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---
layout: post
title: 'Universal logic gates'
permalink: '/universal_logic_gates/'
---
<div class="lead">
<p>
It is common wisdom that NAND and NOR are universal logic gates: any logic circuit can be
created using any one of these two gates. This property is called "functional completeness".
</p>
<p>
But are these the only two (binary) gates with this property? <a href="http://en.wikipedia.org/wiki/Functional_completeness" class="external">This wikipedia page</a> states there aren't any others: <blockquote>In digital electronics terminology, the binary NAND gate and the binary NOR gate are the only binary universal logic gates.</blockquote>
</p>
</div>
<section>
<div class="page-header"><h3>How many binary logic gates are there?</h3></div>
<p>
Given two signals, A and B, there are 4 possible cominations: AB, A&#773;B, AB&#773, A&#773B&#773. This
implies there are 16 possible binary gates. Some like AND, OR, XOR, NAND, NOR have names. There are 5 gates that
don't have any names, so we'll just number them G1-G5:
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<section>
<div class="page-header"><h3>G1, G2, G4, G5 are universal</h3></div>
<p>Let's assume that a gate is universal if it can be used as to create a NOT and an AND gate. As
long as we agree that NAND is a universal gate, we don't need to get into too much formalisme</p>
<p>Using G1, you can create a NOT by using the 1 constant. Constants always exist in a circuit, how else
can a gate invert a 0 into a 1?</p>
<ul>
<li>NOT: 1 G1 A => A&#773;</li>
<li>AND: A G1 (1 G1 B) => A & B</li>
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<p>G2 is the mirror of G1.</p>
<ul>
<li>NOT: A G2 1 => A&#773;</li>
<li>AND: (A G2 1) G2 B => A & B</li>
</ul>
<p>In a similar way, you can use the 0 constant with G4.</p>
<ul>
<li>NOT: 0 G4 A => A&#773;</li>
<li>AND: 0 G4 ((0 G4 A) G4 B) => A & B</li>
</ul>
<p>And G5 is the mirror of G4.</p>
<ul>
<li>NOT: A G5 0 => A&#773;</li>
<li>AND: (A (B G5 0)) G5 0 => A & B</li>
</ul>
<p>Notice how G1, G2, G5 and G4 form a pattern on the 16 possible gates?</p>
</section>