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// Copyright Materialize, Inc. and contributors. All rights reserved.
//
// Use of this software is governed by the Business Source License
// included in the LICENSE file.
//
// As of the Change Date specified in that file, in accordance with
// the Business Source License, use of this software will be governed
// by the Apache License, Version 2.0.
//! An analysis that reports all known-equivalent expressions for each relation.
//!
//! Expressions are equivalent at a relation if they are certain to evaluate to
//! the same `Datum` for all records in the relation.
//!
//! Equivalences are recorded in an `EquivalenceClasses`, which lists all known
//! equivalences classes, each a list of equivalent expressions.
use std::collections::BTreeMap;
use mz_expr::{Id, MirRelationExpr, MirScalarExpr};
use mz_repr::{ColumnType, Datum};
use crate::analysis::Analysis;
use crate::analysis::{Arity, RelationType};
use crate::analysis::{Derived, DerivedBuilder};
/// Pulls up and pushes down predicate information represented as equivalences
#[derive(Debug, Default)]
pub struct Equivalences;
impl Analysis for Equivalences {
type Value = EquivalenceClasses;
fn announce_dependencies(builder: &mut DerivedBuilder) {
builder.require::<Arity>();
builder.require::<RelationType>(); // needed for expression reduction.
}
fn derive(
expr: &MirRelationExpr,
_index: usize,
results: &[Self::Value],
depends: &Derived,
) -> Self::Value {
let mut equivalences = match expr {
MirRelationExpr::Constant { rows, typ } => {
// Trawl `rows` for any constant information worth recording.
// Literal columns may be valuable; non-nullability could be too.
let mut equivalences = EquivalenceClasses::default();
if let Ok([(row, _cnt), rows @ ..]) = rows.as_deref() {
// Vector of `Option<Datum>` which becomes `None` once a column has a second datum.
let len = row.iter().count();
let mut common = Vec::with_capacity(len);
common.extend(row.iter().map(Some));
for (row, _cnt) in rows.iter() {
for (datum, common) in row.iter().zip(common.iter_mut()) {
if Some(datum) != *common {
*common = None;
}
}
}
for (index, common) in common.into_iter().enumerate() {
if let Some(datum) = common {
equivalences.classes.push(vec![
MirScalarExpr::Column(index),
MirScalarExpr::literal_ok(
datum,
typ.column_types[index].scalar_type.clone(),
),
]);
}
}
}
equivalences
}
MirRelationExpr::Get { id, .. } => {
let mut equivalences = EquivalenceClasses::default();
// Find local identifiers, but nothing for external identifiers.
if let Id::Local(id) = id {
if let Some(offset) = depends.bindings().get(id) {
// It is possible we have derived nothing for a recursive term
if let Some(result) = results.get(*offset) {
equivalences.clone_from(result);
}
}
}
equivalences
}
MirRelationExpr::Let { .. } => results.last().unwrap().clone(),
MirRelationExpr::LetRec { .. } => results.last().unwrap().clone(),
MirRelationExpr::Project { outputs, .. } => {
// restrict equivalences, and introduce equivalences for repeated outputs.
let mut equivalences = results.last().unwrap().clone();
equivalences.project(outputs.iter().cloned());
equivalences
}
MirRelationExpr::Map { scalars, .. } => {
// introduce equivalences for new columns and expressions that define them.
