GHC/Core/Multiplicity.hs

    {-# 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