Terms, types, and functions

Many programming languages have both terms and types. Terms are also sometimes called expressions.

Roughly speaking, terms, like 3 or true, denote the data being manipulated, while types, like Number or Boolean, describe what operations are permitted on terms.

For instance, if you have a term of type Number, you might be able to do things like add or subtract with other Numbers. And if you have a term of type Boolean (aka true or false), you could do things like negate it or use it to branch with an if construct.

Common in most programming languages are functions, to which one may pass a term and get back a term. For instance, we could define the function is_zero, which takes a term of type Number and returns a term of type Boolean.

Given the existence of both types and terms, though, we can consider four distinct varieties of function:

  1. From terms to terms
  2. From types to terms
  3. From types to types
  4. From terms to types

Let us examine each in detail, and how each is uniquely useful.

Terms to terms

As mentioned, the most common variety of function is the one that takes a term and returns a term, like is_zero.

Types to terms

Consider the identity function, which is the function that returns its argument unchanged.

The implementation of the identity function is identical for any choice of parameter/return type: just return the term passed in. So, it would be convenient if, instead of having to define a "different" identity function for every type, we could have a single identity function that would work for any type. This is sometimes called a generic function.

What we can do is allow the identity function to take a type argument. Let us call it T. We then take a term argument whose type is T and return it.

The identity "function", then, is actually defined with two functions. First is a function that takes a type T and then returns a term. That term is also a function. It takes a term of type T, and then returns that term.

Types to types

Commonly found alongside generic functions are generic types.

For instance, many programming languages provide a list data structure, which is a ordered sequence of elements that can be dynamically added to and removed from. Different programming languages call this data structure different things: list, array, vector, sequence, and so on, but the general idea is the same.

We would like a list data structure to permit the elements stored to be any fixed type. That is, instead of separately defining NumberList and BooleanList, we would like to just define List, and have it work for any element type.

Thus, List itself is not a type, but rather a function that takes a type (the type of the elements) and returns a type (the type of lists of that element type). So, if T is a type, then List<T> is the type of a list of Ts.

Terms to types

Some languages have a fixed-length array type. This is a type which is a bit like a list, but its length is fixed, and thus part of the type itself. Languages like C and Rust permit types like this.

For instance, in Rust, the definition

const A: [u32; 3] = [2, 4, 6];

defines A to be an array, with a fixed length of 3, of 32-bit unsigned integers.

This is a limited form of allowing terms in types, since here, the term 3 is used in the type [u32; 3].

However, Rust rejects the following function type:

fn foo(n: usize) -> [u32; n]

With the following error:

error[E0435]: attempt to use a non-constant value in a constant
 --> src/
1 | fn foo(n: usize) -> [u32; n]
  |        -                  ^
  |        |
  |        this would need to be a `const`

We can take the Rust compiler's suggestion and make n a "const parameter" (and also capitalize it, to conform to style guidelines):

fn foo<const N: usize>() -> [u32; N]

But now, because N is a const parameter, we can only pass values for it that are known at compile time.

Types like [u32; N] that contain, or "depend on", terms, are called dependent types. Not many programming languages fully support dependent types, likely due to their incredible expressive power. As seen in the example, Rust only permits limited usage of these types.

The lambda cube

To reiterate, the four varieties of functions are:

  1. From terms to terms
  2. From types to terms
  3. From types to types
  4. From terms to types

Most languages have term-term functions, but choose to allow or disallow the other three varieties of functions. There are three yes-or-no choices to make, and thus 23=82^3 = 8 possible configurations.

We may visualize the three choices as dimensions, and thus organize the possibilities into a cube. The vertices of the cube represent languages that arise from choosing combinations of allowing or disallowing the three varieties of function. All vertices on the cube allow for term-to-term functions.

The lambda cube

Some commonly-known vertices on the cube are shown below. Columns 1-4 correspond to the 4 varieties of function discussed.

λ ⁣\lambda\!\rightarrowSimply typed lambda calculus×××
λ2\lambda 2System FF××
λω\lambda \omegaSystem FωF\omega×
λC\lambda CCalculus of constructions

Once we reach the calculus of constructions (CoC), the distinction between types and terms somewhat disappears, since each may freely appear in both themselves and the other. Indeed, as powerful as the CoC is, it has a very sparse syntax of terms, fully described by the following context-free grammar:

t::= Propbase type Typetype of Prop xvariable t(t)application λ(x:t) tabstraction Π(x:t) tforall\begin{aligned} t ::= \ & \mathsf{Prop} && \text{base type} \\ | \ & \mathsf{Type} && \text{type of $\mathsf{Prop}$} \\ | \ & x && \text{variable} \\ | \ & t(t') && \text{application} \\ | \ & \lambda (x: t) \ t' && \text{abstraction} \\ | \ & \Pi (x: t) \ t' && \text{forall} \end{aligned}

There is no separate syntax for types in the CoC: all terms and types are represented with just the above syntax.

I wrote up an implementation of the CoC in Rust for edification.

The calculus of constructions serves as the foundation for many dependently-typed programming languages, like Coq. Using the CoC as a foundation, Coq is able to express and prove mathematical theorems like the four-color theorem.

It's rather remarkable to me that functions and variables, the most basic realization of the concept of "abstraction", can be so powerful in allowing all different types of language features. In the words of jez, on variables:

I think variables are just so cool!

And functions:

I think it's straight-up amazing that something so simple can at the same time be that powerful. Functions!