Probdb slides
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df72083f76
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@ -8,7 +8,7 @@ end
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def data_table(schema, data, params = {})
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tablename_style = params.fetch(:tablename_style, { "font-weight" => "bold", "text-decoration" => "underline" })
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rownum_style = params.fetch(:rownum_style, { "font-size" => "50%", "font-face" => "Courier, fixed-width", "vertical-align"=> "middle" })
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annotation_style = params.fetch(:annotation_style, rownum_style)
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annotation_style = params.fetch(:annotation_style, rownum_style.merge("text-align" => :left, "font-size" => "70%"))
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separator_style = params.fetch(:separator_style, { "border-right" => "1px solid black" })
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num_cols = data.map { |row| row.size if row.is_a? Array }.compact.max
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@ -25,10 +25,10 @@ def data_table(schema, data, params = {})
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raise "Unknown rowtype: #{row}"
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end
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end
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if params.has_key? :annotations
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if params.has_key? :annotations and not params[:annotations].nil?
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data = data.zip(params[:annotations]).map do |row, annot|
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if annot.nil? then row
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else row+[tag("td", "→ "+annot.to_s, annotation_style)] end
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else row+[tag("td", "→ "+annot.to_s, style: annotation_style)] end
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end
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num_cols += 1
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end
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@ -298,6 +298,93 @@ dependencies:
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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>Does this work? <span class="fragment">... not quite</span></p>
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<p class="fragment">Is it a true provenance polynomial?</p>
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</section>
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<% [0, 1, 2].each do |plug_in_values| %>
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<section>
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<p>Plug in "F" for $R_1$ (i.e., simulate removing $R_1$ from the input)</p>
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<%= data_table(["A","B", "C"],
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[
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[1, 2, 1],
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[1, 2, 2],
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[1, 3, 3],
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[2, 3, 3],
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[2, 1, 4],
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[4, 5, 5]
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],
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annotations:
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case plug_in_values
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when 0 then [
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"$R_1 \\wedge S_1$",
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"$R_1 \\wedge S_2$",
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"$R_2 \\wedge S_3$",
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"$R_3 \\wedge S_3$",
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"$T_1$",
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"$T_2$"
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]
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when 1 then [
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"$F \\wedge S_1$",
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"$F \\wedge S_2$",
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"$T \\wedge S_3$",
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"$T \\wedge S_3$",
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"$T$",
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"$T$"
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]
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when 2 then [
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"$F$",
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"$F$",
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"$T$",
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"$T$",
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"$T$",
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"$T$"
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]
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end,
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name: "$Q_1'$") %>
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</section>
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<% end %>
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<section>
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<%= data_table(["A","B","C","$\\phi_1$","$\\phi_2$"],
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[
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[1, 3, 3, "$R_2$", "$S_3$"],
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[2, 3, 3, "$R_3$", "$S_3$"],
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[2, 1, 4, "$T_1$", "<code>NULL</code>"],
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[4, 5, 5, "$T_2$", "<code>NULL</code>"]
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],
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name: "$Q_1'$") %>
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</section>
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<section>
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<%= data_table(["A","COUNT"],
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[
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[1, "<span class='fragment highlight-red' data-fragment-index='2'>1</span>", "$R_2$", "$S_3$"],
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[2, 2, "$R_3$", "$S_3$"],
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[2, 2, "$T_1$", "<code>NULL</code>"],
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[4, 1, "$T_2$", "<code>NULL</code>"]
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],
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name: "$_A\\gamma_{COUNT}(Q_1')$") %>
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<div class="fragment" data-fragment-index='1'>
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<p>vs</p>
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<%= data_table(["A","COUNT"],
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[
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[1, "<span class='fragment highlight-red' data-fragment-index='2'>3</span>", "$R_2$", "$S_3$"],
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[2, 2, "$R_3$", "$S_3$"],
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[2, 2, "$T_1$", "<code>NULL</code>"],
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[4, 1, "$T_2$", "<code>NULL</code>"]
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],
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name: "$_A\\gamma_{COUNT}(Q_1)$") %>
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</div>
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</section>
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<section>
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<p>Not quite a provenance polynomial...</p>
