{-# LANGUAGE PatternSynonyms #-} {-| This module defines the semi-ring of multiplicities, and associated functions. Multiplicities annotate arrow types to indicate the linearity of the arrow (in the sense of linear types). Mult is a type synonym for Type, used only when its kind is Multiplicity. To simplify dealing with multiplicities, functions such as mkMultMul perform simplifications such as Many * x = Many on the fly. -} module GHC.Core.Multiplicity ( Mult , pattern One , pattern Many , isMultMul , mkMultAdd , mkMultMul , mkMultSup , Scaled(..) , scaledMult , scaledThing , unrestricted , linear , tymult , irrelevantMult , mkScaled , scaledSet , scaleScaled , IsSubmult(..) , submult , mapScaledType) where import GHC.Prelude import GHC.Utils.Outputable import GHC.Core.TyCo.Rep import {-# SOURCE #-} GHC.Builtin.Types ( multMulTyCon ) import GHC.Core.Type import GHC.Builtin.Names (multMulTyConKey) import GHC.Types.Unique (hasKey) {- Note [Linear types] ~~~~~~~~~~~~~~~~~~~ This module is the entry point for linear types. The detailed design is in the _Linear Haskell_ article [https://arxiv.org/abs/1710.09756]. Other important resources in the linear types implementation wiki page [https://gitlab.haskell.org/ghc/ghc/wikis/linear-types/implementation], and the proposal [https://github.com/ghc-proposals/ghc-proposals/pull/111] which describes the concrete design at length. For the busy developer, though, here is a high-level view of linear types is the following: - Function arrows are annotated with a multiplicity (as defined by type `Mult` and its smart constructors in this module) - Because, as a type constructor, the type of function now has an extra argument, the notation (->) is no longer suitable. We named the function type constructor `FUN`. - (->) retains its backward compatible meaning: `(->) a b = a -> b`. To achieve this, `(->)` is defined as a type synonym to `FUN Many` (see below). - Multiplicities can be reified in Haskell as types of kind `GHC.Types.Multiplicity` - Ground multiplicity (that is, without a variable) can be `One` or `Many` (`Many` is generally rendered as ω in the scientific literature). Functions whose type is annotated with `One` are linear functions, functions whose type is annotated with `Many` are regular functions, often called “unrestricted” to contrast them with linear functions. - A linear function is defined as a function such that *if* its result is consumed exactly once, *then* its argument is consumed exactly once. You can think of “consuming exactly once” as evaluating a value in normal form exactly once (though not necessarily in one go). The _Linear Haskell_ article (see infra) has a more precise definition of “consuming exactly once”. - Data types can have unrestricted fields (the canonical example being the `Unrestricted` data type), then these don't need to be consumed for a value to be consumed exactly once. So consuming a value of type `Unrestricted` exactly once means forcing it at least once. - Why “at least once”? Because if `case u of { C x y -> f (C x y) }` is linear (provided `f` is a linear function). So we might as well have done `case u of { !z -> f z }`. So, we can observe constructors as many times as we want, and we are actually allowed to force the same thing several times because laziness means that we are really forcing a the value once, and observing its constructor several times. The type checker and the linter recognise some (but not all) of these multiple forces as indeed linear. Mostly just enough to support variable patterns. - Multiplicities form a semiring. - Multiplicities can also be variables and we can universally quantify over these variables. This is referred to as “multiplicity polymorphism”. Furthermore, multiplicity can be formal semiring expressions combining variables. - Contrary to the paper, the sum of two multiplicities is always `Many`. This will have to change, however, if we want to add a multiplicity for 0. Whether we want to is still debated. - Case expressions have a multiplicity annotation too. A case expression with multiplicity `One`, consumes its scrutinee exactly once (provided the entire case expression is consumed exactly once); whereas a case expression with multiplicity `Many` can consume its scrutinee as many time as it wishes (no matter how much the case expression is