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//! Traits used to represent [lattices] for use as the domain of a dataflow analysis. //! //! # Overview //! //! The most common lattice is a powerset of some set `S`, ordered by [set inclusion]. The [Hasse //! diagram] for the powerset of a set with two elements (`X` and `Y`) is shown below. Note that //! distinct elements at the same height in a Hasse diagram (e.g. `{X}` and `{Y}`) are //! *incomparable*, not equal. //! //! ```text //! {X, Y} <- top //! / \ //! {X} {Y} //! \ / //! {} <- bottom //! //! ``` //! //! The defining characteristic of a lattice—the one that differentiates it from a [partially //! ordered set][poset]—is the existence of a *unique* least upper and greatest lower bound for //! every pair of elements. The lattice join operator (`∨`) returns the least upper bound, and the //! lattice meet operator (`∧`) returns the greatest lower bound. Types that implement one operator //! but not the other are known as semilattices. Dataflow analysis only uses the join operator and //! will work with any join-semilattice, but both should be specified when possible. //! //! ## `PartialOrd` //! //! Given that they represent partially ordered sets, you may be surprised that [`JoinSemiLattice`] //! and [`MeetSemiLattice`] do not have [`PartialOrd`][std::cmp::PartialOrd] as a supertrait. This //! is because most standard library types use lexicographic ordering instead of set inclusion for //! their `PartialOrd` impl. Since we do not actually need to compare lattice elements to run a //! dataflow analysis, there's no need for a newtype wrapper with a custom `PartialOrd` impl. The //! only benefit would be the ability to check that the least upper (or greatest lower) bound //! returned by the lattice join (or meet) operator was in fact greater (or lower) than the inputs. //! //! [lattices]: https://en.wikipedia.org/wiki/Lattice_(order) //! [set inclusion]: https://en.wikipedia.org/wiki/Subset //! [Hasse diagram]: https://en.wikipedia.org/wiki/Hasse_diagram //! [poset]: https://en.wikipedia.org/wiki/Partially_ordered_set use rustc_index::bit_set::BitSet; use rustc_index::vec::{Idx, IndexVec}; /// A [partially ordered set][poset] that has a [least upper bound][lub] for any pair of elements /// in the set. /// /// [lub]: https://en.wikipedia.org/wiki/Infimum_and_supremum /// [poset]: https://en.wikipedia.org/wiki/Partially_ordered_set pub trait JoinSemiLattice: Eq { /// Computes the least upper bound of two elements, storing the result in `self` and returning /// `true` if `self` has changed. /// /// The lattice join operator is abbreviated as `∨`. fn join(&mut self, other: &Self) -> bool; } /// A [partially ordered set][poset] that has a [greatest lower bound][glb] for any pair of /// elements in the set. /// /// Dataflow analyses only require that their domains implement [`JoinSemiLattice`], not /// `MeetSemiLattice`. However, types that will be used as dataflow domains should implement both /// so that they can be used with [`Dual`]. /// /// [glb]: https://en.wikipedia.org/wiki/Infimum_and_supremum /// [poset]: https://en.wikipedia.org/wiki/Partially_ordered_set pub trait MeetSemiLattice: Eq { /// Computes the greatest lower bound of two elements, storing the result in `self` and /// returning `true` if `self` has changed. /// /// The lattice meet operator is abbreviated as `∧`. fn meet(&mut self, other: &Self) -> bool; } /// A `bool` is a "two-point" lattice with `true` as the top element and `false` as the bottom: /// /// ```text /// true /// | /// false /// ``` impl