cek post revive

src/components/PostList.astro

@@ -6,26 +6,36 @@ import { sortBy } from "lodash-es";
 interface Props {
   basePath: string;
   drafts?: "exclude" | "include" | "only";
+  filteredPosts?: Post[];
 }
 
 type Post = CollectionEntry<"posts">;
-const { basePath, drafts: includeDrafts } = Astro.props;
+const { basePath, drafts: includeDrafts, filteredPosts } = Astro.props;
 
-let filter;
-switch (includeDrafts) {
-  case "exclude":
-  case undefined:
-    filter = (post: Post) => !post.data.draft;
-    break;
-  case "include":
-    filter = (_: Post) => true;
-    break;
-  case "only":
-    filter = (post: Post) => post.data.draft === true;
-    break;
+type FilterFn = (_: Post) => boolean;
+
+function unreachable(): never {
+  throw new Error("unreachable");
 }
 
-const allPosts = await getCollection("posts", filter);
+function getFilter(): FilterFn {
+  switch (includeDrafts) {
+    case "exclude":
+    case undefined:
+      return (post: Post) => !post.data.draft;
+    case "include":
+      return (_: Post) => true;
+    case "only":
+      return (post: Post) => post.data.draft === true;
+  }
+  return unreachable();
+}
+
+const filter = getFilter();
+let allPosts;
+if (filteredPosts) allPosts = filteredPosts.filter(filter);
+else allPosts = await getCollection("posts", filter);
+
 const sortedPosts = sortBy(allPosts, (post) => -post.data.date);
 ---
 

