercise</h2>
<ul>
  <li>Consider the DFA <code class="language-plaintext highlighter-rouge">$M = (Q, \Sigma, \delta, q_0, F)$</code> defined by the following diagram</li>
</ul>

</div>

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`$Q = \{q, r\} $`<br />
`$\Sigma = \{0, 1\}$`<br />
`$q_0 = q$`<br />
`$F = \{r\}$`

</div>
<div>

`$\delta(q, 0) = q$`<br />
`$\delta(q, 1) = r$`<br />
`$\delta(r, 0) = r$`<br />
`$\delta(r, 1) = q$`

</div>
</div>

<h2 id="yet-another-dfa-exercise">Yet Another DFA Exercise</h2>

<p>Consider the following two DFAs</p>

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</div>

</div>
<div>

`$$A = \{ w \mid \text{ # of 0's in }w\text{ is even} \}$$`

</div>
</div>

<hr />

</div>

</div>
<div>

`$$B = \{ w1 \mid w \in \{0,1\}^\ast \}$$`

</div>
</div>

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<p>Construct a DFA for <code class="language-plaintext highlighter-rouge">$$A\cap B = \{ w1 \mid\text{# of 0's in }w\text{ is even}\}$$</code></p>

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