blog/content/posts/2022-02-02-formal-cek-machine-in-agda/_index.md

+++ title = "a formal cek machine in agda" draft = true date = 2022-02-02 tags = ["computer-science", "programming-languages", "formal-verification", "lambda-calculus"] languages = ["agda"] layout = "single" toc = true +++

Last semester, I took a course on reasoning about programming languages using Agda, 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.

My lambda calculus implemented call/cc on top of a CEK machine.

Foreword

Why is this interesting? 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!"

Unfortunately, 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."

Lambda calculus

The lambda calculus is a mathematical structure for describing computation. At the most basic level, it defines 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:

• 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. 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.

Interestingly, the lambda calculus is Turing-complete, so any computation can 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 recursively like this:

• 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 have a set of "base" values from which to build off, such as unit values, booleans, and natural numbers (having numbers in the language means we don't need the s and z dance to refer to them).

Turing completeness

As I noted above, the lambda calculus is Turing-complete. One feature of Turing complete systems is that they have a (provably!) unsolvable "halting" problem.

Simply-typed lambda calculus (STLC)

CEK machine

A CEK machine