Nondeterministic Finite Automata | CS 139: Theory of Computation

<h1 id="nondeterministic-finite-automata">Nondeterministic Finite Automata</h1>

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<h2 id="cs-candidate-research-talks">CS Candidate Research Talks</h2>
<ul>
  <li>CS Candidate Research Talks
    <ul>
      <li>Matt Behrens</li>
      <li>Wednesday, 4:30 PM</li>
      <li>…</li>
    </ul>
  </li>
  <li>Assignment 3 posted</li>
</ul>

<h1 id="dfa-recap">DFA Recap</h1>

<h2 id="deterministic-finite-automata">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>As input, it receives a <strong>string</strong></li>
  <li>It outputs one of two things:
    <ul>
      <li><strong>ACCEPT</strong>: akin to “YES” or “True”</li>
      <li><strong>REJECT</strong>: akin to “NO” or “False”</li>
    </ul>
  </li>
  <li>
    <p>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.</p>
  </li>
  <li class="fragment">A language is <strong>regular</strong> if there exists some DFA that recognizes it</li>
</ul>

<h2 id="deterministic-finite-automata-1">Deterministic Finite Automata</h2>
<ul>
  <li>Visually, a DFA <code class="language-plaintext highlighter-rouge">$M$</code> looks like the following:</li>
</ul>

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

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

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

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<p class="fragment">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>

<h2 id="closure-properties">Closure Properties</h2>
<ul>
  <li>The class of regular languages is <strong>closed under intersection</strong></li>
  <li class="fragment">This means that if <code class="language-plaintext highlighter-rouge">$A$</code> and <code class="language-plaintext highlighter-rouge">$B$</code> are regular languages, then so is the language <code class="language-plaintext highlighter-rouge">$A\cap B$</code></li>
</ul>

<h2 id="proof-idea">Proof Idea</h2>
<ul>
  <li>Given two DFAs
    <ul>
      <li><code class="language-plaintext highlighter-rouge">$M_1 = (Q_1, \Sigma, \delta_1, q_1, F_1)$</code></li>
      <li><code class="language-plaintext highlighter-rouge">$M_2 = (Q_2, \Sigma, \delta_2, q_2, F_2)$</code></li>
    </ul>
  </li>
  <li>Need to show that <code class="language-plaintext highlighter-rouge">$L(M_1)\cap L(M_2)$</code> is regular</li>
</ul>

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<ul>
  <li class="fragment">Can define DFA <code class="language-plaintext highlighter-rouge">$M_3 = (Q_3, \Sigma, \delta_3, q_3, F_3)$</code> by
    <ul>
      <li class="fragment"><code class="language-plaintext highlighter-rouge">$Q_3 = Q_1\times Q_2$</code></li>
      <li class="fragment"><code class="language-plaintext highlighter-rouge">$\delta_3\big((r_1,r_2),a\big) = \big(\delta_1(r_1,a), \delta_2(r_2, a)\big)$</code></li>
      <li class="fragment"><code class="language-plaintext highlighter-rouge">$q_3 = (q_1, q_2)$</code></li>
      <li class="fragment"><code class="language-plaintext highlighter-rouge">$F_3 = \{(r_1, r_2)\mid r_1\in F_1\text{ and }r_2\in F_2\}$</code></li>
    </ul>
  </li>
</ul>

<h1 id="nondeterministic-finite-automata-1">Nondeterministic Finite Automata</h1>

<h2 id="why-nfas">Why NFAs?</h2>
<ul>
  <li class="fragment">“Syntactical sugar”
    <ul>
      <li>Same reason Python has <code class="language-plaintext highlighter-rouge">for</code> loops—for <strong>convenience</strong>!</li>
    </ul>
  </li>
  <li class="fragment"><strong>Nondeterminism</strong> foreshadows some important concepts we will cover later in the course (<code class="language-plaintext highlighter-rouge">$P$</code> versus <code class="language-plaintext highlighter-rouge">$NP$</code>)</li>
</ul>

<h2 id="how-are-nfas-different">How are NFAs different?</h2>

<ul>
  <li>DFAs are <strong>deterministic</strong> in the following sense:
    <ul>
      <li class="fragment">When a DFA is given an input string, its behavior is <strong>fully determined</strong></li>
      <li class="fragment">In other words, there is only <strong>a single path</strong> through the machine from the start state to an accepting state</li>
    </ul>
  </li>
  <li class="fragment">NFAs are <strong>nondeterministic</strong> in the sense that there are <strong>multiple ways</strong> the machine can process its input</li>
</ul>

<h2 id="nfa-example">NFA Example</h2>

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<ul>
  <li>What does this machine do on input <code class="language-plaintext highlighter-rouge">$w = 00110$</code>?
    <ul>
      <li class="fragment"><code class="language-plaintext highlighter-rouge">$q_0$</code> has two transitions for <code class="language-plaintext highlighter-rouge">$0$</code> and <code class="language-plaintext highlighter-rouge">$1$</code></li>
      <li class="fragment"><code class="language-plaintext highlighter-rouge">$q_2$</code> has <strong>zero</strong> transitions</li>
    </ul>
  </li>
</ul>

