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<!DOCTYPE HTML PUBLIC "-//W3C//DTD HTML 4.01//EN" "http://www.w3.org/TR/html4/strict.dtd">	<html>		<head>			<title>page_478</title>			<link rel="stylesheet" href="reset.css" type="text/css" media="all">			<meta http-equiv="Content-Type" content="text/html; charset=UTF-8" />		</head>		<body>		<table summary="top nav" border="0" width="100%">			<tr>				<td align="left" width="30%" style="background: #EEF3E2"><a style="color: blue; font-size: 120%; font-weight: bold; text-decoration: none; font-family: verdana;" href="page_477.html">&lt;&nbsp;previous page</a></td>				<td align="center" width="40%" style="background: #EEF3E2"><strong style="color: #2F4F4F; font-size: 120%;">page_478</strong></td>				<td align="right" width="30%" style="background: #EEF3E2"><a style="color: blue; font-size: 120%; font-weight: bold; text-decoration: none; font-family: verdana;" href="page_479.html">next page&nbsp;&gt;</a></td>			</tr>					<tr>				<td align="left" colspan="3" style="background: #ffffff; padding: 20px;">    <table border="0" width="100%" cellpadding="0"><tr><td align="center">  <table border="0" cellpadding="2" cellspacing="0" width="100%"><tr><td align="left"></td>  <td align="right"></td>  </tr></table></td></tr><tr><td align="left"><p></p><table border="0" cellspacing="0" cellpadding="0" width="100%"><tr><td align="right"><font face="Times New Roman, Times, Serif" size="2" color="#FF0000">Page 478</font></td></tr></table><table border="0" cellspacing="0" cellpadding="0"><tr><td rowspan="5"></td>  <td colspan="3" height="12"></td>  <td rowspan="5"></td></tr><tr><td colspan="3"></td></tr><tr><td></td>  <td><font face="Times New Roman, Times, Serif" size="3">is trivially true because there are no integers in this range. The second part of the invariant is also truenamely, that </font><font face="Courier New, Courier, Mono New, Courier, Mono" size="3">i</font><font face="Times New Roman, Times, Serif" size="3"> is in the range 1 through 51, inclusive. The first loop iteration prints the value 1 and increments </font><font face="Courier New, Courier, Mono New, Courier, Mono" size="3">i</font><font face="Times New Roman, Times, Serif" size="3"> to 2. Prior to the loop test, the invariant is true; the value 1 has been output, and </font><font face="Courier New, Courier, Mono New, Courier, Mono" size="3">i</font><font face="Times New Roman, Times, Serif" size="3"> is in the stated range. On the final iteration, the value 50 is printed and </font><font face="Courier New, Courier, Mono New, Courier, Mono" size="3">i</font><font face="Times New Roman, Times, Serif" size="3"> becomes 51. Prior to the loop test, you can see that the invariant is true. After the loop test, control exits the loop and the invariant is still true.</font></td><td></td></tr><tr><td colspan="3"></td></tr><tr><td colspan="3" height="1"></td></tr></table><table border="0" cellspacing="0" cellpadding="0"><tr><td rowspan="5"></td>  <td colspan="3" height="12"></td>  <td rowspan="5"></td></tr><tr><td colspan="3"></td></tr><tr><td></td>  <td><font face="Times New Roman, Times, Serif" size="3">When loops don't use counters, loop invariants are harder to write. For example, what is the invariant for the following loop?</font></td><td></td></tr><tr><td colspan="3"></td></tr><tr><td colspan="3" height="1"></td></tr></table><table border="0" cellspacing="0" cellpadding="0"><tr><td rowspan="5"></td>  <td colspan="3" height="12"></td>  <td rowspan="5"></td></tr><tr><td colspan="3"></td></tr><tr><td></td>  <td><font face="Courier New, Courier, Mono New, Courier, Mono" size="2">do<br />{<br />聽聽聽聽cout聽&lt;&lt;聽Enter聽your聽age:聽;<br />聽聽聽聽cin聽&gt;&gt;聽age;<br />}聽while聽(age聽&lt;=聽0)聽;</font></td><td></td></tr><tr><td colspan="3"></td></tr><tr><td colspan="3" height="1"></td></tr></table><table border="0" cellspacing="0" cellpadding="0"><tr><td rowspan="5"></td>  <td colspan="3" height="12"></td>  <td rowspan="5"></td></tr><tr><td colspan="3"></td></tr><tr><td></td>  <td><font face="Times New Roman, Times, Serif" size="3">This loop continues to iterate as long as </font><font face="Courier New, Courier, Mono New, Courier, Mono" size="3">age聽<u>&lt;</u>聽0</font><font face="Times New Roman, Times, Serif" size="3">. We might be tempted to write the invariant this way:</font></td><td></td></tr><tr><td colspan="3"></td></tr><tr><td colspan="3" height="1"></td></tr></table><table border="0" cellspacing="0" cellpadding="0"><tr><td rowspan="5"></td>  <td colspan="3" height="12"></td>  <td rowspan="5"></td></tr><tr><td colspan="3"></td></tr><tr><td></td>  <td><font face="Courier New, Courier, Mono New, Courier, Mono" size="2">do<br />{<br /><br />聽聽聽聽cout聽&lt;&lt;聽Enter聽your聽age:聽;<br />聽聽聽聽cin聽&gt;&gt;聽age;<br /><br />聽聽聽聽聽聽聽聽//聽Invariant聽(WRONG)聽:<br />聽聽聽聽聽聽聽聽//聽聽聽聽聽age聽has聽been聽input<br />聽聽聽聽聽聽聽聽//聽聽&amp;&amp;聽age聽&lt;=聽0<br /><br />}聽while聽(age聽&lt;=聽0);</font></td><td></td></tr><tr><td colspan="3"></td></tr><tr><td colspan="3" height="1"></td></tr></table><table border="0" cellspacing="0" cellpadding="0"><tr><td rowspan="5"></td>  <td colspan="3" height="12"></td>  <td rowspan="5"></td></tr><tr><td colspan="3"></td></tr><tr><td></td>  <td><font face="Times New Roman, Times, Serif" size="3">This invariant is true for every iteration <i>except</i> the final iteration, which occurs when the user finally types a positive number. Suppose that the user types 5 as the input value. Immediately after the input statement and before the loop test, the invariant is false. This is not the correct invariant.</font></td><td></td></tr><tr><td colspan="3"></td></tr><tr><td colspan="3" height="1"></td></tr></table><table border="0" cellspacing="0" cellpadding="0"><tr><td rowspan="5"></td>  <td colspan="3" height="12"></td>  <td rowspan="5"></td></tr><tr><td colspan="3"></td></tr><tr><td></td>  <td><font face="Times New Roman, Times, Serif" size="3">With loops that input data and immediately test the data in the loop condition, there are two keys to expressing a loop invariant correctly. The first is to think about the <i>history</i> of the loopwhat has happened so far since the loop began. The second is to exclude the current input value from any claim about its value. In the above loop, we can't claim that </font><font face="Courier New, Courier, Mono New, Courier, Mono" size="3">age</font><font face="Times New Roman, Times, Serif" size="3"> is negative or 0 or positive immediately after reading it inthat's for the loop test to determine. Here is a reasonable way to write the invariant for the above loop:</font></td><td></td></tr><tr><td colspan="3"></td></tr><tr><td colspan="3" height="1"></td></tr></table><table border="0" cellspacing="0" cellpadding="0"><tr><td rowspan="5"></td>  <td colspan="3" height="12"></td>  <td rowspan="5"></td></tr><tr><td colspan="3"></td></tr><tr><td></td>  <td><font face="Courier New, Courier, Mono New, Courier, Mono" size="2">do<br />{<br />聽聽聽聽cout聽&lt;&lt;聽Enter聽your聽age:聽;<br /></font></td><td></td></tr><tr><td colspan="3"></td></tr><tr><td colspan="3" height="1"></td></tr></table></td></tr></table><p><font size="0"></font></p>聽  </td>			</tr>				<tr>				<td align="left" width="30%" style="background: #EEF3E2"><a style="color: blue; font-size: 120%; font-weight: bold; text-decoration: none; font-family: verdana;" href="page_477.html">&lt;&nbsp;previous page</a></td>				<td align="center" width="40%" style="background: #EEF3E2"><strong style="color: #2F4F4F; font-size: 120%;">page_478</strong></td>				<td align="right" width="30%" style="background: #EEF3E2"><a style="color: blue; font-size: 120%; font-weight: bold; text-decoration: none; font-family: verdana;" href="page_479.html">next page&nbsp;&gt;</a></td>			</tr>		</table>		</body>	</html>
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