let mut equivalences = results.last().unwrap().clone();
let input_arity = depends.results::<Arity>().unwrap()[results.len() - 1];
for (pos, expr) in scalars.iter().enumerate() {
equivalences
.classes
.push(vec![MirScalarExpr::Column(input_arity + pos), expr.clone()]);
}
equivalences
}
MirRelationExpr::FlatMap { .. } => results.last().unwrap().clone(),
MirRelationExpr::Filter { predicates, .. } => {
let mut equivalences = results.last().unwrap().clone();
let mut class = predicates.clone();
class.push(MirScalarExpr::literal_ok(
Datum::True,
mz_repr::ScalarType::Bool,
));
equivalences.classes.push(class);
equivalences
}
MirRelationExpr::Join { equivalences, .. } => {
// Collect equivalences from all inputs;
let expr_index = results.len();
let mut children = depends
.children_of_rev(expr_index, expr.children().count())
.collect::<Vec<_>>();
children.reverse();
let arity = depends.results::<Arity>().unwrap();
let mut columns = 0;
let mut result = EquivalenceClasses::default();
for child in children.into_iter() {
let input_arity = arity[child];
let mut equivalences = results[child].clone();
let permutation = (columns..(columns + input_arity)).collect::<Vec<_>>();
equivalences.permute(&permutation);
result.classes.extend(equivalences.classes);
columns += input_arity;
}
// Fold join equivalences into our results.
result.classes.extend(equivalences.iter().cloned());
result
}
MirRelationExpr::Reduce {
group_key,
aggregates: _,
..
} => {
let input_arity = depends.results::<Arity>().unwrap()[results.len() - 1];
let mut equivalences = results.last().unwrap().clone();
// Introduce keys column equivalences as a map, then project to them as a projection.
for (pos, expr) in group_key.iter().enumerate() {
equivalences
.classes
.push(vec![MirScalarExpr::Column(input_arity + pos), expr.clone()]);
}
// TODO: MIN, MAX, ANY, ALL aggregates pass through all certain properties of their columns.
// They also pass through equivalences of them and other constant columns (e.g. key columns).
// However, it is not correct to simply project onto these columns, as relationships amongst
// aggregate columns may no longer be preserved. MAX(col) != MIN(col) even though col = col.
// TODO: COUNT ensures a non-null value.
equivalences.project(input_arity..(input_arity + group_key.len()));
equivalences
}
MirRelationExpr::TopK { .. } => results.last().unwrap().clone(),
MirRelationExpr::Negate { .. } => results.last().unwrap().clone(),
MirRelationExpr::Threshold { .. } => results.last().unwrap().clone(),
MirRelationExpr::Union { .. } => {
// TODO: `EquivalenceClasses::union` takes references, and we could probably skip much of this cloning.
let expr_index = results.len();
depends
.children_of_rev(expr_index, expr.children().count())
.map(|c| results[c].clone())
.reduce(|e1, e2| e1.union(&e2))
.unwrap()
}
MirRelationExpr::ArrangeBy { .. } => results.last().unwrap().clone(),
};
let expr_type = &depends.results::<RelationType>().unwrap()[results.len()];
equivalences.minimize(expr_type);
equivalences
}
}
/// A compact representation of classes of expressions that must be equivalent.
///
/// Each "class" contains a list of expressions, each of which must be `Eq::eq` equal.
/// Ideally, the first element is the "simplest", e.g. a literal or column reference,
/// and any other element of that list can be replaced by it.
///
/// The classes are meant to be minimized, with each expression as reduced as it can be,
/// and all classes sharing an element merged.
#[derive(Clone, Eq, PartialEq, Ord, PartialOrd, Default, Debug)]
pub struct EquivalenceClasses {
/// Multiple lists of equivalent expressions, each representing an equivalence class.
///
/// The first element should be the "canonical" simplest element, that any other element
/// can be replaced by.
/// These classes are unified whenever possible, to minimize the number of classes.
pub classes: Vec<Vec<MirScalarExpr>>,
}
impl EquivalenceClasses {
/// Sorts and deduplicates each class, and the classes themselves.
fn tidy(&mut self) {
for class in self.classes.iter_mut() {
// Remove all literal errors, as they cannot be equated to other things.
