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

+++ title = "building 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.

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!"

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.

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. Most of the simple term 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:

Y = λf.(λx.f(x(x)))(λx.f(x(x)))

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.

Because there are terms that may not terminate, the untyped lambda calculus is not very useful for logical reasoning. Instead, we add some constraints on it that makes evaluation total, at the cost of losing Turing-completeness.

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. 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 make this term completely unrepresentable at all. To say you want to apply 5 to 6 would not be a legal STLC term. That's because 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.

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 seem like much has changed. But all of a sudden, every term 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.

CEK machine

A CEK machine is responsible for evaluating a lambda calculus term.