Multi-Level IR Compiler Framework

Linalg OpDSL

Python based DSL for authoring Linalg op definitions and generating linalg.generic IR based on them for samples.

The Linalg OpDSL is a high level DSL for constructing structured op definitions in a way that can be exported to built-in, named structured ops via YAML-based definitions or used interactively to emit corresponding linalg.generic IR for the composition.

Basic usage 

The tool is bundled with the MLIR Python bindings. To use from the CMake build tree, MLIR must be build with Python bindings enabled (-DMLIR_ENABLE_BINDINGS_PYTHON=ON). Then add the python directory in the build tree to your PYTHONPATH environment variable (i.e. export PYTHONPATH=$PWD/build/tools/mlir/python_packages/mlir_core). Optionally, use an installed MLIR package, if available, to avoid building.

# Dump the `` module as YAML.
python -m mlir.dialects.linalg.opdsl.dump_oplib .ops.core_named_ops

Alternatively, run the $PWD/build/bin/ script, which is available after building the mlir-linalg-ods-gen target. The tool is meant for use during both development and runtime, but not as a build tool of the core compiler: in order to export static named op definitions to be built as part of the compiler, the corresponding Linalg dialect YAML file must be updated and reviewed. TODO: Develop a script to automate op updates to these files.

Language Guide 

The language presented here is loosely inspired from the Tensor Comprehensions work, adapted to represent linalg structured ops.

This tool is new and rapidly evolving. For language examples, refer to the built-in ops in the package (lib/Bindings/Python/mlir/tools/linalg_opdsl/ops in the repository).

Using a matmul as an example, we will decompose the language:

T1 = TV.T1
T2 = TV.T2

def matmul(A=TensorDef(T1, S.M, S.K),
           B=TensorDef(T2, S.K, S.N),
           C=TensorDef(U, S.M, S.N, output=True)):
  """Performs a matrix multiplication of two 2D inputs.

  Numeric casting is performed on the operands to the inner multiply, promoting
  them to the same data type as the accumulator/output.
  domain(D.m, D.n, D.k)
  C[D.m, D.n] += TypeFn.cast_signed(
      U, A[D.m, D.k]) * TypeFn.cast_signed(U, B[D.k, D.n])

Here we have a simple type polymorphic contraction that takes arguments A and B and outputs C. Each is bound to a TensorDef, which specifies:

  • The symbolic element type (T1, T2, U above).
  • Symbolic shape expressions with symbols that are bound globally for the op ( note that in this simple example, the shape expressions are just symbol references, but they are permitted to be a constrained set of affine expressions).
  • Usage (output=True).

The docstring will be transferred to the op definition verbatim.

An explicit iteration domain dimension order can be declared for the op via domain(D.d0[, D.d1...]).

Special identifying op interfaces can be declared for the op via implements(interface1[, interface2...]).

Extra method definitions can be declared for the op via defines(definition1[, definition2...]).


Structured operations take two types of runtime parameters namely scalars and tensors. While scalars are inputs only, a tensor may be marked as an output. Assignment expressions index the tensor parameters to access the individual elements, while scalars can be accessed directly.

The following example demonstrates the use of the two parameter types:

def copy_and_scale(val=ScalarDef(T),
                   I=TensorDef(T, S.M, S.K),
                   O=TensorDef(T, S.M, S.K, output=True)):
  """Scale the input by the scalar value and store the result"""
  O[D.m, D.n] = I[D.m, D.n] * val

The operation scales the input tensor I scales its elements by the value val and writes the result to the output tensor out. The scalar val is bound to a ScalarDef, which specifies the type of the scalar operand. The tensors are bound to a TensorDef as demonstrated by the matmul example. All parameters appear in the parameter list of the operation:

copy_and_scale(val, in_tensor, outs=[out_tensor])

Index Attributes 

Index attributes are compile-time constant parameters only accessible in index expressions. They can be used to parameterize the access pattern of a structured operation, for example, by setting its strides. They cannot take part in the actual computation.

The following example demonstrates the use of index attributes:

def strided_copy(I=TensorDef(T, S.IH, S.IW),
                 O=TensorDef(T, S.OH, S.OW, output=True),
                 strides=IndexAttrDef(S.SH, S.SW, default=[1, 1])):
  """Copy a subset of the input tensor elements to the output tensor"""
  O[D.oh, D.ow] = I[D.oh * S.SH, D.ow * S.SW]

The operation implements a strided copy from the input tensor I to the output tensor O. The strides attribute is bound to an IndexAttrDef. It defines the symbols S.SH and S.SW, which are used to index the input tensor I. When instantiating the operation, the attribute is set using a named argument:

strided_copy(in_tensor, outs=[out_tensor], strides=[1, 2])

The strides vector elements substitute the symbols S.SH and S.SW in the index expressions of the operation instance. If no strides are provided the default vector elements are used instead.

Index attributes are currently limited to integer vectors and only accessible in index expressions. An operation may have multiple attributes all of them placed at the end of the parameter list after the output tensors.

Shape-Only Tensors 

Structured operations derive the iteration space given the sizes of the input and output tensors. Certain operations need shape-only tensors that are not accessed and exist purely for the sake of specifying the iteration domain. An example is the pooling operation that takes a shape-only tensor to define the iteration space of the reduction. As shape-only tensors have no uses, the TensorDef takes an additional optional index_dims parameter to map the shape to index dimensions.

