# Crate lattices

Expand description

# The lattices Crate

The lattices crate provides ergonomic and composable lattice types. You can also implement custom lattices via a few simple traits.

Lattices are an incredibly powerful mathematical concept which can greatly simplify the trickiness of distributed computing. They align very well with the reality of what happens physically in a distributed system: messages can always arrive out-of-order or duplicated. But if that data is represented as lattices then all machines will always reach the same end result simply by merging the data together. One popular way that lattices are currently used in distributed systems is as the data underlying Conflict-free Replicated Data Types (CRDTs).

Lattices also allow us to harness the power of the CALM Theorem: “a program has a consistent, coordination-free distributed implementation if and only if it is monotonic.” Lattice state is always monotonic, meaning any part of a distributed system built on lattice state can be freely distributed with no coordination overhead. The goal of the Hydro Project is to allow users to write programs that automatically scale and distribute effortlessly.

For more information on the underlying mathematics of lattices and monotonicity, take a look at Lattice Math section of the Hydroflow Book and Section 2 of the Hydroflow Thesis (2021).

Take a look at the lattice rustdocs.

### §Lattices

lattices provides implementations of common lattice types:

*Special implementations which do not obey all lattice properties but are still useful under certain circumstances.

### §Traits

A type becomes a lattice by implementing one or more traits starting with Merge. These traits are already implemented for all the provided lattice types.

#### §Merge

The main trait is Merge, which defines a lattice merge function (AKA “join” or “least upper bound”). Implementors must define the Merge::merge method which does a merge in-place into &mut self. The method must return true if self was modified (i.e. the value got larger in the lattice partial order) and false otherwise (i.e. other was smaller than self). The Merge::merge_owned function, which merges two owned values, is provided.

The merge method must be associative, commutative, and idempotent. This is not checked by the compiler, but the implementor can use the test::check_lattice_properties method to spot-check these properties on a collection of values.

#### §PartialOrd, LatticeOrd, and NaiveLatticeOrd

Rust already has a trait for partial orders, PartialOrd, which should be implemented on lattice types. However that trait is not specific to lattice partial orders, therefore we provide theLatticeOrd<Rhs>: PartialOrd<Rhs> marker trait to denote when a PartialOrd implementation is a lattice partial order. LatticeOrd must always agree with the Merge function.

Additionally, the sealed NaiveLatticeOrd trait is provided on all lattice types that implement Merge and Clone. This trait provides a naive_cmp method which derives a lattice order from the Merge function directly. However the implementation is generally slow and inefficient.

Implementors should use the test::check_partial_ord_properties method to check their PartialOrd implementation, and should use the test::check_lattice_ord to ensure the partial order agrees with the Merge-derived NaiveLatticeOrd order.

#### §LatticeFrom

LatticeFrom is equivalent to the std::convert::From trait but specific to lattices. LatticeFrom should be implemented only between different representations of the same lattice type, e.g. between set_union::SetUnionBTreeSet and set_union::SetUnionHashSet. For compound lattice (lattices with nested lattice types), the LatticeFrom implementation should be recursive for those nested lattices.

#### §IsBot, IsTop, and Default

A bottom (⊥) is strictly less than all other values. A top (⊤) is strictly greater than all other values. IsBot::is_bot and IsTop::is_top determine if a lattice instance is top or bottom respectively.

For lattice types, Default::default() must create a bottom value. IsBot::is_bot(&Default::default()) should always return true for all lattice types.

#### §Atomize

Atomize::atomize converts a lattice point into a bunch of smaller lattice points. When these “atoms” are merged together they will form the original lattice point. See the docs for more precise semantics.

#### §DeepReveal

DeepReveal allows recursive “revealing” of the underlying data within latties. Particularly useful for revealing nested lattices.

## Structs§

• A Conflict lattice, stores a single instance of T and goes to a “conflict” state (None) if inequal T instances are merged together.
• Dominating pair compound lattice.
• A totally ordered max lattice. Merging returns the larger value.
• A totally ordered min lattice. Merging returns the smaller value.
• Pair compound lattice.
• Bimorphism which pairs up the two input lattices.
• A Point lattice, corresponding to a single instance of T.
• Vec-union compound lattice.
• Wraps a lattice in Option, treating None as a new bottom element which compares as less than to all other values.
• Wraps a lattice in Option, treating None as a new top element which compares as greater than to all other values.

## Traits§

• Trait to atomize a lattice into individual elements. For example, a set_union::SetUnion will be broken up into individual singleton elements.
• Trait for recursively revealing the underlying types within lattice types.
• Trait to check if a lattice instance is bottom (⊥).
• Trait to check if a lattice instance is top (⊤) and therefore cannot change any futher.
• Alias trait for lattice types.
• Semilattice bimorphism. Lattice merge must distribute over this binary function, in both arguments.
• Same as From but for lattices.
• Semilattice morphism. Lattice merge must distribute over this unary function.
• Trait for lattice partial order comparison PartialOrd is implemented for many things, this trait can be used to require the type be a lattice.
• Trait for lattice merge (AKA “join” or “least upper bound”).
• Trait for Semiring Multiplication.
• Naive lattice compare, based on the Merge::merge function.
• Trait to define a one in a semiring.
• Alias trait for semirings.
• Trait to check if semiring contains a zero.