2018-01-15 23:43:28 -05:00
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<title>Category Theory for DB-ies</title>
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<meta name="description" content="Mimir">
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<meta name="author" content="Oliver Kennedy">
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<div class="header">
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<!-- Any Talk-Specific Header Content Goes Here -->
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Category Theory for DB-ies
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<div class="footer">
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<!-- Any Talk-Specific Footer Content Goes Here -->
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<div style="float: left; margin-top: 15px; ">
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Exploring <u><b>O</b></u>nline <u><b>D</b></u>ata <u><b>In</b></u>teractions
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</div>
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<img src="graphics/FullText-white.png" height="40" style="float: right;"/>
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</div>
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<div class="slides">
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<section>
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<section>
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<h2>Category Theory for DB-ies</h2>
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2018-01-23 18:23:31 -05:00
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<h4>(Or how I learned that mathematicians are bad at naming)<h4>
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2018-01-15 23:43:28 -05:00
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</section>
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<section>
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<img src="graphics/bake_pi.jpg" height="400px">
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</section>
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<section>
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<p>People always say that SQL is "Declarative"</p>
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</section>
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<section>
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<h3>Why is the following replacement ok?</h3>
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<pre><code class="sql">
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SELECT * FROM R WHERE A = 1 AND B = 2
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</code></pre>
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<pre><code class="sql">
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SELECT * FROM (SELECT * FROM R WHERE A = 1) WHERE B = 2
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</code></pre>
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<p class="fragment">(and how is it the same as computing $x + 1 + 2$?)</p>
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</section>
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</section>
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<section>
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<section>
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<h2>Addition is Cool</h2>
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<p class="fragment" data-fragment-index="1">
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<b class="fragment" data-fragment-index="4">Commutative</b>
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$$a + b = b + a$$
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</p>
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<p class="fragment" data-fragment-index="2">
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<b class="fragment" data-fragment-index="4">Associative</b>
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$$(a + b) + c = a + (b + c)$$
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</p>
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<p class="fragment" data-fragment-index="3">
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<b class="fragment" data-fragment-index="4">Neutral Element</b>
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$$0 + a = a$$
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</p>
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</section>
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<section>
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<h2>Addition is Cool</h2>
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<pre><code class="java">
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public int add(int a, int b) { ... }
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add(a, b) == add(b, a)
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add(add(a, b), c) == add(a, add(b, c))
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add(a, 0) == a
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</code></pre>
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</section>
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<section>
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<h2>Math is about Patterns</h2>
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<table style="font-size: smaller;">
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<tr><th>Commutativity</th><th>Associativity</th><th>Neutral Element</th></tr>
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<tr>
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<td>$a + b = b + a$</td>
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<td>$(a + b) + c = a + (b + c)$</td>
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<td>0</td>
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</tr>
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<tr class="fragment">
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<td>$a \texttt{ AND } b = b \texttt{ AND } a$</td>
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<td>$(a \texttt{ AND } b) \texttt{ AND } c = a \texttt{ AND } (b \texttt{ AND } c)$</td>
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<td>TRUE</td>
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</tr>
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<tr class="fragment">
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<td>$a \texttt{ OR } b = b \texttt{ OR } a$</td>
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<td>$(a \texttt{ OR } b) \texttt{ OR } c = a \texttt{ OR } (b \texttt{ OR } c)$</td>
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<td>FALSE</td>
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</tr>
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<tr class="fragment">
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<td>$A \cup B = B \cup A$</td>
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<td>$(A \cup B) \cup C = A \cup (B \cup C)$</td>
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<td>$\emptyset$</td>
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</tr>
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<tr class="fragment" style="font-size: smaller">
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<td>$min(A, B) = min(B, A)$</td>
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<td>$min(min(A, B), C) = min(A, min(B, C))$</td>
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<td>$\infty$</td>
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</tr>
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</table>
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<p class="fragment">Mathematicians call this a <b>Group</b></p>
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</section>
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<section>
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<pre><code class="java">
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interface Group<K> {
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public K getZero() { ... }
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public K add(K a, K b) { ... }
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}
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</code></pre>
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<br/>
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<p class="fragment">
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$$\left<\; K,\; +,\; 0\;\right>$$
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</p>
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</section>
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<section>
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$$\left<\; \mathbb R,\; +,\; 0\;\right>$$
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$$\left<\; \mathbb N,\; +,\; 0\;\right>$$
