Data.Equality.Analysis

E-class analysis, which allows the concise expression of a program analysis over the e-graph.

An e-class analysis resembles abstract interpretation lifted to the e-graph level, attaching analysis data from a semilattice to each e-class.

The e-graph maintains and propagates this data as e-classes get merged and new e-nodes are added.

Analysis data can be used directly to modify the e-graph, to inform how or if rewrites apply their right-hand sides, or to determine the cost of terms during the extraction process.

References: https://arxiv.org/pdf/2004.03082.pdf

Documentation

class Eq domain => Analysis domain (l :: Type -> Type) where

An e-class analysis with domain domain defined for a language l.

The domain is the type of the domain of the e-class analysis, that is, the type of the data stored in an e-class according to this e-class analysis

Minimal complete definition

makeA, joinA

Methods

makeA :: l domain -> domain

When a new e-node is added into a new, singleton e-class, construct a new value of the domain to be associated with the new e-class, by accessing the associated data of the node's children

The argument is the e-node term populated with its children data

Example

-- domain = Maybe Double
makeA :: Expr (Maybe Double) -> Maybe Double
makeA = case
    BinOp Div e1 e2 -> liftA2 (/) e1 e2
    BinOp Sub e1 e2 -> liftA2 (-) e1 e2
    BinOp Mul e1 e2 -> liftA2 (*) e1 e2
    BinOp Add e1 e2 -> liftA2 (+) e1 e2
    Const x -> Just x
    Sym _ -> Nothing

joinA :: domain -> domain -> domain

When e-classes c1 c2 are being merged into c, join d_c1 and d_c2 into a new value d_c to be associated with the new e-class c

modifyA

Arguments

:: ClassId

Id of class c whose new data d_c triggered the modify call

-> EGraph domain l

E-graph where class c being modified exists

-> EGraph domain l

E-graph resulting from the modification

Optionally modify the e-class c (based on d_c), typically by adding an e-node to c. Modify should be idempotent if no other changes occur to the e-class, i.e., modify(modify(c)) = modify(c)

Example

Pruning an e-class with a constant value of all its nodes except for the leaf values, and adding a constant value node

modifyA cl eg0
  = case eg0^._class cl._data of
      Nothing -> eg0
      Just d  ->
            -- Add constant as e-node
        let (new_c,eg1) = represent (Fix (Const d)) eg0
            (rep, eg2)  = merge cl new_c eg1
            -- Prune all except leaf e-nodes
         in eg2 & _class rep._nodes %~ S.filter (F.null .unNode)

Instances

Analysis () l

The simplest analysis that defines the domain to be () and does nothing otherwise

Defined in Data.Equality.Analysis

Methods

makeA :: l () -> ()

joinA :: () -> () -> ()

modifyA :: ClassId -> EGraph () l -> EGraph () l

(Language l, Analysis a l, Analysis b l) => Analysis (a, b) l

This instance is not necessarily well behaved for any two analysis, so care must be taken when using it.

A possible criterion is:

For any two analysis, where modifyA is called m1 and m2 respectively, this instance is well behaved if m1 and m2 commute, and the analysis only change the e-class being modified.

That is, if m1 and m2 satisfy the following law: m1 . m2 = m2 . m1

A simple criterion that should suffice for commutativity. If: * The modify function only depends on the analysis value, and * The modify function doesn't change the analysis value Then any two such functions commute.

Note: there are weaker (or at least different) criteria for this instance to be well behaved.

Defined in Data.Equality.Analysis

Methods

makeA :: l (a, b) -> (a, b)

joinA :: (a, b) -> (a, b) -> (a, b)

modifyA :: ClassId -> EGraph (a, b) l -> EGraph (a, b) l