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+    	Banach-Tarski - A basic magic of group theory
+    </h1>
+    <span class="post-meta">
+    	<time datetime="2013-05-02T00:00:00-04:00" itemprop="datePublished">
+    		2 May 2013
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+        - note
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+  <article class="post-content" itemscope itemtype="http://schema.org/BlogPosting">
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+      <p>A while ago I wrote up these notes describing the Banach-Tarski “paradox”.</p>
+
+<p>Banach-Tarski shows <em>how to conjure things out of thin air</em> …with the axiom of choice (you also need an infinitely divisible object, but only on short loan, because after you cut it up and reassemble it, you’ll have two!).  Basically, it gives a method for taking a ball, cutting it up into pieces, and putting the pieces back together to make two balls just identical the one you started with.</p>
+
+<p>PDF <a href="/assets/pdfs/2013.05.02-banach-tarski.pdf">here</a>.</p>
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+  <title>An equivalence between generative automata and logical constraints</title>
+  <meta name="description" content="I gave a presentation to MCQLL, on an an example of the duality of generative systems and constraint-based systems.  The particular example I discussed was a...">
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+    	An equivalence between generative automata and logical constraints
+    </h1>
+    <span class="post-meta">
+    	<time datetime="2018-11-15T00:00:00-05:00" itemprop="datePublished">
+    		15 Nov 2018
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+        - presentation
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+  <article class="post-content" itemscope itemtype="http://schema.org/BlogPosting">
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+      <p>I gave a presentation to <a href="https://mcqll.org/">MCQLL</a>, on an an example of the duality of generative systems and constraint-based systems.  The particular example I discussed was a proof of the equivalence in expressive power between Finite state automata (FSA) and monadic second order logic (with a successor function; MSOL[<em>S</em>]).  That is, I presented the proof that any language for which there exists an FSA can be described equivalently using constraints written in MSOL[<em>S</em>].</p>
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