class.retain(|e| !e.is_literal_err());
class.sort_by(|e1, e2| match (e1, e2) {
(MirScalarExpr::Literal(_, _), MirScalarExpr::Literal(_, _)) => e1.cmp(e2),
(MirScalarExpr::Literal(_, _), _) => std::cmp::Ordering::Less,
(_, MirScalarExpr::Literal(_, _)) => std::cmp::Ordering::Greater,
(MirScalarExpr::Column(_), MirScalarExpr::Column(_)) => e1.cmp(e2),
(MirScalarExpr::Column(_), _) => std::cmp::Ordering::Less,
(_, MirScalarExpr::Column(_)) => std::cmp::Ordering::Greater,
(x, y) => {
// General expressions should be ordered by their size,
// to ensure we only simplify expressions by substitution.
let x_size = x.size();
let y_size = y.size();
if x_size == y_size {
x.cmp(y)
} else {
x_size.cmp(&y_size)
}
}
});
class.dedup();
}
self.classes.retain(|c| c.len() > 1);
self.classes.sort();
self.classes.dedup();
}
/// Update `self` to maintain the same equivalences which potentially reducing along `Ord::le`.
///
/// Informally this means simplifying constraints, removing redundant constraints, and unifying equivalence classes.
pub fn minimize(&mut self, columns: &Option<Vec<ColumnType>>) {
// Repeatedly, we reduce each of the classes themselves, then unify the classes.
// This should strictly reduce complexity, and reach a fixed point.
// Ideally it is *confluent*, arriving at the same fixed point no matter the order of operations.
self.tidy();
// We continue as long as any simplification has occurred.
// An expression can be simplified, a duplication found, or two classes unified.
let mut stable = false;
while !stable {
stable = self.minimize_once(columns);
}
// TODO: remove these measures once we are more confident about idempotence.
let prev = self.clone();
self.minimize_once(columns);
mz_ore::soft_assert_eq_or_log!(self, &prev, "Equivalences::minimize() not idempotent");
}
/// A single iteration of minimization, which we expect to repeat but benefit from factoring out.
fn minimize_once(&mut self, columns: &Option<Vec<ColumnType>>) -> bool {
// We are complete unless we experience an expression simplification, or an equivalence class unification.
let mut stable = true;
// 0. Reduce each expression
//
// This is optional in that `columns` may not be provided (`reduce` requires type information).
if let Some(columns) = columns {
for class in self.classes.iter_mut() {
for expr in class.iter_mut() {
let prev_expr = expr.clone();
expr.reduce(columns);
if &prev_expr != expr {
stable = false;
}
}
}
}
// 1. Reduce each class.
// Each class can be reduced in the context of *other* classes, which are available for substitution.
for class_index in 0..self.classes.len() {
for index in 0..self.classes[class_index].len() {
let mut cloned = self.classes[class_index][index].clone();
// Use `reduce_child` rather than `reduce_expr` to avoid entire expression replacement.
let reduced = self.reduce_child(&mut cloned);
if reduced {
self.classes[class_index][index] = cloned;
stable = false;
}
}
}
// 2. Unify classes.
// If the same expression is in two classes, we can unify the classes.
// This element may not be the representative.
// TODO: If all lists are sorted, this could be a linear merge among all.
// They stop being sorted as soon as we make any modification, though.
// But, it would be a fast rejection when faced with lots of data.
for index1 in 0..self.classes.len() {
for index2 in 0..index1 {
if self.classes[index1]
.iter()
.any(|x| self.classes[index2].iter().any(|y| x == y))
{
let prior = std::mem::take(&mut self.classes[index2]);
self.classes[index1].extend(prior);
stable = false;
}
}
}
// 3. Identify idioms
// E.g. If Eq(x, y) must be true, we can introduce classes `[x, y]` and `[false, IsNull(x), IsNull(y)]`.
let mut to_add = Vec::new();
for class in self.classes.iter_mut() {
if class.iter().any(|c| c.is_literal_true()) {
for expr in class.iter() {
// If Eq(x, y) must be true, we can introduce classes `[x, y]` and `[false, IsNull(x), IsNull(y)]`.