src/content/posts/2023-09-01-formal-cek-machine-in-agda.md

@@ -0,0 +1,215 @@
+---
+title: Building a formal CEK machine in agda
+draft: true
+date: 2023-09-01T13:53:23.974Z
+tags: ["computer-science", "programming-languages", "formal-verification", "lambda-calculus"]
+---
+
+Back in 2022, I took a special topics course, CSCI 8980, on [reasoning about
+programming languages using Agda][plfa], a dependently typed meta-language. For
+the term project, we were to implement a simply-typed lambda calculus with
+several extensions, along with proofs of certain properties.
+
+[plfa]: https://plfa.github.io/
+
+My lambda calculus implemented `call/cc` on top of a CEK machine.
+
+<details>
+  <summary><b>Why is this interesting?</b></summary>
+
+Reasoning about languages is one way of ensuring whole-program correctness.
+Building up these languages from foundations grounded in logic helps us
+achieve our goal with more rigor.
+
+As an example, suppose I wrote a function that takes a list of numbers and
+returns the maximum value. Mathematically speaking, this function would be
+_non-total_; an input consisting of an empty set would not produce reasonable
+output. If this were a library function I'd like to tell people who write code
+that uses this function "don't give me an empty list!"
+
+But just writing this in documentation isn't enough. What we'd really like is
+for a tool (like a compiler) to tell any developer who is trying to pass an
+empty list into our maximum function "You can't do that." Unfortunately, most
+of the popular languages being used today have no way of describing "a list
+that's not empty."
+
+We still have a way to prevent people from running into this problem, though
+it involves pushing the problem to runtime rather than compile time. The
+maximum function could return an "optional" maximum. Some languages'
+implementations of optional values force programmers to handle the "nothing"
+case, while others ignore it silently. But in the more optimistic case, even
+if the list was empty, the caller would have handled it and treated it
+accordingly.
+
+This isn't a pretty way to solve this problem. _Dependent types_ gives us
+tools to solve this problem in an elegant way, by giving the type system the
+ability to contain values. This also opens its own can of worms, but for
+questions about program correctness, it is more valuable than depending on
+catching problems at runtime.
+
+</details>
+
+## Crash course on the lambda calculus
+
+The [lambda calculus] is a mathematical abstraction for computation. The core
+mechanism it uses is a concept called a _term_. Everything that can be
+represented in a lambda calculus is some combination of terms. A term can have
+several constructors:
+
+[lambda calculus]: https://en.wikipedia.org/wiki/Lambda_calculus
+
+- **Var.** This is just a variable, like `x` or `y`. During evaluation, a
+  variable can resolve to a value in the evaluation environment by name. If
+  the environment says `{ x = 5 }`, then evaluating `x` would result in 5.
+
+- **Abstraction, or lambda (λ).** An _abstraction_ is a term that describes some
+  other computation. From an algebraic perspective, it can be thought of as a
+  function with a single argument (i.e f(x) = 2x is an abstraction, although
+  it would be written `(λx.2x)`)
+
+- **Application.** Application is sort of the opposite of abstraction, exposing
+  the computation that was abstracted away. From an algebraic perspective,
+  this is just function application (i.e applying `f(x) = 2x` to 3 would
+  result in 2\*3. Note that only a simple substitution has been done and
+  further evaluation is required to reduce 2\*3)
+
+### Why?
+
+The reason it's set up this way is so we can reason about terms inductively.
+Rather than having lots of syntax for making it easier for programmers to write
+a for loop as opposed to a while loop, or constructing different kinds of
+values, the lambda calculus focuses on function abstraction and calls, and
+strips everything else away.
+
+The idea is that because terms are just nested constructors, we can describe the
+behavior of any term by just defining the behavior of these 3 constructors. The
+flavorful features of other programming languages can be implemented on top of
+the function call rules in ways that don't disrupt the basic function of the
+evaluation.
+
+In fact, the lambda calculus is Turing-complete, so any computation can
+technically be reduced to those 3 constructs. I used numbers liberally in the
+examples above, but in a lambda calculus without numbers, you could define
+integers using a recursive strategy called [Church numerals]. It looks like this:
+
+[church numerals]: https://en.wikipedia.org/wiki/Church_encoding
+
+- Let `z` represent zero.
+- Let `s` represent a "successor", or increment function. `s(z)` represents 1,
+  `s(s(z))` represents 2, and so on.
+
+In lambda calculus terms, this would look like:
+
+- 0 = `λs.(λz.z)`
+- 1 = `λs.(λz.s(z))`
+- 2 = `λs.(λz.s(s(z)))`
+- 3 = `λs.(λz.s(s(s(z))))`
+
+In practice, many lambda calculus incorporate higher level constructors such as
+numbers or lists to make it so we can avoid having to represent them using only
+a series of function calls. However, any time we add more syntax to a language,
+we increase its complexity in proofs, so for now let's keep it simple.
+
+### The Turing completeness curse
+
+As I noted above, the lambda calculus is [_Turing-complete_][tc]. One feature of
+Turing complete systems is that they have a (provably!) unsolvable "halting"
+problem. Most of the simple terms shown above terminate predictably. But as an
+example of a term that doesn't halt, consider the _Y combinator_, an example of
+a fixed-point combinator:
+
+[tc]: https://en.wikipedia.org/wiki/Turing_completeness
+
+    Y = λf.(λx.f(x(x)))(λx.f(x(x)))
+
+That's quite a mouthful. If you tried calling Y on some term, you will find that
+evaluation will quickly expand infinitely. That makes sense given its purpose:
+to find a _fixed point_ of whatever function you pass in.
+
+> As an example, the fixed-point of the function f(x) = sqrt(x) is 1. That's
+> because f(1) = 1. The Y combinator attempts to find the fixed point by simply
+> applying the function multiple times. In the untyped lambda calculus, this can
+> be used to implement simple (but possibly unbounded) recursion.
+
+This actually proves disastrous for trying to reason about the logic of a
+program. If we don't even know for sure if something will halt, how can we know
+that it'll produce the correct value? In fact, you can prove false statements
+using infinite recursion as a basis.
+
+This is why we actually prefer _not_ to work with Turing-complete languages when
+doing logical reasoning on program evaluation. Instead, we always want to add
+some constraints on it to make evaluation total, ensuring that we have perfect
+information about our program's behavior.
+
+### Simply-typed lambda calculus
+
+The simply-typed lambda calculus (STLC) adds types to every term. Types are
+crucial to any kind of static program analysis. Suppose I was trying to apply
+the term 5 to 6 (in other words, call 5 with the argument 6 as if 5 was a
+function). As humans we can look at that and instantly recognize that the
+evaluation would be invalid, yet under the untyped lambda calculus, it would be
+completely representable.
+
+To solve this in STLC, we would make this term completely unrepresentable at
+all. To say you want to apply 5 to 6 would not be a legal STLC term. We do this
+by requiring that all STLC terms are untyped lambda calculus terms accompanied
+by a _type_.
+
+This gives us more information about what's allowed before we run the
+evaluation. For example, numbers may have their own type `Nat` (for "natural
+number"), while functions have a special "arrow" type `_ -> _`, where the
+underscores represent other types. A function that takes a number and returns a
+boolean (like isEven) would have the type `Nat -> Bool`, while a function that
+takes a boolean and returns another boolean would be `Bool -> Bool`.
+
+With this, we have a framework for rejecting terms that would otherwise be legal
+in untyped lambda calculus, but would break when we tried to evaluate them. A
+function application would be able to require that the argument is the same type
+as what the function is expecting.
+
+The nice property you get now is that all valid STLC programs will never get
+_stuck_, which is being unable to evaluate due to some kind of error. Each term
+will either be able to be evaluated to a next state, or is done.
+
+A semi-formal definition for STLC terms would look something like this:
+
+- **Var.** Same as before, it's a variable that can be looked up in the
+  environment.
+
+- **Abstraction, or lambda (λ).** This is a function that carries three pieces
+  of information: (1) the name of the variable that its input will be substituted
+  for, (2) the _type_ of the input, and (3) the body in which the substitution
+  will happen.
+
+- **Application.** Same as before.
+
+It doesn't really seem like changing just one term changes the language all that
+much. But as a result of this tiny change, _every_ term now has a type:
+
+- `5 :: Nat`
+- `λ(x:Nat).2x :: Nat -> Nat`
+- `isEven(3) :: (Nat -> Bool) · Nat = Bool`
+
+Notation: (`x :: T` means `x` has type `T`, and `f · x` means `f` applied to
+`x`)
+
+This also means that some values are now unrepresentable:
+
+- `isEven(λx.2x) :: (Nat -> Bool) · (Nat -> Nat)` doesn't work because the type
+  of `λx.2x :: Nat -> Nat` can't be used as an input for `isEven`, which is
+  expecting a `Nat`.
+
+We have a good foundation for writing programs now, but this by itself can't
+qualify as a system for computation. We need an abstract machine of sorts that
+can evaluate these symbols and actually compute on them.
+
+In practice, there's a number of different possible abstract machines that can
+evaluate the lambda calculus. Besides the basic direct implementation, alternate
+implementations such as [interaction nets] have become popular due to being able
+to be parallelized efficiently.
+
+[interaction nets]: https://en.wikipedia.org/wiki/Interaction_nets
+
+## CEK machine
+
+A CEK machine is responsible for evaluating a lambda calculus term.

