Multi-Level IR Compiler Framework

Chapter 3: High-level Language-Specific Analysis and Transformation

Creating a dialect that closely represents the semantics of an input language enables analyses, transformations and optimizations in MLIR that require high-level language information and are generally performed on the language AST. For example, clang has a fairly heavy mechanism for performing template instantiation in C++.

We divide compiler transformations into two categories: local and global. In this chapter, we focus on how to leverage the Toy Dialect and its high-level semantics to perform local pattern-match transformations that would be difficult in LLVM. For this, we use MLIR’s Generic DAG Rewriter.

There are two methods that can be used to implement pattern-match transformations: 1. Imperative, C++ pattern-match and rewrite 2. Declarative, rule-based pattern-match and rewrite using table-driven Declarative Rewrite Rules (DRR). Note that the use of DRR requires that the operations be defined using ODS, as described in Chapter 2.

Optimize Transpose using C++ style pattern-match and rewrite 

Let’s start with a simple pattern and try to eliminate a sequence of two transposes that cancel out: transpose(transpose(X)) -> X. Here is the corresponding Toy example:

def transpose_transpose(x) {
  return transpose(transpose(x));

Which corresponds to the following IR:

toy.func @transpose_transpose(%arg0: tensor<*xf64>) -> tensor<*xf64> {
  %0 = toy.transpose(%arg0 : tensor<*xf64>) to tensor<*xf64>
  %1 = toy.transpose(%0 : tensor<*xf64>) to tensor<*xf64>
  toy.return %1 : tensor<*xf64>

This is a good example of a transformation that is trivial to match on the Toy IR but that would be quite hard for LLVM to figure. For example, today Clang can’t optimize away the temporary array, and the computation with the naive transpose is expressed with these loops:

#define N 100
#define M 100

void sink(void *);
void double_transpose(int A[N][M]) {
  int B[M][N];
  for(int i = 0; i < N; ++i) {
    for(int j = 0; j < M; ++j) {
       B[j][i] = A[i][j];
  for(int i = 0; i < N; ++i) {
    for(int j = 0; j < M; ++j) {
       A[i][j] = B[j][i];

For a simple C++ approach to rewrite, involving matching a tree-like pattern in the IR and replacing it with a different set of operations, we can plug into the MLIR Canonicalizer pass by implementing a RewritePattern:

/// Fold transpose(transpose(x)) -> x
struct SimplifyRedundantTranspose : public mlir::OpRewritePattern<TransposeOp> {
  /// We register this pattern to match every toy.transpose in the IR.
  /// The "benefit" is used by the framework to order the patterns and process
  /// them in order of profitability.
  SimplifyRedundantTranspose(mlir::MLIRContext *context)
      : OpRewritePattern<TransposeOp>(context, /*benefit=*/1) {}

  /// This method is attempting to match a pattern and rewrite it. The rewriter
  /// argument is the orchestrator of the sequence of rewrites. It is expected
  /// to interact with it to perform any changes to the IR from here.
  matchAndRewrite(TransposeOp op,
                  mlir::PatternRewriter &rewriter) const override {
    // Look through the input of the current transpose.
    mlir::Value transposeInput = op.getOperand();
    TransposeOp transposeInputOp = transposeInput.getDefiningOp<TransposeOp>();

    // Input defined by another transpose? If not, no match.
    if (!transposeInputOp)
      return failure();

    // Otherwise, we have a redundant transpose. Use the rewriter.
    rewriter.replaceOp(op, {transposeInputOp.getOperand()});
    return success();

The implementation of this rewriter is in ToyCombine.cpp. The canonicalization pass applies transformations defined by operations in a greedy, iterative manner. To ensure that the canonicalization pass applies our new transform, we set hasCanonicalizer = 1 and register the pattern with the canonicalization framework.