consumed). Note [Usages] ~~~~~~~~~~~~~ In the _Linear Haskell_ paper, you'll find typing rules such as these: Γ ⊢ f : A #π-> B Δ ⊢ u : A --------------------------- Γ + kΔ ⊢ f u : B If you read this as a type-checking algorithm going from the bottom up, this reads as: the algorithm has to find a split of some input context Ξ into an appropriate Γ and a Δ such as Ξ = Γ + kΔ, *and the multiplicities are chosen to make f and u typecheck*. This could be achieved by letting the typechecking of `f` use exactly the variable it needs, then passing the remainder, as `Delta` to the typechecking of u. But what does that mean if `x` is bound with multiplicity `p` (a variable) and `f` consumes `x` once? `Delta` would have to contain `x` with multiplicity `p-1`. It's not really clear how to make that works. In summary: bottom-up multiplicity checking forgoes addition and multiplication in favour of subtraction and division. And variables make the latter hard. The alternative is to read multiplicities from the top down: as an *output* from the typechecking algorithm, rather than an input. We call these output multiplicities Usages, to distinguish them from the multiplicities which come, as input, from the types of functions. Usages are checked for compatibility with multiplicity annotations using an ordering relation. In other words, the usage of x in the expression u is the smallest multiplicity which can be ascribed to x for u to typecheck. Usages are usually group in a UsageEnv, as defined in the UsageEnv module. So, in our function application example, the typechecking algorithm would receive usage environements f_ue from the typechecking of f, and u_ue from the typechecking of u. Then the output would be f_ue + (k * u_ue). Addition and scaling of usage environment is the pointwise extension of the semiring operations on multiplicities. Note [Zero as a usage] ~~~~~~~~~~~~~~~~~~~~~~ In the current presentation usages are not exactly multiplicities, because they can contain 0, and multiplicities can't. Why do we need a 0 usage? A function which doesn't use its argument will be required to annotate it with `Many`: \(x # Many) -> 0 However, we cannot replace absence with Many when computing usages compositionally: in (x, True) We expect x to have usage 1. But when computing the usage of x in True we would find that x is absent, hence has multiplicity Many. The final multiplicity would be One+Many = Many. Oops! Hence there is a usage Zero for absent variables. Zero is characterised by being the neutral element to usage addition. We may decide to add Zero as a multiplicity in the future. In which case, this distinction will go away. Note [Joining usages] ~~~~~~~~~~~~~~~~~~~~~ The usage of a variable is defined, in Note [Usages], as the minimum usage which can be ascribed to a variable. So what is the usage of x in case … of { p1 -> u -- usage env: u_ue ; p2 -> v } -- usage env: v_ue It must be the least upper bound, or _join_, of u_ue(x) and v_ue(x). So, contrary to a declarative presentation where the correct usage of x can be conjured out of thin air, we need to be able to compute the join of two multiplicities. Join is extended pointwise on usage environments. Note [Bottom as a usage] ~~~~~~~~~~~~~~~~~~~~~~ What is the usage of x in case … of {} Per usual linear logic, as well as the _Linear Haskell_ article, x can have every multiplicity. So we need a minimum usage _bottom_, which is also the neutral element for join. In fact, this is not such as nice solution, because it is not clear how to define sum and multiplication with bottom. We give reasonable definitions, but they are not complete (they don't respect the semiring laws, and it's possible to come up with examples of Core transformation which are not well-typed) A better solution would probably be to annotate case expressions with a usage environment, just like they are annotated with a type. Which, probably not coincidentally, is also primarily for empty cases. A side benefit of this approach is that the linter would not need to join multiplicities, anymore; hence would be closer to the presentation in the article. That's because it could use the annotation as the multiplicity for each branch. Note [Data constructors are linear by default] ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ Data constructors defined without -XLinearTypes (as well as data constructors defined with the Haskell 98 in all circumstances) have all their fields linear. That is, in data Maybe a = Nothing | Just a We have Just :: a %1 -> Just a The goal is to maximise reuse of types between linear code and traditional code. This is argued at length in the proposal and the article (links in Note [Linear Types]). Note [Polymorphisation of linear fields] ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ The choice in Note [Data constructors are linear by default] has an impact on backwards compatibility. Consider map Just We have map :: (a -> b) -> f a -> f b Just :: a %1 -> Just a Types don't match, we should get a type error. But this is legal Haskell 98 code! Bad! Bad! Bad! It could be solved with subtyping, but subtyping doesn't combine well with polymorphism. Instead, we generalise the type of Just, when used as term: Just :: forall {p}. a %p-> Just a This is solely a concern for higher-order code like this: when called fully applied linear constructors are more general than constructors with unrestricted fields. In particular, linear constructors can always be eta-expanded to their Haskell 98 type. This is explained in the paper (but there, we had a different strategy to resolve this type mismatch in higher-order code. It turned out to be insufficient, which is explained in the wiki page as well as the proposal). We only generalise linear fields this way: fields with multiplicity Many, or other multiplicity expressions are exclusive to -XLinearTypes, hence don't have backward compatibility implications. The implementation is described in Note [Linear fields generalization]. More details in the proposal. -} {- Note [Adding new multiplicities] ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ To add a new multiplicity, you need to: * Add the new type with Multiplicity kind * Update cases in mkMultAdd, mkMultMul, mkMultSup, submult, tcSubMult * Check supUE function that computes sup of a multiplicity and Zero -} isMultMul :: Mult -> Maybe (Mult, Mult) isMultMul ty | Just (tc, [x, y]) <- splitTyConApp_maybe ty , tc `hasKey` multMulTyConKey = Just (x, y) | otherwise = Nothing {- Note [Overapproximating multiplicities] ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ The functions mkMultAdd, mkMultMul, mkMultSup perform operations on multiplicities. They can return overapproximations: their result is merely guaranteed to be a submultiplicity of the actual value. They should be used only when an upper bound is acceptable. In most cases, they are used in usage environments (UsageEnv); in usage environments, replacing a usage with a larger one can only cause more programs to fail to typecheck. In future work, instead of approximating we might add type families and allow users to write types involving operations on multiplicities. In this case, we could enforce more invariants in Mult, for example, enforce that it is in the form of a sum of products, and even that the sumands and factors are ordered somehow, to have more equalities. -} -- With only two multiplicities One and Many, we can always replace -- p + q by Many. See Note [Overapproximating multiplicities]. mkMultAdd :: Mult -> Mult -> Mult mkMultAdd _ _ = Many mkMultMul :: Mult -> Mult -> Mult mkMultMul One p = p mkMultMul p One = p mkMultMul Many _ = Many mkMultMul _ Many = Many mkMultMul p q = mkTyConApp multMulTyCon [p, q] scaleScaled :: Mult -> Scaled a -> Scaled a scaleScaled m' (Scaled m t) = Scaled (m' `mkMultMul` m) t -- See Note [Joining usages] -- | @mkMultSup w1 w2@ returns a multiplicity such that @mkMultSup w1 -- w2 >= w1@ and @mkMultSup w1 w2 >= w2@. See Note [Overapproximating multiplicities]. mkMultSup :: Mult -> Mult -> Mult mkMultSup = mkMultMul -- Note: If you are changing this logic, check 'supUE' in UsageEnv as well. -- -- * Multiplicity ordering -- data IsSubmult = Submult -- Definitely a submult | Unknown -- Could be a submult, need to ask the typechecker deriving (Show, Eq) instance Outputable IsSubmult where ppr = text . show -- | @submult w1 w2@ check whether a value of multiplicity @w1@ is allowed where a -- value of multiplicity @w2@ is expected. This is a partial order. submult :: Mult -> Mult -> IsSubmult submult _ Many = Submult submult One One = Submult -- The 1 <= p rule submult One _ = Submult submult _ _ = Unknown