540 lines
18 KiB
HTML
540 lines
18 KiB
HTML
<!doctype html>
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<html lang="en">
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<head>
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<meta charset="utf-8">
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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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<meta name="apple-mobile-web-app-capable" content="yes" />
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<meta name="viewport" content="width=device-width, initial-scale=1.0, maximum-scale=1.0, user-scalable=no, minimal-ui">
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<link rel="stylesheet" href="ubodin.css" id="theme">
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<script src="../reveal.js-3.5.0/lib/js/html5shiv.js"></script>
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<![endif]-->
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</head>
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<body>
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<div class="reveal">
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<!-- Any section element inside of this container is displayed as a slide -->
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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>
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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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<h4>(The Math of "Things" and "Stuff")<h4>
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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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$$\left<\; Set,\; \cup,\; \emptyset\;\right>$$
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$$\left<\; Multiset,\; \uplus,\; \emptyset\;\right>$$
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$$\left<\; \mathbb Z^{-\infty},\; max,\; -\infty;\right>$$
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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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<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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// Empty Set union with anything is a no-op
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public Set<int> getZero() { return Collections.emptySet(); }
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// Compute Set Union
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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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// Max(x, -infinity) = x
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public int getZero() { return Integer.MIN_VALUE; }
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// Compute Max of two elements
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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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//Compute the maxium value in a Set
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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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<svg data-src="graphics/homomorphism.svg" class="fragment" style="margin-top:50px"/>
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</section>
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<section>
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<p>Mathematicians call this a <b>homomorphism</b></p>
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<dl class="fragment">
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<dt>Homo</dt>
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<dd>"Same"</dd>
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<dt>Morphism</dt>
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<dd>"Function"</dd>
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</dl>
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</section>
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<section>
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<h1>So what?</h1>
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</section>
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<section>
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<pre><code class="sql">
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SELECT MAX(x) FROM (SELECT * FROM R UNION S)
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</code></pre>
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<pre><code class="sql">
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MAX( (SELECT MAX(x) FROM R), (SELECT MAX(x) FROM S) )
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</code></pre>
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</section>
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<section>
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<h2>Without Homomorphisms</h2>
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<svg data-src="graphics/distributed-1.svg"/>
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<imagecredits>(OpenClipArt.org)</imagecredits>
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</section>
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<section>
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<h2>With Homomorphisms</h2>
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<svg data-src="graphics/distributed-2.svg"/>
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<imagecredits>(OpenClipArt.org)</imagecredits>
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</section>
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</section>
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<section>
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<section>
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<p>Let's dig a little deeper</p>
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</section>
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<section>
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<h2>Multiplication is also Cool!</h2>
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<p class="fragment" data-fragment-index="1">
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<b>Commutative</b>
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$$a \times b = b \times a$$
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</p>
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<p class="fragment" data-fragment-index="2">
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<b>Associative</b>
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$$(a \times b) \times c = a \times (b \times c)$$
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</p>
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<p class="fragment" data-fragment-index="3">
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<b>Neutral Element</b>
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$$1 \times a = a$$
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</p>
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</section>
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<section>
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<h2>Combine for epic coolness!</h2>
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<p class="fragment" data-fragment-index="1">
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<b class="fragment" data-fragment-index="3">Distributive</b>
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$$a \times (b + c) = a \times b + a \times c$$
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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="3">Sphere of Annihilation</b>
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$$0 \times a = 0$$
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</p>
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</section>
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<section>
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<h2>More Patterns!</h2>
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<table>
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<!-- -->
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<tr><th>$K$</th> <th>$+$</th> <th>$\times$</th> <th>$0$</th> <th>$1$</th>
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<td></td></tr>
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<!-- -->
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<tr><td>$\mathbb N$</td> <td>$+$</td> <td>$\times$</td> <td>$0$</td> <td>$1$</td>
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<td style="text-align: left">Natural Number Arithmetic</td></tr>
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<!-- -->
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<tr><td>$\mathbb B$</td> <td>$\vee$</td><td>$\wedge$</td> <td>F</td> <td>T</td>
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<td style="text-align: left">Boolean Algebra</td></tr>
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<!-- -->
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<tr class="fragment highlight-current-blue" data-fragment-index="2">
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<td>Tables</td> <td>$\cup$</td><td>$\bowtie$</td><td>$\emptyset$</td><td>$\left<\right>$</td>
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<td style="text-align: left">SQL</td></tr>
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</table>
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<p class="fragment" data-fragment-index="1">Mathematicians call this a <b>Semiring</b></p>
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</section>
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<section>
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<table><tr>
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<td>
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<table>
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<tr><th>$Students$</th><th>|</th><th>S_ID</th><th>Name</th></tr>
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<tr><td> </td><td>|</td><td>111 </td><td>Alice</td></tr>
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<tr><td> </td><td>|</td><td>222 </td><td>Bob</td></tr>
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<tr><td> </td><td>|</td><td>333 </td><td>Carol</td></tr>