// This substitution replaces a complex expression with several smaller expressions, and cannot
// cycle if we follow that practice.
if let MirScalarExpr::CallBinary {
func: mz_expr::BinaryFunc::Eq,
expr1,
expr2,
} = expr
{
to_add.push(vec![*expr1.clone(), *expr2.clone()]);
to_add.push(vec![
MirScalarExpr::literal_false(),
expr1.clone().call_is_null(),
expr2.clone().call_is_null(),
]);
stable = false;
}
}
// Remove the more complex form of the expression.
class.retain(|expr| {
if let MirScalarExpr::CallBinary {
func: mz_expr::BinaryFunc::Eq,
..
} = expr
{
false
} else {
true
}
});
for expr in class.iter() {
// If TRUE == NOT(X) then FALSE == X is a simpler form.
if let MirScalarExpr::CallUnary {
func: mz_expr::UnaryFunc::Not(_),
expr: e,
} = expr
{
to_add.push(vec![MirScalarExpr::literal_false(), (**e).clone()]);
stable = false;
}
}
class.retain(|expr| {
if let MirScalarExpr::CallUnary {
func: mz_expr::UnaryFunc::Not(_),
..
} = expr
{
false
} else {
true
}
});
}
if class.iter().any(|c| c.is_literal_false()) {
for expr in class.iter() {
// If FALSE == NOT(X) then TRUE == X is a simpler form.
if let MirScalarExpr::CallUnary {
func: mz_expr::UnaryFunc::Not(_),
expr: e,
} = expr
{
to_add.push(vec![MirScalarExpr::literal_true(), (**e).clone()]);
stable = false;
}
}
class.retain(|expr| {
if let MirScalarExpr::CallUnary {
func: mz_expr::UnaryFunc::Not(_),
..
} = expr
{
false
} else {
true
}
});
}
}
self.classes.extend(to_add);
// Tidy up classes, restore representative.
self.tidy();
stable
}
/// Produce the equivalences present in both inputs.
pub fn union(&self, other: &Self) -> Self {
// TODO: seems like this could be extended, with similar concepts to localization:
// We may removed non-shared constraints, but ones that remain could take over
// and substitute in to retain more equivalences.
// For each pair of equivalence classes, their intersection.
let mut equivalences = EquivalenceClasses {
classes: Vec::new(),
};
for class1 in self.classes.iter() {
for class2 in other.classes.iter() {
let class = class1
.iter()
.filter(|e1| class2.iter().any(|e2| e1 == &e2))
.cloned()
.collect::<Vec<_>>();
if class.len() > 1 {
equivalences.classes.push(class);
}
}
}
equivalences.minimize(&None);
equivalences
}
/// Permutes each expression, looking up each column reference in `permutation` and replacing with what it finds.
pub fn permute(&mut self, permutation: &[usize]) {
for class in self.classes.iter_mut() {
for expr in class.iter_mut() {
expr.permute(permutation);
}
}
}
/// Subject the constraints to the column projection, reworking and removing equivalences.
///
/// This method should also introduce equivalences representing any repeated columns.
pub fn project<I>(&mut self, output_columns: I)
where
I: IntoIterator<Item = usize> + Clone,
{
// Retain the first instance of each column, and record subsequent instances as duplicates.
let mut dupes = Vec::new();
let mut remap = BTreeMap::default();
for (idx, col) in output_columns.into_iter().enumerate() {
if let Some(pos) = remap.get(&col) {
dupes.push((*pos, idx));
} else {
remap.insert(col, idx);
}
}
// Some expressions may be "localized" in that they only reference columns in `output_columns`.
// Many expressions may not be localized, but may reference canonical non-localized expressions
// for classes that contain a localized expression; in that case we can "backport" the localized
// expression to give expressions referencing the canonical expression a shot at localization.
//
// Expressions should only contain instances of canonical expressions, and so we shouldn't need
// to look any further than backporting those. Backporting should have the property that the simplest
// localized expression in each class does not contain any non-localized canonical expressions
// (as that would make it non-localized); our backporting of non-localized canonicals with localized
// expressions should never fire a second
// Let's say an expression is "localized" once we are able to rewrite its support in terms of `output_columns`.