// Register our patterns for rewrite by the Canonicalization framework.
void TransposeOp::getCanonicalizationPatterns(
    RewritePatternSet &results, MLIRContext *context) {

We also need to update our main file, toyc.cpp, to add an optimization pipeline. In MLIR, the optimizations are run through a PassManager in a similar way to LLVM:

  mlir::PassManager pm(module->getName());

Finally, we can run toyc-ch3 test/Examples/Toy/Ch3/transpose_transpose.toy -emit=mlir -opt and observe our pattern in action:

toy.func @transpose_transpose(%arg0: tensor<*xf64>) -> tensor<*xf64> {
  %0 = toy.transpose(%arg0 : tensor<*xf64>) to tensor<*xf64>
  toy.return %arg0 : tensor<*xf64>

As expected, we now directly return the function argument, bypassing any transpose operation. However, one of the transposes still hasn’t been eliminated. That is not ideal! What happened is that our pattern replaced the last transform with the function input and left behind the now dead transpose input. The Canonicalizer knows to clean up dead operations; however, MLIR conservatively assumes that operations may have side-effects. We can fix this by adding a new trait, Pure, to our TransposeOp:

def TransposeOp : Toy_Op<"transpose", [Pure]> {...}

Let’s retry now toyc-ch3 test/transpose_transpose.toy -emit=mlir -opt:

toy.func @transpose_transpose(%arg0: tensor<*xf64>) -> tensor<*xf64> {
  toy.return %arg0 : tensor<*xf64>

Perfect! No transpose operation is left - the code is optimal.

In the next section, we use DRR for pattern match optimizations associated with the Reshape op.

Optimize Reshapes using DRR 

Declarative, rule-based pattern-match and rewrite (DRR) is an operation DAG-based declarative rewriter that provides a table-based syntax for pattern-match and rewrite rules:

class Pattern<
    dag sourcePattern, list<dag> resultPatterns,
    list<dag> additionalConstraints = [],
    dag benefitsAdded = (addBenefit 0)>;

A redundant reshape optimization similar to SimplifyRedundantTranspose can be expressed more simply using DRR as follows:

// Reshape(Reshape(x)) = Reshape(x)
def ReshapeReshapeOptPattern : Pat<(ReshapeOp(ReshapeOp $arg)),
                                   (ReshapeOp $arg)>;

The automatically generated C++ code corresponding to each of the DRR patterns can be found under path/to/BUILD/tools/mlir/examples/toy/Ch3/

DRR also provides a method for adding argument constraints when the transformation is conditional on some properties of the arguments and results. An example is a transformation that eliminates reshapes when they are redundant, i.e. when the input and output shapes are identical.

def TypesAreIdentical : Constraint<CPred<"$0.getType() == $1.getType()">>;
def RedundantReshapeOptPattern : Pat<
  (ReshapeOp:$res $arg), (replaceWithValue $arg),
  [(TypesAreIdentical $res, $arg)]>;

Some optimizations may require additional transformations on instruction arguments. This is achieved using NativeCodeCall, which allows for more complex transformations either by calling into a C++ helper function or by using inline C++. An example of such an optimization is FoldConstantReshape, where we optimize Reshape of a constant value by reshaping the constant in place and eliminating the reshape operation.

def ReshapeConstant : NativeCodeCall<"$0.reshape(($1.getType()).cast<ShapedType>())">;
def FoldConstantReshapeOptPattern : Pat<
  (ReshapeOp:$res (ConstantOp $arg)),
  (ConstantOp (ReshapeConstant $arg, $res))>;

We demonstrate these reshape optimizations using the following trivial_reshape.toy program:

def main() {
  var a<2,1> = [1, 2];
  var b<2,1> = a;
  var c<2,1> = b;
module {
  toy.func @main() {
    %0 = toy.constant dense<[1.000000e+00, 2.000000e+00]> : tensor<2xf64>
    %1 = toy.reshape(%0 : tensor<2xf64>) to tensor<2x1xf64>
    %2 = toy.reshape(%1 : tensor<2x1xf64>) to tensor<2x1xf64>
    %3 = toy.reshape(%2 : tensor<2x1xf64>) to tensor<2x1xf64>
    toy.print %3 : tensor<2x1xf64>

We can try to run toyc-ch3 test/Examples/Toy/Ch3/trivial_reshape.toy -emit=mlir -opt and observe our pattern in action:

module {
  toy.func @main() {
    %0 = toy.constant dense<[[1.000000e+00], [2.000000e+00]]> : tensor<2x1xf64>
    toy.print %0 : tensor<2x1xf64>

As expected, no reshape operations remain after canonicalization.

Further details on the declarative rewrite method can be found at Table-driven Declarative Rewrite Rule (DRR).

In this chapter, we saw how to use certain core transformations through always available hooks. In the next chapter, we will see how to use generic solutions that scale better through Interfaces.