src/content/posts/2023-09-01-hindley-milner/index.md

@@ -0,0 +1,12 @@
+---
+title: Hindley-Milner type inference
+date: 2023-09-01T13:06:55.614Z
+tags:
+  - type-theory
+draft: true
+---
+
+Today, a lot of languages have a feature called something along the lines of
+"type inference". The idea is that through the way variables are passed and
+used, you can make guesses about what type it should be, and preemptively point
+out invalid or incompatible uses. Let's talk about how it works.

src/pages/posts/[...slug].astro

@@ -47,6 +47,26 @@ const datestamp = post.data.date.toLocaleDateString(undefined, {
   <div class="post-container">
     <h1 class="post-title">{post.data.title}</h1>
 
+    <span class="tags">
+      {
+        post.data.draft && (
+          <a href="/drafts" class="tag draft">
+            <i class="fa fa-warning" aria-hidden="true" />
+            <span class="text">draft</span>
+          </a>
+        )
+      }
+
+      {
+        post.data.tags.map((tag) => (
+          <a href={`/tags/${tag}`} class="tag">
+            <i class="fa fa-tag" aria-hidden="true" />
+            <span class="text">{tag}</span>
+          </a>
+        ))
+      }
+    </span>
+
     <small class="post-meta">
       Posted on <Timestamp timestamp={post.data.date} />
        - {remarkPluginFrontmatter.minutesRead}

src/pages/tags/[tag].astro

@@ -0,0 +1,28 @@
+---
+import { join, dirname } from "path";
+import BaseLayout from "../../layouts/BaseLayout.astro";
+import PostList from "../../components/PostList.astro";
+import { getCollection } from "astro:content";
+
+export async function getStaticPaths() {
+  const posts = await getCollection("posts");
+  return posts.flatMap((post) =>
+    post.data.tags.map((tag) => ({
+      params: { tag },
+    })),
+  );
+}
+
+const { tag } = Astro.params;
+const posts = await getCollection("posts");
+const taggedPosts = posts.filter((post) => post.data.tags.includes(tag));
+
+const currentUrl = Astro.url;
+const root = dirname(dirname(currentUrl.pathname));
+---
+
+<BaseLayout>
+  <p>Posts tagged with {tag}</p>
+
+  <PostList basePath={join(root, "posts")} filteredPosts={taggedPosts} />
+</BaseLayout>

src/styles/post.scss

@@ -82,6 +82,7 @@
       font-size: 0.9rem;
       padding: 0 30px;
       line-height: 1.5;
+      position: relative;
 
       p:nth-of-type(1) {
         padding-top: 0;
@@ -89,8 +90,13 @@
       }
 
       summary {
+        cursor: pointer;
         padding: 10px 0;
         transition: margin 150ms ease-out;
+
+        position: sticky;
+        top: 0;
+        background-color: var(--background-color);
       }
 
       &[open] summary {
@@ -196,3 +202,29 @@ hr.endline {
   border-width: 1px 0 0 0;
   border-color: var(--hr-color);
 }
+
+.tags {
+  display: flex;
+  gap: 6px;
+  margin-bottom: 6px;
+
+  .tag {
+    font-size: 0.75rem;
+    background-color: var(--tag-color);
+    padding: 2px 7px;
+    border-radius: 4px;
+
+    &.draft {
+      background-color: orange;
+      color: black;
+    }
+
+    .text {
+      text-decoration: none;
+    }
+
+    &:hover {
+      text-decoration: none;
+    }
+  }
+}