Luau is the primary programming language to place the ability of semantic subtyping within the palms of tens of millions of creators.

## Minimizing false positives

One of many points with kind error reporting in instruments just like the Script Evaluation widget in Roblox Studio is *false positives*. These are warnings which might be artifacts of the evaluation, and don’t correspond to errors which may happen at runtime. For instance, this system

native x = CFrame.new() native y if math.random() < 0.5 then y = CFrame.new() else y = Vector3.new() finish native z = x * y

studies a kind error which can not occur at runtime, since `CFrame`

helps multiplication by each `Vector3`

and `CFrame`

. (Its kind is `((CFrame, CFrame) -> CFrame) & ((CFrame, Vector3) -> Vector3)`

.)

False positives are particularly poor for onboarding new customers. If a type-curious creator switches on typechecking and is instantly confronted with a wall of spurious crimson squiggles, there’s a sturdy incentive to right away swap it off once more.

Inaccuracies in kind errors are inevitable, since it’s not possible to determine forward of time whether or not a runtime error will likely be triggered. Kind system designers have to decide on whether or not to stay with false positives or false negatives. In Luau that is decided by the mode: `strict`

mode errs on the aspect of false positives, and `nonstrict`

mode errs on the aspect of false negatives.

Whereas inaccuracies are inevitable, we attempt to take away them each time doable, since they lead to spurious errors, and imprecision in type-driven tooling like autocomplete or API documentation.

## Subtyping as a supply of false positives

One of many sources of false positives in Luau (and lots of different comparable languages like TypeScript or Circulate) is *subtyping*. Subtyping is used each time a variable is initialized or assigned to, and each time a perform is known as: the sort system checks that the kind of the expression is a subtype of the kind of the variable. For instance, if we add varieties to the above program

native x : CFrame = CFrame.new() native y : Vector3 | CFrame if math.random() < 0.5 then y = CFrame.new() else y = Vector3.new() finish native z : Vector3 | CFrame = x * y

then the sort system checks that the kind of `CFrame`

multiplication is a subtype of `(CFrame, Vector3 | CFrame) -> (Vector3 | CFrame)`

.

Subtyping is a really helpful characteristic, and it helps wealthy kind constructs like kind union (`T | U`

) and intersection (`T & U`

). For instance, `quantity?`

is carried out as a union kind `(quantity | nil)`

, inhabited by values which might be both numbers or `nil`

.

Sadly, the interplay of subtyping with intersection and union varieties can have odd outcomes. A easy (however moderately synthetic) case in older Luau was:

native x : (quantity?) & (string?) = nil native y : nil = nil y = x -- Kind '(quantity?) & (string?)' couldn't be transformed into 'nil' x = y

This error is brought on by a failure of subtyping, the outdated subtyping algorithm studies that `(quantity?) & (string?)`

is just not a subtype of `nil`

. This can be a false optimistic, since `quantity & string`

is uninhabited, so the one doable inhabitant of `(quantity?) & (string?)`

is `nil`

.

That is a man-made instance, however there are actual points raised by creators brought on by the issues, for instance https://devforum.roblox.com/t/luau-recap-july-2021/1382101/5. At the moment, these points largely have an effect on creators making use of refined kind system options, however as we make kind inference extra correct, union and intersection varieties will grow to be extra frequent, even in code with no kind annotations.

This class of false positives now not happens in Luau, as we’ve got moved from our outdated method of *syntactic subtyping* to an alternate referred to as *semantic subtyping*.

## Syntactic subtyping

AKA “what we did earlier than.”

Syntactic subtyping is a syntax-directed recursive algorithm. The fascinating circumstances to take care of intersection and union varieties are:

- Reflexivity:
`T`

is a subtype of`T`

- Intersection L:
`(T₁ & … & Tⱼ)`

is a subtype of`U`

each time a number of the`Tᵢ`

are subtypes of`U`

- Union L:
`(T₁ | … | Tⱼ)`

is a subtype of`U`

each time the entire`Tᵢ`

are subtypes of`U`

- Intersection R:
`T`

is a subtype of`(U₁ & … & Uⱼ)`

each time`T`

is a subtype of the entire`Uᵢ`

- Union R:
`T`

is a subtype of`(U₁ | … | Uⱼ)`

each time`T`

is a subtype of a number of the`Uᵢ`

.

For instance:

- By Reflexivity:
`nil`

is a subtype of`nil`

- so by Union R:
`nil`

is a subtype of`quantity?`

- and:
`nil`

is a subtype of`string?`

- so by Intersection R:
`nil`

is a subtype of`(quantity?) & (string?)`

.

Yay! Sadly, utilizing these guidelines:

`quantity`

isn’t a subtype of`nil`

- so by Union L:
`(quantity?)`

isn’t a subtype of`nil`

- and:
`string`

isn’t a subtype of`nil`

- so by Union L:
`(string?)`

isn’t a subtype of`nil`

- so by Intersection L:
`(quantity?) & (string?)`

isn’t a subtype of`nil`

.