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<p>...but still correct for lineage queries</p>
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<p class="fragment">(and there are other solutions)</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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<p><b>Pro:</b> Standardized, relational representation</p>
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@ -0,0 +1,379 @@
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---
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template: templates/cse4562_2019_slides.erb
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title: Incomplete and Probabilistic Databases
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date: April 26, 2019
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textbook: "<a href='https://github.com/UBOdin/mimir/wiki/Concepts-CTables'>PDB Concepts and C-Tables</a>"
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dependencies:
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- lib/slide_utils.rb
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---
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<%
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require "slide_utils.rb"
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%>
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<section>
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<section>
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<img src="graphics/2019-04-31-4or9.png" height="300px" />
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</section>
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<section>
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<img src="graphics/2019-04-31-guacamole.png" class="stretch" />
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<attribution><a href="https://www.anishathalye.com/2017/07/25/synthesizing-adversarial-examples/">https://www.anishathalye.com/2017/07/25/synthesizing-adversarial-examples/</a></attribution>
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</section>
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<section>
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<img src="graphics/2019-04-31-catVSdog.jpg" class="stretch" />
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<attribution><a href="https://www.pyimagesearch.com/pyimagesearch-gurus/?src=post-deep-learning-libs">Deep Learning Demystified</a></attribution>
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</section>
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<section>
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<h3>What happens when you don't know your data precisely?</h3>
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</section>
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<section>
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<pre><code>
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SELECT * FROM Posts WHERE image_class = 'Cat';
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</code></pre>
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<pre class="fragment"><code>
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SELECT COUNT(*) FROM Posts WHERE image_class = 'Cat';
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</code></pre>
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<pre class="fragment"><code>
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SELECT user_id FROM Posts
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WHERE image_class = 'Cat'
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GROUP BY user_id HAVING COUNT(*) > 10;
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</code></pre>
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</section>
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</section>
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<section>
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<section>
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<h3 class="fragment">Incomplete Databases<br/>↓</h3>
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<h3>Probabilistic Databases</h3>
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</section>
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<section>
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<ol>
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<li>Representing Incompleteness</li>
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<li class="fragment">Querying Incomplete Data</li>
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<li class="fragment">Implementing It</li>
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</ol>
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</section>
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<section>
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<table><tr><td>
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<%= data_table(["Name", "ZipCode"], [["Alice", "10003"], ["Bob","1<span class='fragment highlight-current-red' data-fragment-index='1'>4</span>260"]], name: "$R_1$", rowids: true) %>
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</td><td class="fragment" data-fragment-index="3">or</td>
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<td class="fragment highlight-current-grey" data-fragment-index="2">
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<%= data_table(["Name", "Division"], [["Alice", "10003"], ["Bob","1<span class='fragment highlight-current-red' data-fragment-index='1'>9</span>260"]], name: "$R_2$", rowids: true) %>
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</td></tr></table>
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</section>
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<section>
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<p><b>Incomplete Database</b> ($\mathcal D$): A set of <i>possible worlds</i></p>
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<p class="fragment"><b>Possible World</b> ($D \in \mathcal D$): One (of many) database instances</p>
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<p class="fragment">(Require all possible worlds to have the same schema)</p>
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</section>
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<section>
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<p>What does it mean to run a query on an incomplete database?</p>
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<p class="fragment" data-fragment-index="1"><span class="fragment fade-out" data-fragment-index="2">$Q(\mathcal D) = ?$</span></p>
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<p class="fragment" data-fragment-index="2">$Q(\mathcal D) = \{\;Q(D)\;|\;D \in \mathcal D \}$</p>
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</section>
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<section>
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<table><tr><td>
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<%= data_table(["Name", "ZipCode"], [["Alice", "10003"], ["Bob","14260"]], name: "$R_1$", rowids: true) %>
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</td><td>or</td><td>
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<%= data_table(["Name", "Division"], [["Alice", "10003"], ["Bob","19260"]], name: "$R_2$", rowids: true) %>
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</td></tr></table>
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<p class="fragment" style="font-size: 90%">$$Q_1 = \pi_{Name}\big( \sigma_{state = \texttt{'NY'}} (R \bowtie_{zip} ZipLookups) \big)$$</p>
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<table class="fragment"><tr>
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<td style="font-size: 600%; margin: 0px; padding: 0px; height: 0.5em;">{</td>
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<td style="vertical-align: middle;">
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<%= data_table(["Name"], [["Alice"], ["Bob"]], name: "$Q(R_1)$", rowids: true) %>
