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hugo/content/posts/2016-04-21-modular-machines.md

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+---
+lastmod: "2021-09-12T22:47:44.0000000+01:00"
+author: patrick
+categories:
+- mathematical_summary
+comments: true
+date: "2016-04-21T00:00:00Z"
+aliases:
+- /modular-machines/
+title: Modular machines
+---
+
+I've written [a blurb][MM] about what a modular machine is (namely, another Turing-equivalent form of computing machine),
+and how a Turing machine may be simulated in one.
+(In fact, that blurb now contains an overview of how we may use modular machines to produce a group with insoluble word problem,
+and how to use them to embed a recursively presented group into a finitely presented one.)
+
+A modular machine is like a slightly more complicated version of a Turing machine, but it has the advantage
+that it is easier to embed a modular machine into a group than it is to embed a Turing machine directly into a group.
+We can use this embedding to show that there is a group with unsolvable word problem:
+solving the word problem would correspond to determining whether a certain Turing machine halted.
+
+This is as part of my revision process for the Part III course on "Infinite Groups and Decision Problems".
+It's probably more comprehensible if you already know what a modular machine is.
+Below are some notes which are handwritten, because I needed to draw pictures easily; the linked notes are typeset but might be less legible.
+
+![Notes1]
+![Notes2]
+
+[MM]: /misc/ModularMachines/EmbedMMIntoTuringMachine.pdf
+[Notes1]: /images/ModularMachines/ModularMachines1.jpg
+[Notes2]: /images/ModularMachines/ModularMachines2.jpg