Result about Aperiodic Tilings for Finite Sets of Prototiles - and MathJax test
This is a quick blog post to accomplish two things: Get some content up on the blog, finally... Test some \(\rm\LaTeX\) code... So, to get things rolling, here's a brief result from my thesis that is kind of cute. Definitions and Preliminaries So, first up we need the following notation: Let \( \varphi_e \) be the \(e^{th}\) Turing Machine under some effective enumeration of all possible Turing Machines - some Gödel coding of all 5-tuples of all possible TM states. If a machine \(e\) on input \(x\) halts, then we write \(\varphi_e(x) \downarrow\). For a set \(A\) we notate the complement of \(A\) (that is, everything not in \(A\)) by \(\overline{A}\). A formula \(\phi\) is in \(\Sigma^0_1\) if it is of the form \( \exists x \, ( \psi(x) ) \) where \(\psi\) is a formula without any free (read: only bounded) quantifiers over any other variables. A formula \( \phi \) is \( \Pi^0_1\) if it is of the form \( \forall x \, ( \psi(x) ) \) that is, it is the complement of a \