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</td><td style="vertical-align: middle; font-weight: bold;">or</td><td style="vertical-align: middle;">
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<%= data_table(["Name"], [["Alice"]], name: "$Q(R_2)$", rowids: true) %>
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</td>
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<td style="font-size: 600%; margin: 0px; padding: 0px; height: 0.5em;">}</td>
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</tr></table>
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<aside class="notes">
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19260 is Phoenixville, PA
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</aside>
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</section>
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<section>
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<table><tr><td>
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<%= data_table(["Name", "ZipCode"], [["Alice", "10003"], ["Bob","14260"]], name: "$R_1$", rowids: true) %>
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</td><td>or</td><td>
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<%= data_table(["Name", "Division"], [["Alice", "10003"], ["Bob","19260"]], name: "$R_2$", rowids: true) %>
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</td></tr></table>
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<p class="fragment" style="font-size: 80%">$$Q_2 = \pi_{Name}\big( \sigma_{region = \texttt{'Northeast'}} (R \bowtie_{zip} ZipLookups) \big)$$</p>
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<table class="fragment">
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<td style="font-size: 600%; margin: 0px; padding: 0px; height: 0.5em;">{</td>
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<td style="vertical-align: middle;">
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<%= data_table(["Name"], [["Alice"], ["Bob"]], name: "$Q(R_1)$", rowids: true) %>
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</td><td style="vertical-align: middle; font-weight: bold;">or</td><td style="vertical-align: middle;">
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<%= data_table(["Name"], [["Alice"], ["Bob"]], name: "$Q(R_2)$", rowids: true) %>
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</td>
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<td style="font-size: 600%; margin: 0px; padding: 0px; height: 0.5em;">}</td>
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</tr></table>
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</section>
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<section>
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<table><tr><td>
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<%= data_table(["Name", "ZipCode"], [["Alice", "10003"], ["Bob","14260"]], name: "$R_1$", rowids: true) %>
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</td><td>or</td><td>
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<%= data_table(["Name", "Division"], [["Alice", "10003"], ["Bob","19260"]], name: "$R_2$", rowids: true) %>
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</td></tr></table>
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<p style="font-size: 80%">$$Q_2 = \pi_{Name}\big( \sigma_{region = \texttt{'Northeast'}} (R \bowtie_{zip} ZipLookups) \big)$$</p>
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<table><tr>
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<td style="font-size: 600%; margin: 0px; padding: 0px; height: 0.5em;">{</td>
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<td style="vertical-align: middle;">
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<%= data_table(["Name"], [["Alice"], ["Bob"]], name: "$Q(R_1)$ or $Q(R_2)$", rowids: true) %>
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</td>
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<td style="font-size: 600%; margin: 0px; padding: 0px; height: 0.5em;">}</td>
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</tr></table>
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</section>
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<section>
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<img src="graphics/2019-04-31-NormalDB.svg" /><br/>
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<hr class="fragment" data-fragment-index="1"/>
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<svg data-src="graphics/2019-04-31-IncompleteDB.svg" class="fragment" data-fragment-index="1"/>
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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><b>Challenge:</b> There can be <u>lots</u> of possible worlds.</p>
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</section>
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<section>
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<p><b>Observation: </b> Possibilities for database creation break down into lots of independent choices.</p>
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<p class="fragment"><u>Factorize</u> the database.</p>
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</section>
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<section>
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<table><tr><td>
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<%= data_table(["Name", "ZipCode"], [["Alice", "10003"], ["Bob","14260"], ["Carol", "13201"]], name: "$R_1$", rowids: true) %>
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</td><td>
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<%= data_table(["Name", "Division"], [["Alice", "10003"], ["Bob","19260"], ["Carol", "18201"]], name: "$R_2$", rowids: true) %>
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</td></tr>
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<tr><td>
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<%= data_table(["Name", "ZipCode"], [["Alice", "10003"], ["Bob","14260"], ["Carol", "13201"]], name: "$R_3$", rowids: true) %>
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</td><td>
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<%= data_table(["Name", "Division"], [["Alice", "10003"], ["Bob","19260"], ["Carol", "18201"]], name: "$R_4$", rowids: true) %>
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</td></tr></table>
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<p class="fragment">Alice appears in both databases. <br/>The only differences are Bob and Carol's zip codes.</p>
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</section>
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<section>
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<h3>List Out Choices</h3>
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<ul>
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<li>$\texttt{bob}$<span class="fragment" data-fragment-index="1">$ \in \{ 4, 9 \}$</span> (Bob's zip code digit)</li>
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<li>$\texttt{carol}$<span class="fragment" data-fragment-index="1">$ \in \{ 3, 8 \}$</span> (Carol's zip code digit)</li>
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</ul>
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</section>
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<% [false, true].each do |with_annotations| %>
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<section>
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<%= data_table(
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["Name", "ZipCode"],
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[ ["Alice", "10003"],
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["Bob","14260"],
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["Bob","14290"],
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["Carol","13201"],
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["Carol","18201"]
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],
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name: "$\\mathcal R$",