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</table>
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</td>
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<td>
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<table>
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<tr><th>$Courses$</th><th>|</th><th>S_ID</th><th>Course</th></tr>
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<tr><td> </td><td>|</td><td>111 </td><td>CSE-562</td></tr>
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<tr><td> </td><td>|</td><td>111 </td><td>CSE-521</td></tr>
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<tr><td> </td><td>|</td><td>222 </td><td>CSE-562</td></tr>
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</table>
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</td>
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</tr></table>
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</section>
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<section>
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<p>How many courses is each student taking?</p>
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<table class="fragment">
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<tr><th>$\pi_{S\_ID}(Courses)$</th><th>|</th><th>S_ID</th></tr>
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<tr><td> </td><td>|</td><td>111 </td></tr>
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<tr><td> </td><td>|</td><td>111 </td></tr>
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<tr><td> </td><td>|</td><td>222 </td></tr>
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</table>
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<pre class="fragment"><code class="sql">SELECT S_ID FROM COURSES</code></pre>
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</section>
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<section>
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<p>How many courses is each student taking?</p>
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<table>
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<tr><th>$\pi_{S\_ID}(Courses)$</th><th>|</th><th>S_ID</th><th> </th><th>#</th></tr>
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<tr><td> </td><td>|</td><td>111 </td><td>$\rightarrow$</td><td>2</td></tr>
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<tr><td> </td><td>|</td><td>222 </td><td>$\rightarrow$</td><td>1</td></tr>
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<tr class="fragment" style="color: grey"
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><td> </td><td>|</td><td>* </td><td>$\rightarrow$</td><td>0</td></tr>
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</table>
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<pre><code class="sql">SELECT S_ID, COUNT(*) FROM COURSES GROUP BY S_ID</code></pre>
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</section>
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<section>
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<p>How many courses is each student taking?</p>
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<table>
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<tr><th>$\pi_{Name}(Courses \bowtie Students)$</th><th>|</th><th>S_ID</th><th> </th><th>#</th></tr>
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<tr><td> </td><td>|</td><td>111 </td><td>$\rightarrow$</td><td class="fragment highlight-current-blue" data-fragment-index=1>2</td></tr>
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<tr><td> </td><td>|</td><td>222 </td><td>$\rightarrow$</td><td class="fragment highlight-current-blue" data-fragment-index=1>1</td></tr>
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<tr style="color: grey"
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><td> </td><td>|</td><td>* </td><td>$\rightarrow$</td><td class="fragment highlight-current-blue" data-fragment-index=1>0</td></tr>
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</table>
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<pre><code class="sql">SELECT Name, COUNT(*) FROM COURSES GROUP BY S_ID</code></pre>
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</section>
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<section>
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<table>
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<tr>
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<td>$$\left<\mathbb B, \vee, \wedge, F, T\right>$$</td>
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<td>Set Databases (SELECT DISTINCT)</td>
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</tr>
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<tr>
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<td>$$\left<\mathbb N, +, \times, 0, 1\right>$$</td>
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<td>Multiset Databases (Normal SQL)</td>
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</tr>
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</table>
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<p class="fragment"><b>Other Applications</b>: Provenance, Permissions, Differential Privacy</p>
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</section>
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<section>
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<table>
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<tr>
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<th>Database Operation</th>
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<th>Semiring Operation</th>
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</tr>
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<tr>
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<td>Union</td>
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<td>$+$</td>
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</tr>
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<tr>
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<td>Join</td>
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<td>$\times$</td>
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</tr>
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<tr>
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<td>Aggregation</td>
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<td>$+$</td>
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</tr>
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</table>
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</section>
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</section>
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<section>
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<section>
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<pre><code class="sql">
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SELECT * FROM R WHERE A = 1
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</code></pre>
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<p class="fragment" style="margin-top: 100px">
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$$\sigma_{A = 1}(t \rightarrow \#) = \begin{cases} \# & \text{if }t.A = 1 \\ 0 & \text{otherwise}\end{cases}$$
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</p>
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</section>
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<section>
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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 style="margin-top: 100px">
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$$\sigma_{B = 1}(\sigma_{A = 1}(t \rightarrow \#)) = \begin{cases} \sigma_{A = 1}(t \rightarrow \#) & \text{if }t.B = 2 \\ 0 & \text{otherwise}\end{cases}$$
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</p>
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</section>
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<section>
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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 style="margin-top: 100px">
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$$\sigma_{B = 1}(\sigma_{A = 1}(t \rightarrow \#)) = \begin{cases} \# & \text{if }t.A = 1\text{ and }t.B = 2 \\ 0 & \text{if }t.A = 1\text{ and }t.B \neq 2 \\ 0 & \text{otherwise}\end{cases}$$
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</p>
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</section>
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<section>
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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 style="margin-top: 100px">
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$$\sigma_{B = 1}(\sigma_{A = 1}(t \rightarrow \#)) = \begin{cases} \# & \text{if }t.A = 1\text{ and }t.B = 2 \\ 0 & \text{otherwise}\end{cases}$$
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</p>
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</section>
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<section>
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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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|
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<p style="margin-top: 100px">
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$$\sigma_{B = 1}(\sigma_{A = 1}(t \rightarrow \#)) = \begin{cases} \# & \text{if }t.A = 1\text{ and }t.B = 2 \\ 0 & \text{otherwise}\end{cases}$$
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</p>
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|
</section>
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|
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</section>
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|
|
|
<section>
|
|
<h2>Other Resources</h2>
|
|
<dl>
|
|
<dt>How to Bake Pi</dt>
|
|
<dd><img src="graphics/bake_pi.jpg" height="100px">
|
|
<dt>Category Theory for Programmers</dt>
|
|
<dd><a href="https://bartoszmilewski.com/2014/10/28/category-theory-for-programmers-the-preface/">https://bartoszmilewski.com/2014/10/28/category-theory-for-programmers-the-preface/</a>
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</dl>
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|
</section>
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|
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