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 are 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

experiences a sort 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 instantly confronted with a wall of spurious purple squiggles, there’s a sturdy incentive to instantly change it off once more.

Inaccuracies in kind errors are inevitable, since it’s unattainable to resolve forward of time whether or not a runtime error shall 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 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 attainable, 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 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 perform is named: 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 carried out as a union kind `(quantity | nil)`

, inhabited by values which are both numbers or `nil`

.

Sadly, the interplay of subtyping with intersection and union varieties can have odd outcomes. A easy (however relatively 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 brought on by a failure of subtyping, the outdated subtyping algorithm experiences that `(quantity?) & (string?)`

will not be a subtype of `nil`

. This can be a false optimistic, 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 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 widespread, even in code with no kind annotations.

This class of false positives now not happens in Luau, as we’ve moved from our outdated method of *syntactic subtyping* to another referred to as *semantic subtyping*.

## Syntactic subtyping

AKA “what we did earlier than.”

Syntactic subtyping is a syntax-directed recursive algorithm. The attention-grabbing instances to cope with 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 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 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” consequence, it’s appropriate, however when it returns a “no” consequence, it may be unsuitable. The algorithm is a *conservative approximation*, and since a “no” consequence can result in kind errors, this can be a supply of false positives.

## Semantic subtyping

AKA “what we do now.”

Quite than considering 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 sort is a set of values.
- Intersection varieties are regarded as intersections of units.
- Union varieties are regarded as unions of units.
- Subtyping is regarded as 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 accordance with semantic subtyping, `(quantity?) & (string?)`

is equal to `nil`

, however syntactic subtyping solely helps one course.

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

## Semantic subtyping is tough

NP-hard to be exact.

We will 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 consequence as checking for the impossibility of coloring the graph

For instance, coloring a three-node, two shade graph might be completed utilizing varieties:

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

Then a graph might 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’t have the identical shade:

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 shade:

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

Now, `Triangle`

is a subtype of `Uncolorable`

, however `Line`

will not be, for the reason that line might 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 cope with this in two methods:

- we cache varieties to scale back reminiscence footprint, and
- surrender with a “Code Too Complicated” error if the cache of varieties will get too massive.

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 techniques like that of Normal ML, which is EXPTIME-complete, however in follow it’s a must to exit of your option 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 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 test semantic subtyping.

Each kind might 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 (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 affordable 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 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 the 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 assist 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 will not be, 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 software `f(x)`

is `B`

if `f`

has kind `(A) -> B`

and `x`

has kind `A`

. Unexpectedly, this isn’t all the time 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 that perform software 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 relatively 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 assist 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 permit this sort of typing refinement on *check varieties* like `string`

, `perform`

, `Half`

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

, which avoids the widespread case of basic negated varieties.

## Prototyping and verification

Throughout the design of Luau’s semantic subtyping algorithm, there have been adjustments made (for instance initially we thought we have 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 relatively than leaping straight to a manufacturing implementation.

Validating the prototype was essential, since subtyping algorithms can have sudden nook instances. For that reason, 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 useful subset, however this was sufficient to find refined function interactions that will in all probability have surfaced as difficult-to-fix bugs in manufacturing.

Prototyping will not be good, for instance the principle points that we hit in manufacturing have been about efficiency and the C++ customary 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 no less than as easy as a 3kLOC change might be).

## Subsequent steps

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

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

The hunt to take away spurious purple 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 lots of the options of improvement in Roblox Studio. Dr. Jeffrey has over 30 years of expertise with analysis in programming languages, has been an lively member of quite a few open-source software program initiatives, and holds a DPhil from the College of Oxford, England.*