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<center><h2>cs150: Notes 26</h2></center>

<h4>Assignments Due</h4>

<ul>
<li> This week: read <a href="book/ch11-computability.pdf">Chapter 11</a>
<li> <b>Monday, 26 March</b>: <a href="/cs150/ps/ps6/">Problem Set 6</a>
<li> Someday next week, there will be a "surprise" quiz.  Your
quiz grade will be the maximum of your grades on the two quizzes.  Next
week's "surprise" quiz will cover the readings from Neil de Grasse
Tyson; G&ouml;del, Escher, Bach; and the <a href="/cs150/readings/backus.html">article attached to today's notes</a>.
<li> GEB reading (by Friday, 30 March): <em>Aria with Diverse
Variations</em> and Chapter 13. Chapter 13 proves that the halting
problem is not decidable, and introduces the Church-Turing Thesis (which
we will explore more in later classes).   You will not be assigned to
read Chapter 14, but it goes into more depth on G&ouml;del's proof and
is highly recommended.  

</ul>

<p>
<center><font size=+1><b>Notes</b></font></center>
<p>

<b>Halting Problem</b>
<p>
<b>Input:</b> a procedure specification <em>P</em> 
<p>
<b>Output:</b> true if applying <em>P</em> would halt (finishes execution), false
otherwise.
<p>

<blockquote>
<pre>
(define (halts? procedure) <em>...?...</em>)
</pre>
</blockquote>

<blockquote>
<pre>
(define (paradox)
  (if (halts? 'paradox) 
      (infinite-loop)
      150))
</pre>
</blockquote>

What would it mean if <tt>(halts? 'paradox)</tt> evaluates to
<tt>#t</tt>?
<p><br><p><br><p><br><p><br><p><br><p><br>
What would it mean if <tt>(halts? 'paradox)</tt> evaluates to
<tt>#f</tt>?
<p><br><p><br><p><br><p><br><p><br><p><br>

Prove the <tt>halts?</tt> algorithm cannot exist.
<p><br><p><br><p><br><p><br><p><br><p><br>

<b>Proof by Contradiction</b>

<ol>
<li> Show <em>X</em> is non-sensical.
<li> Show that if you have an <em>A</em> you can make <em>X</em>.
<li> Therefore, <em>A</em> cannot exist.
</ol>

<p><br><p><br><p><br>


<b>Evaluates-to-3</b>

<blockquote>
<b>Input:</b> A procedure specification P
<p>
<b>Output:</b> true if evaluating an application of <tt>P</tt> would result in 3; false otherwise. 
</blockquote>

Is the Evaluates-to-3 Problem computable?

<p><br><p><br><p><br><p><br>



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