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center LaTeX stuff in SECD post

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Rachel Lambda Samuelsson 2022-01-27 22:23:00 +01:00
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_posts/2021-12-10-secd1.md
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@ -17,9 +17,9 @@ SECD stands for Stack Environment Control Dump, all of which but the environment
This approach, while sound, suffers from some performance issues. Thankfully, there are a number of improvements which can be made. The first of which is the use of De Brujin indecies. In De Brujin index notation each variable is a number indexing which lambda it was bound at (counting outwards). Here are some examples:
$\lambda x.x \xRightarrow[]{\text{De Brujin}} \lambda \#0$
$$\lambda x.x \xRightarrow[]{\text{De Brujin}} \lambda \#0$$
$\lambda f.\lambda g.\lambda x. f(gx) \xRightarrow[]{\text{De Brujin}} \lambda\lambda\lambda \#2(\#1\#0)$
$$\lambda f.\lambda g.\lambda x. f(gx) \xRightarrow[]{\text{De Brujin}} \lambda\lambda\lambda \#2(\#1\#0)$$
The benefit in using De Brujin notation is that, rather than a balanced binary tree, a stack may be used for the environment. A variable is then an index into the environment. An additional benifit is that the machine does not need to concern itself with alpha equivalence. As a side-note, counting conventionally starts at $\#1$, not $\#0$, however in a virtual machine counting from $\#0$ is, of course, much more natural.
@ -118,11 +118,11 @@ All stacks grow from right to left, that is, the left most element is at the top
Consider the following lambda expression:
$(\lambda f.\lambda g.\lambda x. f(gx)) (\lambda x. x + 1) (\lambda x. x + 1) 0$
$$(\lambda f.\lambda g.\lambda x. f(gx)) (\lambda x. x + 1) (\lambda x. x + 1) 0$$
It composes a lambda which increments its input by one with an identical lambda and then applies 0 to the result. In order to run this on our SECD machine it must first be rewritten with De Brujin index notation.
$(\lambda \lambda \lambda \#2(\#1\#0)) (\lambda 1 + \#1) (\lambda 1 + \#1)0$
$$(\lambda \lambda \lambda \#2(\#1\#0)) (\lambda 1 + \#1) (\lambda 1 + \#1)0$$
Now in order to create an instruction tape for the SECD machine it must be rewritten, using Closure as lambdas, Local $\#i$ for variables and keeping in mind to explicitly apply to and return from lambdas. Additionally it will have to be written in postfix form due to the nature of stack based machines. After performing these actions the expression has been transformed into the following tape: