131 lines
4.3 KiB
Markdown
131 lines
4.3 KiB
Markdown
---
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title: An Idiot's Guide to Lambda Calculus
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published: 2018-08-27
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---
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clarification: this is not a guide to lambda calculus for idiots, this is a guide to lambda calculus written by an idiot.
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imagine for a second you need to explain our numbering system without ***any*** background. any at all, think about that for a second.
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* you have no idea what the number symbols mean
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* you have no idea what plus minus or anything mean
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* you have no idea of anything
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and this isn't limited to just numbering, because almost anything we do on a computer is based around maths, imagine the same for pretty much literally anything.
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what you would need is....
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**is**...
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***is***...
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## a way to describe computability
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and one way of doing that is with lambda calculus. to define lambda calculus we need just a few rules - variables, abstraction, application.
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these concepts aren't too difficult, so awayyyy we go.
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### variables
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variables are simply a name, most of the time we use letters. there are two catches here, if you're used to variables in maths or other languages:
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* unlike pretty much every other language, lambda calc variables don't have a value, only a name. this is because we don't have any concept of what a value is.
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* unlike any language worth using, variables in lambda calculus do not have a *type*. again this is because we don't have a concept of what type is.
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this guide is gonna use lowercase letters for the most part, so the examples are `a b c d e f` etc etc
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### abstraction
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abstractions are the functions of lambda calc, and take the form off:
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```
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(λx.M)
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```
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where `λ` denotes that this is a function
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`x` is an argument name (that is local to this function)
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and `M` is a list of variables that will form the function body
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a quick example of an abstraction is the identity function:
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```
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(λx.x)
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```
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we'll get onto more to do with fuctions in a bit, but next
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### application
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application is what we pretentious people call "actually using a function" and it takes the form of:
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```
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(M N)
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```
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where `M` is an abstraction, and `N` is any lambda term at all. i will refer to `M` as an abstraction and `N` as the application body
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to do the application we just use `N` as the argument for `M`, and then do a find and replace based on the abstraction definition.
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for example, lets do an easy one using the identity function, which is a function that returns its input unchanged:
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if we have the abstraction `(λx.x)` and the variable `y` to use for the application, and then lay the application out as so:
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```
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( (λx.x) y )
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```
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we can then begin applying.
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the identity function has `x` as its argument, so we take the application body, in this case `y`, and every time we see `x` in the abstraction body (the bit past `.`) we replace it with our argument, `y`.
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so our abstraction body is `x`, and `x` is equal to our argument `y`, so therefore our function will return `y`, which is the identity.
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a sligtly more interesting example is something like:
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```
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( (λx.xx) y )
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```
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here, our argument to our abstraction is `y`, which gets bound to `x` in the abstraction body, which is then fed into the `xx` to produce:
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`yy`
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and thats how we go.
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as a spoiler for what the next lambda calc post will cover, heres a bit of an example, imagine we were using the same abstraction:
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```
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(λx.xx)
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```
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and instead of passing it `y`, we passed it **another abstraction**... say, itself?
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```
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( (λx.xx) (λx.xx) )
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```
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here, we bind the entire abstraction `(λx.xx)` to the variable `x` in the first abstraction.
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when we do the substitution into the body `xx` we get left with...
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```
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( (λx.xx) (λx.xx) )
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```
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the same thing, which means this is an ***infinite loop***
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fun stuff fun stuff imo
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## further thoughts
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theres a lot more to go before we get to a numbering system, and at this point it may seem like theres not much use to lambda calculus, but i am not joking when i say ***anything that can be done on a computer can be written in lambda calculus***.
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it may be abstracted, it may be thousands of millions of pages long, but it is possible.
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and thats what makes it so wonderful.
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i'll write more in depth on this at some point in the near future when i'm feeling smarter, less lazy, and don't wanna write edgy armchair psych pieces instead.
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peace out ny'all, if you're unlucky i'll talk about Nock after this.
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