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$$\left<\; \mathbb B,\; \texttt{AND},\; \texttt{TRUE}\;\right>$$
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$$\left<\; \mathbb B,\; \texttt{OR},\; \texttt{FALSE}\;\right>$$
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2018-01-23 18:23:31 -05:00
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$$\left<\; Set,\; \cup,\; \emptyset\;\right>$$
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$$\left<\; Multiset,\; \uplus,\; \emptyset\;\right>$$
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2018-01-15 23:43:28 -05:00
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$$\left<\; \mathbb R,\; min,\; \infty;\right>$$
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</section>
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</section>
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<section>
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<section>
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<h2>Fun With Groups</h2>
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2018-01-23 18:23:31 -05:00
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<p style="margin-top: 100px; margin-bottom: 200px;">
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$$\left<\; Set,\; \cup,\; \emptyset\;\right>$$
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$$\left<\; \mathbb N,\; max,\; -\infty\;\right>$$
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</p>
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</section>
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<section>
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<pre><code class="java">
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class SetGroup implements Group<Set<int>> {
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public Set<int> getZero() { return Collections.emptySet(); }
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public Set<int> add(Set<int> a, Set<int> b) {
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Set<int> tmp = new HashSet<int>(a);
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tmp.addAll(b);
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return tmp;
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}
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}
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</code></pre>
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</section>
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<section>
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<pre><code class="java">
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class IntMaxGroup implements Group<int> {
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public int getZero() { return Integer.MIN_VALUE; }
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public int add(int a, int b) { return Integer.max(a, b) }
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}
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</code></pre>
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</section>
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<section>
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<pre><code class="java">
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public int SetMax(Set<int> set) {
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int ret = Integer.MIN_VALUE
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for(int i : set) {
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ret = Integer.max(i, ret);
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}
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return ret;
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}
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</code></pre>
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<p style="margin-top: 100px;" class="fragment">
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$$\texttt{SetMax} : SetGroup \rightarrow IntGroup$$
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</p>
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</section>
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<section>
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<p>There's something really cool about <tt>SetMax</tt></p>
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</section>
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<section>
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$$\texttt{SetMax}(A + B) \equiv \texttt{SetMax}(A) + \texttt{SetMax}(B)$$
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id="tspan8120"
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y="448.15857"
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x="44.717777"
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sodipodi:role="line"> max(B))</tspan></text>
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</svg>
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|
|
<p class="fragment">Mathematicians call this a <b>homomorphism</b></p>
|
2018-01-15 23:43:28 -05:00
|
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|
|
</section>
|
|
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|
|
|
2018-01-23 18:23:31 -05:00
|
|
|
<section>
|
|
|
|
|
<pre><code class="sql">
|
|
|
|
|
SELECT MAX(x) FROM (SELECT * FROM R UNION S)
|
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|
|
</code></pre>
|
|
|
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|
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|
|
<pre><code class="sql">
|
|
|
|
|
MAX( (SELECT MAX(x) FROM R), (SELECT MAX(x) FROM S) )
|
|
|
|
|
</code></pre>
|
|
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|
|
</section>
|
|
|
|
|
|
2018-01-23 18:25:33 -05:00
|
|
|
<section>
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</section>
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|
|
2018-01-15 23:43:28 -05:00
|
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</section>
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</div></div>
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<script src="../reveal.js-3.5.0/lib/js/head.min.js"></script>
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<script src="../reveal.js-3.5.0/js/reveal.js"></script>
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<script>
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// Full list of configuration options available at:
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// https://github.com/hakimel/../reveal.js#configuration
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Reveal.initialize({
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controls: false,
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progress: true,
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history: true,
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center: true,
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slideNumber: true,
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transition: 'fade', // none/fade/slide/convex/concave/zoom
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// Optional ../reveal.js plugins
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dependencies: [
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{ src: '../reveal.js-3.5.0/lib/js/classList.js', condition: function() { return !document.body.classList; } },
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{ src: '../reveal.js-3.5.0/plugin/math/math.js',
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condition: function() { return true; },
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mathjax: '../reveal.js-3.5.0/js/MathJax.js'
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},
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{ src: '../reveal.js-3.5.0/plugin/markdown/marked.js', condition: function() { return !!document.querySelector( '[data-markdown]' ); } },
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{ src: '../reveal.js-3.5.0/plugin/markdown/markdown.js', condition: function() { return !!document.querySelector( '[data-markdown]' ); } },
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{ src: '../reveal.js-3.5.0/plugin/highlight/highlight.js', async: true, condition: function() { return !!document.querySelector( 'pre code' ); }, callback: function() { hljs.initHighlightingOnLoad(); } },
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{ src: '../reveal.js-3.5.0/plugin/zoom-js/zoom.js', async: true },
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{ src: '../reveal.js-3.5.0/plugin/notes/notes.js', async: true }
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]
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});
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</script>
|
|
|
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</body>
|
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|
|
</html>
|
