Luau is the primary programming language to place the ability of semantic subtyping within the fingers of hundreds of thousands 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 can be artifacts of the evaluation, and don’t correspond to errors which might happen at runtime. For instance, this system

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

studies a kind error which can’t 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 straight away confronted with a wall of spurious pink squiggles, there’s a robust incentive to instantly swap it off once more.

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

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

mode errs on the facet of false negatives.

Whereas inaccuracies are inevitable, we attempt to take away them each time attainable, since they end in 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 plenty of different comparable languages like TypeScript or Movement) is *subtyping*. Subtyping is used each time a variable is initialized or assigned to, and each time a operate 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()) 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 function, and it helps wealthy kind constructs like kind union (`T | U`

) and intersection (`T & U`

). For instance, `quantity?`

is applied as a union kind `(quantity | nil)`

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

.

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

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

This error is attributable to a failure of subtyping, the previous subtyping algorithm studies that `(quantity?) & (string?)`

isn’t a subtype of `nil`

. It is a false constructive, since `quantity & string`

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

is `nil`

.

That is a man-made instance, however there are actual points raised by creators attributable to the issues, for instance https://devforum.roblox.com/t/luau-recap-july-2021/1382101/5. At present, 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 change into extra widespread, even in code with no kind annotations.

This class of false positives not happens in Luau, as we’ve got moved from our previous method of *syntactic subtyping* to another known as *semantic subtyping*.

## Syntactic subtyping

AKA “what we did earlier than.”

Syntactic subtyping is a syntax-directed recursive algorithm. The attention-grabbing 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 among the`Tᵢ`

are subtypes of`U`

- Union L:
`(T₁ | … | Tⱼ)`

is a subtype of`U`

each time all the`Tᵢ`

are subtypes of`U`

- Intersection R:
`T`

is a subtype of`(U₁ & … & Uⱼ)`

each time`T`

is a subtype of all the`Uᵢ`

- Union R:
`T`

is a subtype of`(U₁ | … | Uⱼ)`

each time`T`

is a subtype of among 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 may be incorrect. 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.”

Quite than considering of subtyping as being syntax-directed, we first take into account its semantics, and later return to how the semantics is applied. 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 considered set inclusion.

For instance:

Sort | 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 keeping with semantic subtyping, `(quantity?) & (string?)`

is equal to `nil`

, however syntactic subtyping solely helps one route.

That is all superb and good, but when we need to use semantic subtyping in instruments, we’d like an algorithm, and it seems checking semantic subtyping is non-trivial.

## Semantic subtyping is difficult

NP-hard to be exact.

We are able to cut 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 coloration graph could be completed utilizing varieties:

kind Crimson = "pink" kind Blue = "blue" kind Shade = Crimson | Blue kind Coloring = (Shade) -> (Shade) -> (Shade) -> boolean kind Uncolorable = (Shade) -> (Shade) -> (Shade) -> false

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

and supertype `Coloring`

, as an overloaded operate which returns `false`

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

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

A triangle is analogous, however the finish factors additionally can’t have the identical coloration:

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

Now, `Triangle`

is a subtype of `Uncolorable`

, however `Line`

isn’t, 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 Complicated” error if the cache of varieties will get too massive.

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

## Sort normalization

The algorithm used to resolve semantic subtyping is *kind normalization*. Quite than being directed by syntax, we first rewrite varieties to be normalized, then test 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 operate kind (both
`by no means`

or an intersection of operate varieties) and so forth

As soon as varieties are normalized, it’s simple to test semantic subtyping.

Each kind could be normalized (sigh, with some technical restrictions round generic kind packs). The necessary 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 (advanced code went from typechecking in lower than a minute to operating in a single day). The explanation 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 inexpensive pointer equality test. Sort 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 test 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 completely different from what’s applied in Luau, as a result of it requires fashions to be *set-theoretic*, which requires that inhabitants of operate varieties “act like capabilities.” There are two the explanation why we drop this requirement.

**Firstly**, we normalize operate 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 operate:

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

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

Our normalization depends on rewriting away unions of operate 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 operate software `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 need to burden customers with the concept that operate software has a particular nook case, particularly since that nook case can solely come up in useless 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 fairly 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 widespread 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 type of typing refinement on *check varieties* like `string`

, `operate`

, `Half`

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

, which avoids the widespread case of common negated varieties.

## Prototyping and verification

In the course of the design of Luau’s semantic subtyping algorithm, there have been modifications made (for instance initially we thought we have been going to have the ability to use set-theoretic subtyping). Throughout this time of fast change, it was necessary to have the ability to iterate rapidly, so we initially applied a prototype fairly than leaping straight to a manufacturing implementation.

Validating the prototype was necessary, since subtyping algorithms can have surprising 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 delicate function interactions that will most likely have surfaced as difficult-to-fix bugs in manufacturing.

Prototyping isn’t excellent, for instance the primary points that we hit in manufacturing have been about efficiency and the C++ normal library, that are by no means going to be caught by a prototype. However the manufacturing implementation was in any other case pretty simple (or not less than as simple 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 operate purposes and operators
- Property entry on expressions of advanced kind
- Learn-only properties of tables
- Variables that change kind over time (aka typestates)

The search to take away spurious pink squiggles continues!

## Acknowledgments

Due to Giuseppe Castagna and Ben Greenman for useful feedback on drafts of this submit.

*Alan coordinates the design and implementation of the Luau kind system, which helps drive most of the options of improvement 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.*