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<center><h2>Class 17 Notes</h2></center>

<div class=reading>
<center>
<a href="../slides/class17-2x3.pdf">Lecture Slides, Suitable For Printing</a> 
</center> 
</div> 

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



G&ouml;del's Statement:<br>
<em>G</em>: This statment does not have any proof in the system.
<p>
What would it mean if <em>G</em> is true?
<p><br><p><br><p><br>
What would it mean if <em>G</em> is false?
<p><br><p><br><p><br>


<b>Proof:</b> A proof of <em>S</em> in an axiomatic system is a sequence of strings,
T<sub>0</sub>, T<sub>1</sub>, ..., T<sub>n</sub> where:
<ul>
<li> The first string is the axioms
<li> For all <em>i</em> from 1 to <em>n</em>, T<sub>n</sub> is the
result of applying one of the inference rules to T<sub>n-1</sub>
<li> T<sub>n</sub> is <em>S</em>
</ul>

<center>
<h4>Computability</h4>
</center>
<p>
<b>Procedure</b>: A precise (mechanizable) description of a process.
<p>
<b>Alrogithm</b>: A procedure that always terminates.
<p>

<p>
Is there an algorithm (a procedure that always terminates) that solves
the problem?
<p>
A problem is <b>computable</b> (decidable) is there exists an algorithm that can
solve the problem for all possible inputs.  It is not necessary to know
what that algorithm is to say a problem is computable, only to know that
some algorithm to solve it must exist.  For example, chess is a
computable problem, even if we do not yet know a practical algorithm that solves
it (yes).
<p>

A problem is <b>uncomputable</b> if there is no algorithm that can solve
the problem.  There might be a procedure, but it is not guaranteed to
terminate.
<p>

What is the Halting Problem?

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

Is it possible to define a procedure that solves the Halting Problem?

<pre>
(define (contradict-halts)
  (if (halts? contradict-halts) 
      (infinite-loop) 
</pre>

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

<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>Alan Turing Links</b>

<ul>
<li> <a href="http://www.turing.org.uk/turing/">Alan Turing</a>
<li> <a href="http://www.abelard.org/turpap2/tp2-ie.asp"><em>On
Computable Numbers, with an Application to the
Entscheidungsproblem</em></a> &mdash; Turing's original (1936) paper
that introduced the Halting Problem 
<li> <a href="http://us.imdb.com/Title?0115749"><em>Breaking the
Code</em></a> &mdash; movie about Alan Turing
<li> <a
href="http://www.ai.mit.edu/operations/social/sports/halting-problem-index.html">Halting
Problem D-League Hockey Team</a> 
(<a
href="http://www.ai.mit.edu/operations/social/sports/halting-problem95/game5_results.html">Crushing
defeat of the Media Lab "Freeze Frames"</a>)

<li> <a
href="http://www.time.com/time/time100/scientist/profile/turing.html">Time
100: Alan Turing</a>
</ul>

<b>John Backus Links</b>
<ul>
<li> <a
href="http://www.nytimes.com/2007/03/19/obituaries/20cnd-backus.html?ex=1332043200&en=adde3ee5a1875330&ei=5124&partner=permalink&exprod=permalink">New
York Times</a>
<li> <a
href="http://www.washingtonpost.com/wp-dyn/content/article/2007/03/20/AR2007032000605.html">Washington Post</a>

<li> Garrison Keillor, <a
href="http://www.chicagotribune.com/news/columnists/chi-0703210090mar21,1,5634865.column?coll=chi-technology-hed"><em>Let
us now heap praise on the worker ants</em></a>, March 21, 2007.
<blockquote>
<em>There are thousands of statues of lousy generals and blowhard statesmen
and enormous temples erected for the worship of presidents, and not much
recognition of people such as Mr. Backus who did the work that actually
made life better.</em>
</blockquote>
<li> <a
href="http://www-03.ibm.com/ibm/history/exhibits/builders/builders_backus.html">IBM
Builders Page</a>

<li> <a
href="http://www.stanford.edu/class/cs242/readings/backus.pdf"><Em>Can
Programming Be Liberated from the von Neumann Style? A Functional Style
and Its Algebra of Programs</em></a> (Turing Award speech, 1977)

<li>
<a
href="http://scienceblogs.com/goodmath/2007/03/backuss_idea_of_functional_pro_1.php">Backus'
Functional Programming</a> 


<blockquote>
<p align=right>
<font size="-1">
<em>Much of my work has come from being lazy.</em><br>
John Backus<br>
(not a good approach for current UVa students!)

<p>
<em>You need the willingness to fail all the time.  You have to generate
many ideas and then you have to work very hard only to discover that
they don't work.  And you keep doing that over and over until you find
one that does work.</em><br>
John Backus<br>
(better approach for current UVa students!)
</font>
</p>
</blockquote>



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