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rowids: true,
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annotations: if with_annotations then [
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"always",
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"if $\\texttt{bob} = 4$",
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"if $\\texttt{bob} = 9$",
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"if $\\texttt{carol} = 3$",
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"if $\\texttt{carol} = 8$"
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] else nil end
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) %>
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<div class="fragment">
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<div style="font-size: 200%">+</div>
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<p>$\big[\;\texttt{bob} \in \{4, 9\},\; \texttt{carol} \in \{3, 8\}\;\big]$</p>
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</div>
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</section>
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<% end %>
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<section>
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<%= data_table(
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["Name", "ZipCode"],
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[ ["Alice", "10003"],
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["Bob","14260"],
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["Bob","14290"],
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["Carol","13201"],
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["Carol","18201"]
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],
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name: "$\\mathcal R$",
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rowids: true,
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annotations: [
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"a",
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"b",
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"c",
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"d",
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"e"
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]
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) %>
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<div style="font-size: 200%">+</div>
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<p>Pick one of each: $\big[\;\{a\},\; \{b, c\},\; \{d, e\}\;\big]$</p>
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<p>Set those variables to $T$ and all others to $F$</p>
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</section>
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<section>
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<p>$R_1 \equiv \big[a \rightarrow T, b \rightarrow T, d \rightarrow T, * \rightarrow F\big]$</p>
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<%= data_table(
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["Name", "ZipCode"],
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[ ["Alice", "10003"],
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["Bob","14260"],
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["Bob","14290"],
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["Carol","13201"],
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["Carol","18201"]
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],
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name: "$\\mathcal R$",
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rowids: true,
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annotations: [
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"T (a)",
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"T (b)",
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"F (c)",
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"T (d)",
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"F (e)"
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]
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) %>
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</section>
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<section>
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<p>Use provenance as before...</p>
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<p class="fragment">... but what about aggregates?</p>
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</section>
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<section>
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<pre><code>
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SELECT COUNT(*)
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FROM R NATURAL JOIN ZipCodeLookup
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WHERE State = 'NY'
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</code></pre>
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<p style="font-size: 70%" class="fragment">
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$$= \begin{cases}
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1 & \textbf{if } \texttt{bob} = 9 \wedge \texttt{carol} = 8\\
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2 & \textbf{if } \texttt{bob} = 4 \wedge \texttt{carol} = 8 \\&\; \vee\; \texttt{bob} = 9 \wedge \texttt{carol} = 3\\
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3 & \textbf{if } \texttt{bob} = 4 \wedge \texttt{carol} = 3
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\end{cases}$$</p>
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<p class="fragment"><b>Problem: </b> A combinatorial explosion of possibilities</p>
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</section>
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<section>
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<p><b>Idea: </b> Simplify the problem</p>
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<ol>
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<li class="fragment">Is a particular tuple <i>Possible</i>?</li>
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<li class="fragment">Is a particular tuple <i>Certain</i>?</li>
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</ol>
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</section>
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<section>
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<dl>
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<div class="fragment">
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<dt>Certain Tuple</dt>
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<dd>A tuple that appears in all possible worlds</dd>
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<dd class="fragment">$\forall D \in \mathcal D : t \in D$</dd>
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</div>
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<div class="fragment">
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<dt>Possible Tuple</dt>
|
||||
<dd>A tuple that appears in at least one possible world</dd>
|
||||
<dd class="fragment">$\exists D \in \mathcal D : t \in D$</dd>
|
||||
</div>
|
||||
</dl>
|
||||
</section>
|
||||
|
||||
<section>
|
||||
<h3>Non-aggregate queries</h3>
|
||||
<dl>
|
||||
<dt>Is a tuple Certain?</dt>
|
||||
<dd class="fragment">Is the provenance polynomial a tautology?</dd>
|
||||
|
||||
<dt>Is a tuple Possible?</dt>
|
||||
<dd class="fragment">Is the provenance polynomial a contradiction?</dd>
|
||||
</dl>
|
||||
<p class="fragment">Pick your favorite SAT solver, plug in and go</p>
|
||||
</section>
|
||||
|
||||
<section>
|
||||
<h3>Aggregate queries</h3>
|
||||
|
||||
<p style="margin-top: 50px; margin-bottom: 50px;">
|
||||
As before, factorize the possible outcomes
|
||||
</p>
|
||||
<p class="fragment">
|
||||
$$1 + \{\;1\;\textbf{if}\;\texttt{bob} = 4\;\} + \{\;1\;\textbf{if}\;\texttt{carol} = 3\;\}$$
|
||||
</p>
|
||||
<p style="margin-top: 50px;" class="fragment">
|
||||
Not bigger than the aggregate input...