That is typical of syntactic subtyping: when it returns a “sure” end result, it’s right, however when it returns a “no” end result, it is likely to be flawed. The algorithm is a *conservative approximation*, and since a “no” end result can result in kind errors, it is a supply of false positives.

## Semantic subtyping

AKA “what we do now.”

Somewhat than pondering of subtyping as being syntax-directed, we first contemplate its semantics, and later return to how the semantics is carried out. For this, we undertake semantic subtyping:

- The semantics of a kind is a set of values.
- Intersection varieties are considered intersections of units.
- Union varieties are considered unions of units.
- Subtyping is regarded as set inclusion.

For instance:

Kind | Semantics |
---|---|

`quantity` |
{ 1, 2, 3, … } |

`string` |
{ “foo”, “bar”, … } |

`nil` |
{ nil } |

`quantity?` |
{ nil, 1, 2, 3, … } |

`string?` |
{ nil, “foo”, “bar”, … } |

`(quantity?) & (string?)` |
{ nil, 1, 2, 3, … } ∩ { nil, “foo”, “bar”, … } = { nil } |

and since subtypes are interpreted as set inclusions:

Subtype | Supertype | As a result of |
---|---|---|

`nil` |
`quantity?` |
{ nil } ⊆ { nil, 1, 2, 3, … } |

`nil` |
`string?` |
{ nil } ⊆ { nil, “foo”, “bar”, … } |

`nil` |
`(quantity?) & (string?)` |
{ nil } ⊆ { nil } |

`(quantity?) & (string?)` |
`nil` |
{ nil } ⊆ { nil } |

So in line with semantic subtyping, `(quantity?) & (string?)`

is equal to `nil`

, however syntactic subtyping solely helps one route.

That is all tremendous and good, but when we wish to use semantic subtyping in instruments, we want an algorithm, and it seems checking semantic subtyping is non-trivial.

## Semantic subtyping is tough

NP-hard to be exact.

We are able to scale back graph coloring to semantic subtyping by coding up a graph as a Luau kind such that checking subtyping on varieties has the identical end result as checking for the impossibility of coloring the graph

For instance, coloring a three-node, two colour graph could be accomplished utilizing varieties:

kind Crimson = "crimson" kind Blue = "blue" kind Coloration = Crimson | Blue kind Coloring = (Coloration) -> (Coloration) -> (Coloration) -> boolean kind Uncolorable = (Coloration) -> (Coloration) -> (Coloration) -> false

Then a graph could be encoded as an overload perform kind with subtype `Uncolorable`

and supertype `Coloring`

, as an overloaded perform which returns `false`

when a constraint is violated. Every overload encodes one constraint. For instance a line has constraints saying that adjoining nodes can not have the identical colour:

kind Line = Coloring & ((Crimson) -> (Crimson) -> (Coloration) -> false) & ((Blue) -> (Blue) -> (Coloration) -> false) & ((Coloration) -> (Crimson) -> (Crimson) -> false) & ((Coloration) -> (Blue) -> (Blue) -> false)

A triangle is comparable, however the finish factors additionally can not have the identical colour:

kind Triangle = Line & ((Crimson) -> (Coloration) -> (Crimson) -> false) & ((Blue) -> (Coloration) -> (Blue) -> false)

Now, `Triangle`

is a subtype of `Uncolorable`

, however `Line`

is just not, for the reason that line could be 2-colored. This may be generalized to any finite graph with any finite variety of colours, and so subtype checking is NP-hard.

We take care of this in two methods:

- we cache varieties to cut back reminiscence footprint, and
- hand over with a “Code Too Advanced” error if the cache of varieties will get too giant.

Hopefully this doesn’t come up in follow a lot. There may be good proof that points like this don’t come up in follow from expertise with kind methods like that of Normal ML, which is EXPTIME-complete, however in follow it’s a must to exit of your approach to code up Turing Machine tapes as varieties.

## Kind normalization

The algorithm used to determine semantic subtyping is *kind normalization*. Somewhat than being directed by syntax, we first rewrite varieties to be normalized, then verify subtyping on normalized varieties.

A normalized kind is a union of:

- a normalized nil kind (both
`by no means`

or`nil`

) - a normalized quantity kind (both
`by no means`

or`quantity`

) - a normalized boolean kind (both
`by no means`

or`true`

or`false`

or`boolean`

) - a normalized perform kind (both
`by no means`

or an intersection of perform varieties) and so forth

As soon as varieties are normalized, it’s easy to verify semantic subtyping.

Each kind could be normalized (sigh, with some technical restrictions round generic kind packs). The essential steps are:

- eradicating intersections of mismatched primitives, e.g.
`quantity & bool`

is changed by`by no means`

, and - eradicating unions of capabilities, e.g.
`((quantity?) -> quantity) | ((string?) -> string)`

is changed by`(nil) -> (quantity | string)`

.

For instance, normalizing `(quantity?) & (string?)`

removes `quantity & string`

, so all that’s left is `nil`

.

