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‎2013/05/02/banach-tarski.html

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Banach-Tarski - A basic magic of group theory
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<time datetime="2013-05-02T00:00:00-04:00" itemprop="datePublished">
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2 May 2013
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- note
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<p>The Banach-Tarski paradox is a curious theorem from group theory. I was quite taken by it when I first learned of it, and wrote up <a href="/assets/pdfs/2013.05.02-banach-tarski.pdf">these self-contained notes (PDF)</a>.</p>
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<p>Banach-Tarski shows <em>how to conjure things out of thin air</em> …but not with magic, just with the <a href="https://encyclopediaofmath.org/wiki/Axiom_of_choice">axiom of choice</a>. 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 many tiny pieces, and then putting the pieces back together to make two balls just identical the one you started with.</p>
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‎2018/11/15/eq-FSA-MSOL.html

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An equivalence between generative automata and logical constraints
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<time datetime="2018-11-15T00:00:00-05:00" itemprop="datePublished">
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15 Nov 2018
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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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<p>My slides are <a href="/assets/pdfs/2018.11.15-generative_vs_constraint.pdf">here</a>.</p>
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