From 5ecc5f8eed35691f87fed401dd7a2083ed919f6a Mon Sep 17 00:00:00 2001 From: Michael Zhang Date: Fri, 1 Sep 2023 09:24:12 -0500 Subject: [PATCH] update --- src/content/config.ts | 1 + .../header.jpg | Bin 0 -> 644598 bytes .../index.md} | 25 +++++++++++++----- 3 files changed, 19 insertions(+), 7 deletions(-) create mode 100644 src/content/posts/2023-09-01-formal-cek-machine-in-agda/header.jpg rename src/content/posts/{2023-09-01-formal-cek-machine-in-agda.md => 2023-09-01-formal-cek-machine-in-agda/index.md} (92%) diff --git a/src/content/config.ts b/src/content/config.ts index 692be81..9563b52 100644 --- a/src/content/config.ts +++ b/src/content/config.ts @@ -26,6 +26,7 @@ const posts = defineCollection({ tags: z.array(z.string()), draft: z.boolean().default(false), + math: z.boolean().default(false), }), }); diff --git a/src/content/posts/2023-09-01-formal-cek-machine-in-agda/header.jpg b/src/content/posts/2023-09-01-formal-cek-machine-in-agda/header.jpg new file mode 100644 index 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similarity index 92% rename from src/content/posts/2023-09-01-formal-cek-machine-in-agda.md rename to src/content/posts/2023-09-01-formal-cek-machine-in-agda/index.md index 301572f..3ac442b 100644 --- a/src/content/posts/2023-09-01-formal-cek-machine-in-agda.md +++ b/src/content/posts/2023-09-01-formal-cek-machine-in-agda/index.md @@ -1,8 +1,16 @@ --- -title: Building a formal CEK machine in agda +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"] +tags: + - computer-science + - programming-languages + - formal-verification + - lambda-calculus + +heroImage: ./header.jpg +heroAlt: gears spinning wallpaper +math: true --- Back in 2022, I took a special topics course, CSCI 8980, on [reasoning about @@ -58,9 +66,10 @@ 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. +- **Var.** This is just a variable, like `x` or `y`. By itself it holds no + meaning, but during evaluation, the evaluation _environment_ holds a mapping + from variable names to the values. 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 @@ -87,7 +96,7 @@ 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 +In fact, the lambda calculus is [Turing-complete][tc], 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: @@ -143,13 +152,15 @@ information about our program's behavior. ### Simply-typed lambda calculus -The simply-typed lambda calculus (STLC) adds types to every term. Types are +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. +[simply-typed lambda calculus]: https://en.wikipedia.org/wiki/Simply_typed_lambda_calculus + 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