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Anisomorphism
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[[Mechanics (Book)]] | |||
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I do math | I do math | ||
= Algebraic Geometry of Computing = | = Algebraic Geometry of Computing = | ||
Finite state machines appear in a variety of instantiations: mechanical, electronic, fluidic. The physical mechanisms involved necessitate that the design is described by differential equations, but ultimately the manipulation of abstracted "logical" states is the final goal. Thus we can describe the architecture of a general finite state machine with <math> \mathbb{Z}/2\mathbb{Z} </math> algebra (or other finite rings too). | Finite state machines appear in a variety of instantiations: mechanical, electronic, fluidic. The physical mechanisms involved necessitate that the design is described by differential equations, but ultimately the manipulation of abstracted "logical" states is the final goal. Thus we can describe the architecture of a general finite state machine with <math> \mathbb{Z}/2\mathbb{Z} </math> algebra (or other finite rings too). | ||
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XOR is only "true" or 1 when x or y but not both, are 1. | XOR is only "true" or 1 when x or y but not both, are 1. Disjunctive normal form says that we can view the x, y entries as unary operators which return the input with no change, combine these as given on the lines which evaluate to 1, and take the OR of all of them for the total connective form of the truth table. Here is the third line: <math> x\and\neg y</math>. | ||
<br> The total is: | |||
<math> | |||
\begin{align*} | |||
&(x\and\neg y)\or (\neg x\and y) \\ | |||
=&(x\or(\neg x\and y))\and(\neg y\or(\neg x\and y)) \\ | |||
=&(x\or\neg x)\and (x\or y)\and(\neg y\or\neg x)\and(\neg y\or y) \\ | |||
=&(x\or y)\and (\neg y\or\neg x) \\ | |||
=&(x\or y)\and\neg (y\and x) | |||
\end{align*} | |||
</math> | |||
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This process can be viewed as a sum of "elementary functions" which are only 1 on one line each, and building a general function/table. | |||
= Read prototype = | = Read prototype = | ||