The following example demonstrates the index dimension annotation:

def pooling_poly(
    I=TensorDef(T1, S.N, S.H, S.W, S.C),
    K=TensorDef(T2, S.KH, S.KW, index_dims=[,]),
    O=TensorDef(U, S.N, S.OH, S.OW, S.C, output=True),
    strides=IndexAttrDef(S.SH, S.SW, default=[1, 1]),
    dilations=IndexAttrDef(S.DH, S.DW, default=[1, 1])):
  O[D.n, D.oh, D.ow, D.c] += TypeFn.cast_signed(U,
          I[D.n, D.oh * S.SH + * S.DH, D.ow * S.SW + * S.DW, D.c])

The pooling operation does not access the shape-only tensor K. Instead, the shapes S.KH and S.KW specify the iteration domain for the reduction dimensions and


The bulk of language consists of assignment expressions of the form above. The iteration dimension order is determined lexically based on the order encountered in the expression (following operator precedence if math operators are used). TODO: Introduce a directive to fix the dimension bindings.

Reduction dimensions are inferred to be any dimensions on the RHS that are not on the LHS.

A number of unary and binary arithmetic functions are supported:

  • BinaryFn.add(a, b) (also via overloading the binary + operator)
  • BinaryFn.mul(a, b) (also via overloading the binary * operator)
  • BinaryFn.max_signed(a, b)
  • BinaryFn.min_signed(a, b)
  • BinaryFn.sub(a, b) (also via overloading the binary - operator)
  • BinaryFn.max_unsigned(a, b)
  • BinaryFn.min_unsigned(a, b)
  • UnaryFn.exp(a)
  • UnaryFn.log(a)

As the integer types are signless, signedness is implement by different functions that treat integers as signed or unsigned values.

A subset of the arithmetic functions are supported in reductions. These reduction functions can appear as the outermost function on the RHS:

  • ReduceFn.add (also overloading the inplace += on a LHS)
  • ReduceFn.mul
  • ReduceFn.max_signed
  • ReduceFn.min_signed
  • ReduceFn.max_unsigned
  • ReduceFn.min_unsigned

As the integer types are signless, signedness is implement by different functions that treat integers as signed or unsigned values.

Additionally, type conversion functions cast an operand to a target type:

  • TypeFn.cast_signed(TypeVar, operand)
  • TypeFn.cast_unsigned(TypeVar, operand)

As the integer types are signless, signedness is implement by different functions that treat integers as signed (TypeFn.cast_signed) or unsigned (TypeFn.cast_unsigned) values.

There are also special forms:

  • const(value) returns a constant value.
  • index(dim) returns the iteration index in the given dimension dim.

Function Attributes 

Function attributes are compile-time constant function parameters. They can be used to parameterize the computation performed by a structured operation, for example, to support signed and unsigned computations.

The following example demonstrates the use of function attributes:

def elemwise_binary(
    O=TensorDef(U, output=True),
  O[None] = fun(cast(U, lhs[None]), cast(U, rhs[None]))

The fun and cast function attributes by default are aliases for their default values BinaryFn.add and TypeFn.cast_signed, respectively. When instantiating the operation, the function attributes may be set to other functions using optional named arguments:

elemwise_binary(lhs, rhs, outs=[out_tensor],
                fun=BinaryFn.mul, cast=TypeFn.cast_unsigned)

In the example, the fun and cast arguments adapt the body of the operation to implement multiplication and unsigned casts instead of addition and signed casts.

OpDSL supports unary, binary, and type conversion function attributes. An operation can take multiple attributes of different kinds placed at the end of the parameter list.


All types in assignment expressions are late bound based on actual input and output types of constructed ops. An exception are predefined types such as I32, I64, F32, and F64. These hardwired types enable intermediate computations with a type that is independent of the input and output types. For example, parts of floating point computation may require double precision arithmetic despite all inputs and outputs being single precision values. Assignment expressions with no TypeFn.cast_signed calls will generally require uniform types throughout and will fail to verify if violated. The presence of a TypeFn.cast_signed or TypeFn.cast_unsigned allows for a limited form of numeric type conversion between element types that can be derived from inputs and outputs (and in the future, attributes). TypeFn.cast_signed calls with a TypeVar first argument are emitted as type_fn primitives in the YAML definition.

Casting will perform int<->float and index->int type conversions and will perform any necessary extension or truncation within the type family. The integer types themselves are signless and signedness is implemented by functions/operations. The TypeFn.cast_signed function treats all integers as signed, while TypeFn.cast_unsigned treats them as unsigned.

The following examples illustrate the lowering of signed and unsigned functions:

  • cast_signed(I32 -> I64) -> arith.ExtSIOp
  • cast_signed(F32 -> I32) -> arith.FPToSIOp
  • cast_unsigned(I32 -> I64) -> arith.ExtUIOp
  • cast_unsigned(F32 -> I32) -> arith.FPToUIOp
  • max_signed -> arith.MaxSIOp
  • max_unsigned -> arith.MaxUIOp

Not all functions are applicable for all numeric types, and on mismatch, op verification will fail.

Pointwise Computations 

Pointwise computations are expressible in a rank polymorphic form that supports arbitrary ranked operands - all of them need to have the same rank - with a single operation definition.

An example for a rank polymorphic operation is fill:

def fill(value=ScalarDef(T1),
         O=TensorDef(U, output=True)):
  O[None] = TypeFn.cast_signed(U, value)

The operation sets the elements of the output tensor O to value. All operands are either scalars or rank zero tensors that are accessed using the index None. The operation thus performs a scalar computation that trivially extends to a multi-dimensional pointwise computation. As a result, we may use fill with arbitrary ranked output tensors:

tensor_2d = tensor.EmptyOp([4, 8], f32)
tensor_3d = tensor.EmptyOp([4, 8, 16], f32)
fill(value, outs=[tensor_2d])
fill(value, outs=[tensor_3d])