Our first try at implementing kind normalization utilized it liberally, however this resulted in dreadful efficiency (complicated code went from typechecking in lower than a minute to operating in a single day). The rationale for that is annoyingly easy: there’s an optimization in Luau’s subtyping algorithm to deal with reflexivity (`T`

is a subtype of `T`

) that performs an affordable pointer equality verify. Kind normalization can convert pointer-identical varieties into semantically-equivalent (however not pointer-identical) varieties, which considerably degrades efficiency.

Due to these efficiency points, we nonetheless use syntactic subtyping as our first verify for subtyping, and solely carry out kind normalization if the syntactic algorithm fails. That is sound, as a result of syntactic subtyping is a conservative approximation to semantic subtyping.

## Pragmatic semantic subtyping

Off-the-shelf semantic subtyping is barely totally different from what’s carried out in Luau, as a result of it requires fashions to be *set-theoretic*, which requires that inhabitants of perform varieties “act like capabilities.” There are two explanation why we drop this requirement.

**Firstly**, we normalize perform varieties to an intersection of capabilities, for instance a horrible mess of unions and intersections of capabilities:

((quantity?) -> quantity?) | (((quantity) -> quantity) & ((string?) -> string?))

normalizes to an overloaded perform:

((quantity) -> quantity?) & ((nil) -> (quantity | string)?)

Set-theoretic semantic subtyping doesn’t help this normalization, and as an alternative normalizes capabilities to *disjunctive regular kind* (unions of intersections of capabilities). We don’t do that for ergonomic causes: overloaded capabilities are idiomatic in Luau, however DNF is just not, and we don’t wish to current customers with such non-idiomatic varieties.

Our normalization depends on rewriting away unions of perform varieties:

((A) -> B) | ((C) -> D) → (A & C) -> (B | D)

This normalization is sound in our mannequin, however not in set-theoretic fashions.

**Secondly**, in Luau, the kind of a perform utility `f(x)`

is `B`

if `f`

has kind `(A) -> B`

and `x`

has kind `A`

. Unexpectedly, this isn’t at all times true in set-theoretic fashions, attributable to uninhabited varieties. In set-theoretic fashions, if `x`

has kind `by no means`

then `f(x)`

has kind `by no means`

. We don’t wish to burden customers with the concept perform utility has a particular nook case, particularly since that nook case can solely come up in lifeless code.

In set-theoretic fashions, `(by no means) -> A`

is a subtype of `(by no means) -> B`

, it doesn’t matter what `A`

and `B`

are. This isn’t true in Luau.

For these two causes (that are largely about ergonomics moderately than something technical) we drop the set-theoretic requirement, and use *pragmatic* semantic subtyping.

## Negation varieties

The opposite distinction between Luau’s kind system and off-the-shelf semantic subtyping is that Luau doesn’t help all negated varieties.

The frequent case for wanting negated varieties is in typechecking conditionals:

-- initially x has kind T if kind(x) == "string" then -- on this department x has kind T & string else -- on this department x has kind T & ~string finish

This makes use of a negated kind `~string`

inhabited by values that aren’t strings.

In Luau, we solely enable this sort of typing refinement on *take a look at varieties* like `string`

, `perform`

, `Half`

and so forth, and *not* on structural varieties like `(A) -> B`

, which avoids the frequent case of normal negated varieties.

## Prototyping and verification

In the course of the design of Luau’s semantic subtyping algorithm, there have been adjustments made (for instance initially we thought we had been going to have the ability to use set-theoretic subtyping). Throughout this time of speedy change, it was essential to have the ability to iterate shortly, so we initially carried out a prototype moderately than leaping straight to a manufacturing implementation.

Validating the prototype was essential, since subtyping algorithms can have sudden nook circumstances. Because of this, we adopted Agda because the prototyping language. In addition to supporting unit testing, Agda helps mechanized verification, so we’re assured within the design.

The prototype doesn’t implement all of Luau, simply the practical subset, however this was sufficient to find refined characteristic interactions that might in all probability have surfaced as difficult-to-fix bugs in manufacturing.

Prototyping is just not good, for instance the principle points that we hit in manufacturing had been about efficiency and the C++ commonplace library, that are by no means going to be caught by a prototype. However the manufacturing implementation was in any other case pretty easy (or not less than as easy as a 3kLOC change could be).

## Subsequent steps

Semantic subtyping has eliminated one supply of false positives, however we nonetheless have others to trace down:

- Overloaded perform purposes and operators
- Property entry on expressions of complicated kind
- Learn-only properties of tables
- Variables that change kind over time (aka typestates)

The search to take away spurious crimson squiggles continues!

## Acknowledgments

Because of Giuseppe Castagna and Ben Greenman for useful feedback on drafts of this put up.

*Alan coordinates the design and implementation of the Luau kind system, which helps drive most of the options of growth in Roblox Studio. Dr. Jeffrey has over 30 years of expertise with analysis in programming languages, has been an energetic member of quite a few open-source software program tasks, and holds a DPhil from the College of Oxford, England.*