// Not all expressions can be localized, although some of them may be equivalent to localized expressions.
// As we find localized expressions, we can replace uses of their equivalent representative with them,
// which may allow further expression localization.
// We continue the process until no further classes can be localized.
// A map from representatives to our first localization of their equivalence class.
let mut localized = false;
while !localized {
localized = true;
for class in self.classes.iter_mut() {
if !class[0].support().iter().all(|c| remap.contains_key(c)) {
if let Some(pos) = class
.iter()
.position(|e| e.support().iter().all(|c| remap.contains_key(c)))
{
class.swap(0, pos);
localized = false;
}
}
}
// attempt to replace representatives with equivalent localizeable expressions.
for class_index in 0..self.classes.len() {
for index in 0..self.classes[class_index].len() {
let mut cloned = self.classes[class_index][index].clone();
// Use `reduce_child` rather than `reduce_expr` to avoid entire expression replacement.
let reduced = self.reduce_child(&mut cloned);
if reduced {
self.classes[class_index][index] = cloned;
}
}
}
// NB: Do *not* `self.minimize()`, as we are developing localizable rather than canonical representatives.
}
// Localize all localizable expressions and discard others.
for class in self.classes.iter_mut() {
class.retain(|e| e.support().iter().all(|c| remap.contains_key(c)));
for expr in class.iter_mut() {
expr.permute_map(&remap);
}
}
self.classes.retain(|c| c.len() > 1);
// If column repetitions, introduce them as equivalences.
// We introduce only the equivalence to the first occurrence, and rely on minimization to collect them.
for (col1, col2) in dupes {
self.classes.push(vec![
MirScalarExpr::Column(col1),
MirScalarExpr::Column(col2),
]);
}
self.minimize(&None);
}
/// Perform any exact replacement for `expr`, report if it had an effect.
fn replace(&self, expr: &mut MirScalarExpr) -> bool {
for class in self.classes.iter() {
// TODO: If `class` is sorted We only need to iterate through "simpler" expressions;
// we can stop once x > expr.
// We need to be careful with that reasoning, as it interferes with self-improvement;
// if we are modifying `self` we must restore the invariant before relying on it again.
if class[1..].iter().any(|x| x == expr) {
expr.clone_from(&class[0]);
return true;
}
}
false
}
/// Perform any simplification, report if effective.
pub fn reduce_expr(&self, expr: &mut MirScalarExpr) -> bool {
let mut simplified = false;
simplified = simplified || self.reduce_child(expr);
simplified = simplified || self.replace(expr);
simplified
}
/// Perform any simplification on children, report if effective.
pub fn reduce_child(&self, expr: &mut MirScalarExpr) -> bool {
let mut simplified = false;
match expr {
MirScalarExpr::CallBinary { expr1, expr2, .. } => {
simplified = self.reduce_expr(expr1) || simplified;
simplified = self.reduce_expr(expr2) || simplified;
}
MirScalarExpr::CallUnary { expr, .. } => {
simplified = self.reduce_expr(expr) || simplified;
}
MirScalarExpr::CallVariadic { exprs, .. } => {
for expr in exprs.iter_mut() {
simplified = self.reduce_expr(expr) || simplified;
}
}
MirScalarExpr::If { cond: _, then, els } => {
// Do not simplify `cond`, as we cannot ensure the simplification
// continues to hold as expressions migrate around.
simplified = self.reduce_expr(then) || simplified;
simplified = self.reduce_expr(els) || simplified;
}
_ => {}
}
simplified
}
/// True if any equivalence class contains two distinct non-error literals.
pub fn unsatisfiable(&self) -> bool {
for class in self.classes.iter() {
let mut literal_ok = None;
for expr in class.iter() {
if let MirScalarExpr::Literal(Ok(row), _) = expr {
if literal_ok.is_some() && literal_ok != Some(row) {
return true;
} else {
literal_ok = Some(row);
}
}
}
}
false
}
}