JoinSemiLattice for bool { fn join(&mut self, other: &Self) -> bool { if let (false, true) = (*self, *other) { *self = true; return true; } false } } impl MeetSemiLattice for bool { fn meet(&mut self, other: &Self) -> bool { if let (true, false) = (*self, *other) { *self = false; return true; } false } } /// A tuple (or list) of lattices is itself a lattice whose least upper bound is the concatenation /// of the least upper bounds of each element of the tuple (or list). /// /// In other words: /// (A₀, A₁, ..., Aₙ) ∨ (B₀, B₁, ..., Bₙ) = (A₀∨B₀, A₁∨B₁, ..., Aₙ∨Bₙ) impl<I: Idx, T: JoinSemiLattice> JoinSemiLattice for IndexVec<I, T> { fn join(&mut self, other: &Self) -> bool { assert_eq!(self.len(), other.len()); let mut changed = false; for (a, b) in self.iter_mut().zip(other.iter()) { changed |= a.join(b); } changed } } impl<I: Idx, T: MeetSemiLattice> MeetSemiLattice for IndexVec<I, T> { fn meet(&mut self, other: &Self) -> bool { assert_eq!(self.len(), other.len()); let mut changed = false; for (a, b) in self.iter_mut().zip(other.iter()) { changed |= a.meet(b); } changed } } /// A `BitSet` represents the lattice formed by the powerset of all possible values of /// the index type `T` ordered by inclusion. Equivalently, it is a tuple of "two-point" lattices, /// one for each possible value of `T`. impl<T: Idx> JoinSemiLattice for BitSet<T> { fn join(&mut self, other: &Self) -> bool { self.union(other) } } impl<T: Idx> MeetSemiLattice for BitSet<T> { fn meet(&mut self, other: &Self) -> bool { self.intersect(other) } } /// The counterpart of a given semilattice `T` using the [inverse order]. /// /// The dual of a join-semilattice is a meet-semilattice and vice versa. For example, the dual of a /// powerset has the empty set as its top element and the full set as its bottom element and uses /// set *intersection* as its join operator. /// /// [inverse order]: https://en.wikipedia.org/wiki/Duality_(order_theory) #[derive(Clone, Copy, Debug, PartialEq, Eq)] pub struct Dual<T>(pub T); impl<T> std::borrow::Borrow<T> for Dual<T> { fn borrow(&self) -> &T { &self.0 } } impl<T> std::borrow::BorrowMut<T> for Dual<T> { fn borrow_mut(&mut self) -> &mut T { &mut self.0 } } impl<T: MeetSemiLattice> JoinSemiLattice for Dual<T> { fn join(&mut self, other: &Self) -> bool { self.0.meet(&other.0) } } impl<T: JoinSemiLattice> MeetSemiLattice for Dual<T> { fn meet(&mut self, other: &Self) -> bool { self.0.join(&other.0) } } /// Extends a type `T` with top and bottom elements to make it a partially ordered set in which no /// value of `T` is comparable with any other. A flat set has the following [Hasse diagram]: /// /// ```text /// top /// / / \ \ /// all possible values of `T` /// \ \ / / /// bottom /// ``` /// /// [Hasse diagram]: https://en.wikipedia.org/wiki/Hasse_diagram #[derive(Clone, Copy, Debug, PartialEq, Eq)] pub enum FlatSet<T> { Bottom, Elem(T), Top, } impl<T: Clone + Eq> JoinSemiLattice for FlatSet<T> { fn join(&mut self, other: &Self) -> bool { let result = match (&*self, other) { (Self::Top, _) | (_, Self::Bottom) => return false, (Self::Elem(a), Self::Elem(b)) if a == b => return false, (Self::Bottom, Self::Elem(x)) => Self::Elem(x.clone()), _ => Self::Top, }; *self = result; true } } impl<T: Clone + Eq> MeetSemiLattice for FlatSet<T> { fn meet(&mut self, other: &Self) -> bool { let result = match (&*self, other) { (Self::Bottom, _) | (_, Self::Top) => return false, (Self::Elem(ref a), Self::Elem(ref b)) if a == b => return false, (Self::Top, Self::Elem(ref x)) => Self::Elem(x.clone()), _ => Self::Bottom, }; *self = result; true } }