|
||||
</p>
|
||||
<p class="fragment">
|
||||
...but at least it only reduces to bin-packing <br/>(or a similarly NP problem.)
|
||||
</p>
|
||||
</section>
|
||||
|
||||
<section>
|
||||
<p>In short, incomplete databases are limited, but have some uses.</p>
|
||||
<p class="fragment">What about probabilities?</p>
|
||||
</section>
|
||||
</section>
|
||||
|
||||
<section>
|
||||
<section>
|
||||
<p><b>Idea: </b> Make $\texttt{bob}$ and $\texttt{carol}$ random variables.</p>
|
||||
</section>
|
||||
|
||||
<section>
|
||||
<p>$$\texttt{bob} = \begin{cases} 4 & p = 0.8 \\ 9 & p = 0.2\end{cases}$$</p>
|
||||
<p>$$\texttt{carol} = \begin{cases} 3 & p = 0.4 \\ 8 & p = 0.6\end{cases}$$</p>
|
||||
</section>
|
||||
|
||||
<section>
|
||||
<p style="font-size: 70%">
|
||||
$$Q(\mathcal D) = \begin{cases}
|
||||
1 & \textbf{if } \texttt{bob} = 9 \wedge \texttt{carol} = 8\\
|
||||
2 & \textbf{if } \texttt{bob} = 4 \wedge \texttt{carol} = 8 \\&\; \vee\; \texttt{bob} = 9 \wedge \texttt{carol} = 3\\
|
||||
3 & \textbf{if } \texttt{bob} = 4 \wedge \texttt{carol} = 3
|
||||
\end{cases}$$</p>
|
||||
<p style="font-size: 90%" class="fragment">
|
||||
$$ = \begin{cases}
|
||||
1 & p = 0.2 \times 0.6\\
|
||||
2 & p = 0.8 \times 0.6 + 0.2 \times 0.4\\
|
||||
3 & p = 0.8 \times 0.4 \end{cases}$$
|
||||
</p>
|
||||
<p class="fragment">
|
||||
$$ = \begin{cases}
|
||||
1 & p = 0.12\\
|
||||
2 & p = 0.56\\
|
||||
3 & p = 0.32\end{cases}$$
|
||||
</p>
|
||||
</section>
|
||||
|
||||
<section>
|
||||
<p>
|
||||
$$Q(\mathcal D) = \begin{cases}
|
||||
1 & p = 0.12\\
|
||||
2 & p = 0.56\\
|
||||
3 & p = 0.32\end{cases}$$
|
||||
</p>
|
||||
<p class="fragment" style="margin-top: 50px;">$E\left[Q(\mathcal D)\right] = 0.12+1.12+0.96 = 2.20$</p>
|
||||
<p class="fragment" style="margin-top: 50px;">$P\left[Q(\mathcal D) \geq 2\right] = 0.56+0.32 = 0.88$</p>
|
||||
</section>
|
||||
|
||||
<section>
|
||||
<p>In general, computing probabilities exactly is <code>#P</code></p>
|
||||
|
||||
<p style="margin-top: 50px;" class="fragment">... so we approximate</p>
|
||||
</section>
|
||||
|
||||
<section>
|
||||
<p><b>Idea 1</b>: Sample. Pick 10 random possible worlds and compute results for each.</p>
|
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</section>
|
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</section>
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Reference in New Issue