Type system

Numbat is a language with a special type system that treats physical dimensions as types. A type checker infers types for every expression in the program and ensures that everything is correct in terms of physical dimensions, which implies correctness in terms of physical units. For example, the expression 2 meter has a type of Length. The expression 3 inch also has a type of Length. The combined expression 2 meter + 3 inch is therefore well-typed. On the other hand, 2 meter + 3 second is ill-typed, as 3 second is of type Time.

The type system is static which means that the correctness of a Numbat program is verified before the program starts executing. Note that certain runtime errors (like division-by-zero) can still occur.

Algebra of types

Types in Numbat can be combined in various ways to produce new types. In its most general form, a type can be thought of as a product of physical (base) dimensions \( D_k \) with exponents \( \alpha_k \in \mathbb{Q} \): \[ \prod_k D_k^{\alpha_k} \] For example, the type Energy can be represented as Mass¹ × Length² × Time⁻².

Multiplication

This naturally allows us to multiply types (by combining the factors of both products into a single product). We can use the * operator to construct types for physical dimensions that are products of two or more (base) dimensions. For example:

dimension Time
dimension Current
dimension ElectricCharge = Current * Time

Exponentiation

We can also raise units to arbitrary powers \( n \in \mathbb{Q} \), by simply multiplying each \( \alpha_k \) with \( n \). The syntax uses the ^ exponentiation operator:

dimension Length
dimension Volume = Length^3

dimension Time
dimension Frequency = Time^(-1)

Division

Once we have multiplication and exponentiation, we can define the division of two types as

TypeA / TypeB ≡ TypeA * TypeB^(-1)

This is mostly for convenience. It allows us to write definitions like

dimension Power = Energy / Time

Note: When we talk about products of types in this section, we mean actual, literal products. Type theory also has the notion of product types which denote something else: compound types — like tuples or structs — that are built by combining two or more types. If we think of types in terms of the sets of all possible values that they represent, then product types represent the Cartesian product of those.

Type inference and type annotations

The type checker can infer the types of (most) expressions without explicitly declaring them. For example, the following definition does not mention any types:

let E_pot = 80 kg × 9.8 m/s² × 5 m

However, it is often helpful to specify the type anyway. This way, we can make sure that no mistakes were made:

let E_pot: Energy = 80 kg × 9.8 m/s² × 5 m

The type checker will compare the inferred type with the specified type and raise an error in case of inconsistency.

Function definitions also allow for type annotations, both for the parameters as well as the return type. The following example shows a function that takes a quantity of type Length and returns a Pressure:

let p0: Pressure = 101325 Pa
let t0: Temperature = 288.15 K

let gradient = 0.65 K / 100 m

fn air_pressure(height: Length) -> Pressure = p0 · (1 - gradient · height / t0)^5.255

Generic types

Numbat’s type system also supports generic types (type polymorphism). These can be used for functions that work regardless of the physical dimension of the argument(s). For example, the type signature of the absolute value function is given by

fn abs<D>(x: D) -> D

where the angle brackets after the function name introduce new type parameters (D). This can be read as: abs takes an arbitrary physical quantity of dimension D and returns a quantity of the same physical dimension D.

As a more interesting example, we can look at the sqrt function. Its type signature can be written as

fn sqrt<D>(x: D^2) -> D

Alternatively, it could also be specified as fn sqrt<D>(x: D) -> D^(1/2).

Limitations

The static type system also has some limitations. Let’s look at an exponentiation expression like

expr1 ^ expr2

where expr1 and expr2 are arbitrary expressions. In order for that expression to properly type check, the type of expr2 must be Scalar — something like 2^meter does not make any sense. If the type of expr1 is also Scalar, everything is well and the type of the total expression is also Scalar. An example for this trivial case is an expression like e^(-x²/σ²). As long as the type of x is the same as the type of σ, this is fine.

A more interesting case arises if expr1 is dimensionfull, as in meter^3. Here, things become difficult: in order to compute the type of the total expression expr1 ^ expr2, we need to know the value of expr2. For the meter^3 example, the answer is Length^3. This seems straightforward. However, the syntax of the language allows arbitrary expressions in the exponent. This is important to support use cases like the above e^(-x²/σ²). But it poses a problem for the type checker. In order to compute the type of expr1 ^ expr2, we need to fully evaluate expr2 at compile time. This is not going to work in general. Just think of a hypothetical expression like meter^f() where f() could do anything. Maybe even get some input from the user at runtime.

Numbat’s solution to this problem looks like this: If expr1 is not dimensionless, we restrict expr2 to a small subset of allowed operations that can be fully evaluated at compile time (similar to constexpr expressions in C++, const expressions in Rust, etc). Expressions like meter^(2 * (2 + 1) / 3) are completely fine and can be typechecked (Length^2), but things like function calls are not allowed and will lead to a compile time error.

To summarize: Given an exponentiation expression like expr1 ^ expr2, the type checker requires that:

• expr2 is of type Scalar
• One of the following:
  • expr1 is also of type Scalar
  • expr2 can be evaluated at compile time and yields a rational number.

Remark

We would probably need to enter the world of dependent types if we wanted to fully support exponentiation expressions without the limitations above. For example, consider the function f(x, n) = x^n. The return type of that function depends on the value of the parameter n.