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<h2 id="when-an-nfa-accepts">When an NFA Accepts</h2>
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<ul>
  <li>An NFA <strong>accepts</strong> a string <code class="language-plaintext highlighter-rouge">$w\in\Sigma^\ast$</code> if there is some <strong>path</strong> following transitions according to the symbols in <code class="language-plaintext highlighter-rouge">$w$</code> that ends in one of the accepting states</li>
  <li class="fragment">Whenever a transition with a 0 or 1 on it above is traversed, a 0 or 1 from the input <strong>must be consumed</strong>
    <ul>
      <li class="fragment">This is why 1011 is <strong>rejected</strong>—there’s no way to get to <code class="language-plaintext highlighter-rouge">$q_2$</code> or <code class="language-plaintext highlighter-rouge">$q_4$</code> when the last two bits are 1</li>
    </ul>
  </li>
</ul>

<h2 id="epsilon-transitions"><code class="language-plaintext highlighter-rouge">$\epsilon$</code>-transitions</h2>
<ul>
  <li>Another convenient feature of NFAs is the <strong><code class="language-plaintext highlighter-rouge">$\epsilon$</code>-transition</strong></li>
  <li><code class="language-plaintext highlighter-rouge">$\epsilon$</code>-transitions can be traversed <strong>without consuming a symbol of the input</strong></li>
</ul>

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<h2 id="epsilon-transitions-1"><code class="language-plaintext highlighter-rouge">$\epsilon$</code>-transitions</h2>
<ul>
  <li>Suppose I change the previous NFA to:</li>
</ul>

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<ul>
  <li>What does it recognize now?</li>
  <li class="fragment"><code class="language-plaintext highlighter-rouge">$L(M) = \{00,11\}^\ast = \{\epsilon, 00, 11, 0000, 0011, \ldots\}$</code></li>
</ul>

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

<hr />

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

<h2 id="formal-definition-of-nfas-1">Formal Definition of NFAs</h2>

<p><code class="language-plaintext highlighter-rouge">$$M = (Q, \Sigma, \delta, q_0, F)$$</code></p>

<hr />

<ul>
  <li><code class="language-plaintext highlighter-rouge">$\delta:Q\times\Sigma_\epsilon\rightarrow\mathbb{P}(Q)$</code> is the <strong>transition function</strong></li>
  <li class="fragment"><code class="language-plaintext highlighter-rouge">$\Sigma_\epsilon = \Sigma \cup \{\epsilon\}$</code> allows <code class="language-plaintext highlighter-rouge">$\epsilon$</code>’s on transitions</li>
  <li class="fragment"><code class="language-plaintext highlighter-rouge">$\mathbb{P}(Q)$</code> is the <strong>powerset</strong> of <code class="language-plaintext highlighter-rouge">$Q$</code>
    <ul>
      <li class="fragment">Allows for <strong>multiple</strong> arrows with the <strong>same symbol</strong> coming out of the same state</li>
    </ul>
  </li>
</ul>

<h2 id="formal-nfa-exercise">Formal NFA Exercise</h2>

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<hr />

`$Q = \{q_0, q_1, q_2, q_3, r_1, r_2, r_3, s\}\quad$`<br />
`$\Sigma = \{0,1\}$`<br />
`$q_0 = q_0$`<br />
`$F = \{q_0\}$`

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<div>

`$\delta(q_0, 0) = \emptyset$`<br />
`$\delta(q_0, 1) = \emptyset$`<br />
`$\delta(q_0, \epsilon) = \{q_1, r_1\}$`<br />
`$\delta(q_1, 0) = \{q_2\}$`<br />
`$\delta(q_1, 1) = \emptyset$`<br />
`$\delta(q_1, \epsilon) = \emptyset$`<br />
`$\ldots$`

</div>
</div>

<h2 id="more-nfa-exercises">More NFA Exercises</h2>
<ul>
  <li>Construct an NFA for each of the following languages</li>
</ul>

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

<p><code class="language-plaintext highlighter-rouge">$L(M) = \{w\in\{0,1\}^\ast\mid\text{ the length of }w\text{ is 4} \}$</code></p>

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<h2 id="exercise-2">Exercise 2</h2>
<p><code class="language-plaintext highlighter-rouge">$L(M) = \{w\mid w\text{ starts with a 0 or ends with a 1}\}$</code></p>

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</div>

<h2 id="exercise-3">Exercise 3</h2>

<p><code class="language-plaintext highlighter-rouge">$L(M) = \{w\mid w\text{ contains the substring }\texttt{abba}\}$</code></p>

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