Semantic Subtyping in Luau – Roblox Weblog

Luau is the primary programming language to place the ability of semantic subtyping within the arms of thousands and 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 are 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()) then
  y = CFrame.new()
else
  y = Vector3.new()
finish
native z = x * y

experiences a sort 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 straight away confronted with a wall of spurious crimson squiggles, there’s a sturdy incentive to instantly swap it off once more.

Inaccuracies in kind errors are inevitable, since it’s not possible to resolve forward of time whether or not a runtime error will likely be triggered. Kind system designers have to decide on whether or not to dwell 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 potential, 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 related languages like TypeScript or Circulation) is subtyping. Subtyping is used each time a variable is initialized or assigned to, and each time a operate 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 sorts 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 are both numbers or nil.

Sadly, the interplay of subtyping with intersection and union sorts 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 -- Kind '(quantity?) & (string?)' couldn't be transformed into 'nil'
x = y

This error is attributable to a failure of subtyping, the outdated subtyping algorithm experiences that (quantity?) & (string?) shouldn’t be a subtype of nil. This can be a false constructive, since quantity & string is uninhabited, so the one potential 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 sorts will turn into 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 another 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 cope with intersection and union sorts are:

• Reflexivity: T is a subtype of T
• Intersection L: (T₁ & … & Tⱼ) is a subtype of U each time a few 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 few 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 appropriate, however when it returns a “no” end result, it could be improper. 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.”

Slightly than pondering of subtyping as being syntax-directed, we first think about its semantics, and later return to how the semantics is applied. For this, we undertake semantic subtyping:

• The semantics of a sort is a set of values.
• Intersection sorts are regarded as intersections of units.
• Union sorts are regarded as unions of units.
• Subtyping is considered set inclusion.

For instance:

and since subtypes are interpreted as set inclusions:

So in line with semantic subtyping, (quantity?) & (string?) is equal to nil, however syntactic subtyping solely helps one path.

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

For instance, coloring a three-node, two coloration graph will be accomplished utilizing sorts:

kind Pink = "crimson"
kind Blue = "blue"
kind Shade = Pink | Blue
kind Coloring = (Shade) -> (Shade) -> (Shade) -> boolean
kind Uncolorable = (Shade) -> (Shade) -> (Shade) -> false

Then a graph will 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 not have the identical coloration:

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

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

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

Now, Triangle is a subtype of Uncolorable, however Line shouldn’t be, for the reason that line will 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 sorts to cut back reminiscence footprint, and
• surrender with a “Code Too Advanced” error if the cache of sorts will get too giant.

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

Kind normalization

The algorithm used to resolve semantic subtyping is kind normalization. Slightly than being directed by syntax, we first rewrite sorts to be normalized, then verify subtyping on normalized sorts.

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 sorts) and so forth

As soon as sorts are normalized, it’s simple to verify semantic subtyping.

Each kind will 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 features, 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 working in a single day). The rationale for that is annoyingly easy: there may be an optimization in Luau’s subtyping algorithm to deal with reflexivity (T is a subtype of T) that performs an inexpensive pointer equality verify. Kind normalization can convert pointer-identical sorts into semantically-equivalent (however not pointer-identical) sorts, 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 applied in Luau, as a result of it requires fashions to be set-theoretic, which requires that inhabitants of operate sorts “act like features.” There are two the explanation why we drop this requirement.

Firstly, we normalize operate sorts to an intersection of features, for instance a horrible mess of unions and intersections of features:

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

normalizes to an overloaded operate:

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

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

Our normalization depends on rewriting away unions of operate sorts:

((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 utility 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, as a consequence of uninhabited sorts. 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 operate 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 fairly than something technical) we drop the set-theoretic requirement, and use pragmatic semantic subtyping.

Negation sorts

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

The frequent case for wanting negated sorts 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 type of typing refinement on take a look at sorts like string, operate, Half and so forth, and not on structural sorts like (A) -> B, which avoids the frequent case of normal negated sorts.

Prototyping and verification

Through 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 necessary to have the ability to iterate shortly, so we initially applied a prototype fairly than leaping straight to a manufacturing implementation.

Validating the prototype was necessary, since subtyping algorithms can have sudden nook circumstances. 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 may most likely have surfaced as difficult-to-fix bugs in manufacturing.

Prototyping shouldn’t be excellent, for instance the principle points that we hit in manufacturing had 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 simple (or at the least as simple as a 3kLOC change will 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 hunt 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 lots of the options of growth 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 tasks, and holds a DPhil from the College of Oxford, England.