98 lines
6.2 KiB
HTML
98 lines
6.2 KiB
HTML
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<h2>Circular Logic</h2><div class="info" style="cursor:help;width:200px;margin-bottom:10px;"><h3>Problem 209</h3><span style="width:300px;color:#666;">Published on Friday, 19th September 2008, 06:00 pm; Solved by 1429</span></div>
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<div class="problem_content" role="problem">
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<p>A <var>k</var>-input <i>binary truth table</i> is a map from <var>k</var> input bits
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(binary digits, 0 [false] or 1 [true]) to 1 output bit. For example, the 2-input binary truth tables for the logical AND and XOR functions are:</p>
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<table cellspacing="0" cellpadding="2" border="1" align="left">
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<tr style="background-color:#c1daf9;">
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<td width="30" align="center"><var>x</var></td>
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<td width="30" align="center"><var>y</var></td>
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<td><var>x</var> AND <var>y</var></td></tr>
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<tr><td align="center">0</td><td align="center">0</td><td align="center">0</td></tr>
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<tr><td align="center">0</td><td align="center">1</td><td align="center">0</td></tr>
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<tr><td align="center">1</td><td align="center">0</td><td align="center">0</td></tr>
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<tr><td align="center">1</td><td align="center">1</td><td align="center">1</td></tr>
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</table>
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<tr style="background-color:#c1daf9;">
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<td width="30" align="center"><var>x</var></td>
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<td width="30" align="center"><var>y</var></td>
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<td><var>x</var> XOR <var>y</var></td></tr>
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<tr><td align="center">0</td><td align="center">0</td><td align="center">0</td></tr>
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<tr><td align="center">0</td><td align="center">1</td><td align="center">1</td></tr>
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<tr><td align="center">1</td><td align="center">0</td><td align="center">1</td></tr>
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<tr><td align="center">1</td><td align="center">1</td><td align="center">0</td></tr>
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</table>
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</div><br clear="all" />
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<p>How many 6-input binary truth tables, τ, satisfy the formula</p>
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<div style="text-align:center;">
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τ(<var>a</var>, <var>b</var>, <var>c</var>, <var>d</var>, <var>e</var>, <var>f</var>) AND τ(<var>b</var>, <var>c</var>, <var>d</var>, <var>e</var>, <var>f</var>, <var>a</var> XOR (<var>b</var> AND <var>c</var>)) = 0
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</div><br />
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<p>for all 6-bit inputs (<var>a</var>, <var>b</var>, <var>c</var>, <var>d</var>, <var>e</var>, <var>f</var>)?
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