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-title: Lambda calculus part 1
-date: 2024-09-10T10:16:12.486Z
-type: default
-draft: true
-tags: [pl-theory]
----
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+---
+title: The lambda calculus
+date: 2024-09-10T10:16:12.486Z
+type: default
+draft: true
+tags: [pl-theory]
+---
+
+If you want to analyze something, one useful approach is to try to start with
+some smaller, minimal version of it, making observations about it, and then
+generalizing those observations.
+
+One particularly useful tool for analyzing and developing programming languages
+is the **lambda calculus**, an amazingly simple abstract machine that can
+represent all [Turing-complete] computation.
+
+Due to its simplicity, there are numerous ways to extend it. In fact, the
+standard way for researchers to introduce new language features is to extend the
+lambda calculus with their feature, and prove properties about it.
+
+Not only is it useful as a reasoning tool, some compilers may use a variant of
+lambda calculus as an intermediate language for compilation. GHC famously uses a
+variant of System F as its Core language.
+
+[Turing-complete]: https://en.wikipedia.org/wiki/Turing_completeness
+
+In this blog post series, I'll introduce the lambda calculus in its most basic
+form, introduce some conventions when it comes to defining languages like this,
+writing an interpreter for it, and then writing some extensions.
+
+## Abstract machines
+
+Firstly, what is an abstract machine? We think of real machines as mechanical
+beings, that behave according to some rules. If I punch in 1 + 1 into my
+calculator, and hit enter, I expect it to return 2.
+
+In reality, there's all sorts of reasons that calculation could possibly fail,
+battery too low, the chip is overheating, [cosmic rays], just to name a few.
+From a mathematical point of view, these are all uninteresting failure cases we
+would rather just ignore.
+
+[cosmic rays]: https://en.wikipedia.org/wiki/Single-event_upset
+
+What we could do is distill _just_ the rules into some form we can reason about.
+We could just simulate its execution in our heads and confirm that the rules we
+created are nice.
+
+## The lambda calculus
+
+The lambda calculus is essentially an abstract machine where the only operation
+that's defined is function call. The only operations you need are:
+
+- Make a function (which we call a "lambda", or $\lambda$)
+- Call a function
+
+That's it. That's all there is to it.
+
+Here's the identity function:
+
+$$
+\lambda x . x
+$$
+
+Here's a function that calls its first input on its second:
+
+$$
+\lambda f . (\lambda x . f (x))
+$$
+
+Taking in multiple arguments can be represented by just nesting lambdas.
+
+## Representing numbers
+
+With a language this simple, how can you possibly do anything with it? Can you
+even do arithmetic?
+
+It turns out the answer is yes, as long as you _encode_ numbers in a particular
+way. One clever way is by using [Church encoding]. Here are some of the first
+few natural numbers written with Church encoding:
+
+[Church encoding]: https://en.wikipedia.org/wiki/Church_encoding
+
+- 0 is $\lambda s . \lambda z . z$
+- 1 is $\lambda s . \lambda z . s(z)$
+- 2 is $\lambda s . \lambda z . s(s(z))$
+- 3 is $\lambda s . \lambda z . s(s(s(z)))$
+
+See the pattern? $s$ represents a "successor" function, and $z$ represents zero.
+Note that what we pass in for $s$ and $z$ don't matter, as long as $s$ is a
+function.
+
+With this, we can imagine a "plus" operator that simply takes 2 numbers $a$ and
+$b$, then calls $a$ (since it's a function) using $b$ as the "zero" value. Here it is:
+
+$$
+\mathsf{plus} :\equiv \lambda a . \lambda b . \lambda s . \lambda z . a(s)(b(s)(z))
+$$
+
+If this doesn't make sense, try plugging 2 and 3 from above and simplifying a
+bit to convince yourself that this function works.
+
+There's also ways to encode different things like rational numbers, booleans,
+pairs, and lists. See the [Church encoding] Wikipedia page for more.
+
+## Formal definition
+
+Now that we have the intuition down, let's go on to a more formal definition.
+
+The lambda calculus is a _language_. In this language we have terms that can get
+evaluated. Terms are commonly described with a [grammar], which tell us how
+terms in this language can be constructed. The grammar for the lambda calculus
+is incredibly simple:
+
+[grammar]: https://en.wikipedia.org/wiki/Formal_grammar
+
+$$
+\mathsf{Term} \; t ::= x \; | \; \lambda x . t \; | \; t(t)
+$$
+
+This says that there are only 3 possible terms:
+
+1. $x$, which is our convention for some kind of variable or name
+2. $\lambda x . t$, which creates a lambda function that takes an input called $x$ and returns another term $t$
+3. $t(t)$ which is just function application, one term calling another
+
+All of the terms above, including $\mathsf{plus}$, can be represented as a
+combination of these constructors.
+
+Great, but we can't really do anything if all we know is how to write down some
+terms. We need some rules to tell us how to evaluate it. Something like this:
+
+$$
+(\lambda x . t_1) (t_2) \longmapsto \mathsf{subst}_{x \mapsto t_2} t_1
+$$
+
+Slightly inventing notation here, but essentially whenever we see a lambda being
+applied to something, we should substitute the thing applied into the body of
+the lambda everywhere we see that $x$. This isn't really a formal definition,
+just kind of a vague descriptor. There's too many loose ends; what is
+substitution? How do we keep track of what variables have which values? Where
+does the result go? Let's go back and introduce a few more concepts to clean up
+the formal definition.
+
+### Contexts
+
+### Substitution
+
+### Values
+
+### Judgments
+
+### Rules
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