llvm-project/mlir/docs/Dialects/Linalg.md

# 'linalg' Dialect

[TOC]

## Rationale

<img width="90" align="left" alt="MLIR Codegen Flow" src="https://user-images.githubusercontent.com/10148468/73613629-c5586580-45c5-11ea-94b7-074aeea94c7b.png">

Linalg is designed to solve the High-level Hierarchical Optimization (HHO box)
in MLIR and to interoperate nicely within a *Mixture Of Expert Compilers*
environment (i.e. the *CGSel* box).

The [Rationale Document](../Rationale/RationaleLinalgDialect.md) goes into
significantly more design and architectural decision details.

## Set of Key Transformations<a name="key_transformations"></a>

The following key transformations have been central to driving the design of
Linalg. They are all implemented in terms of the properties of the
`linalg.generic` OpInterface and avoid the pitfall of relying on hardcoded
one-off op knowledge.

The textual form description of these transformations is left for future work.
Still, it is useful to list the key transformations that are performed on the
Linalg IR and that have influenced its design:

1.  Progressive Buffer Allocation.
1.  Parametric Tiling.
1.  Promotion to Temporary Buffer in Fast Memory.
1.  Tiled Producer-Consumer Fusion with Parametric Tile-And-Fuse.
1.  Map to Parallel and Reduction Loops and Hardware.
1.  Vectorization: Rewrite in Vector Form.
1.  Lower to Loops (Affine, Generic, and Parallel).
1.  Lower to Library Calls or Special Instructions, Intrinsics or ISA.
1.  Partially Lower to Iterations Over a Finer-Grained Linalg Op.

## High-Level Description of Linalg Ops<a name="linalg_ops"></a>

Linalg takes at least some inspiration from all previously
[listed prior art](#prior_art). The design enables the definition of
***CustomOps*** with generic properties that enable
[key transformations](#key_transformations), including lowering to scalar
load/store and other operations or to external library calls and intrinsics.

These ops can have ***either tensor or buffer*** as both input and output
operands. Output tensors operands serve the purpose of providing a unifying
abstraction and give a shape to the results. Output tensors can come in 2
flavors and are always associated with a corresponding op result:

1.  an "init tensor" output value which provides an initial value for a tensor
    that is created by iteratively updating the result (also called "destructive
    updates"). Such tensor is always materialized in some form. If enough fusion
    occurs it may end up being materialized only as a register-level SSA value.
    It is expected (but not required) that the destructive update pattern can be
    rewritten as an inplace update on buffers.

2.  a "shape-only" tensor output value whose underlying elements are not used in
    the payload computation and only serves the purpose of carrying shape
    information to lower levels of abstraction. In the future this will be
    replaced by an appropriate shape type when it is available as a builtin type
    (see the discourse discussion
    [Linalg and Shapes](https://llvm.discourse.group/t/linalg-and-shapes/2421)
    for more details).

### Payload-Carrying Ops<a name="payload_ops"></a>

Linalg defines two payload carrying operations that implement the
[structured ops](https://docs.google.com/presentation/d/1P-j1GrH6Q5gLBjao0afQ-GfvcAeF-QU4GXXeSy0eJ9I/edit#slide=id.p)
abstraction on tensors and buffers. This is architected as two generic
operations `linalg.generic` (resp. `linalg.indexed_generic`) that can express
custom operations with *index-free semantics* (resp. *indexing semantics*). The
properties of these generic ops are the result of applying the guiding
principles described in the
[Rationale Document](../Rationale/RationaleLinalgDialect.md). They are listed
next, with a brief example and discussion for each.

#### Property 1: Input and Output Operands Define The Iteration Space<a name="prop1"></a>

A `linalg.generic` op fully *derives* the specification of its iteration space
from its operands. The property enforces that a localized IR element (the op)
*has* all the information needed to synthesize the control-flow required to
iterate over its operands, according to their type. This notion of IR
localization bears some resemblance to
[URUK](http://icps.u-strasbg.fr/~bastoul/research/papers/GVBCPST06-IJPP.pdf).

Consider the following fully specified `linalg.generic` example. Here, the first
operand is a `memref` of `f32` scalar elements that has an ordinary identity
layout, and the second one is a `memref` of 4-element vectors with a 2-strided,
1-offset layout.

```mlir
// File name: example1.mlir
#accesses = [
  affine_map<(m) -> (m)>,
  affine_map<(m) -> (m)>
]

#attrs = {
  indexing_maps = #accesses,
  iterator_types = ["parallel"]
}

// memory layouts
#identity = affine_map<(d0) -> (d0)>

func @example(%A: memref<?xf32, #identity>,
              %B: memref<?xvector<4xf32>, offset: 1, strides: [2]>) {
  linalg.generic #attrs
  ins(%A: memref<?xf32, #identity>)
  outs(%B: memref<?xvector<4xf32>, offset: 1, strides: [2]>) {
  ^bb0(%a: f32, %b: vector<4xf32>):
    %c = "some_compute"(%a, %b): (f32, vector<4xf32>) -> (vector<4xf32>)
    linalg.yield %c: vector<4xf32>
  }
  return
}
```

The property "*Input and Output Operands Define The Iteration Space*" is
materialized by a lowering into a form that will resemble:

```mlir
// Run: mlir-opt example1.mlir -allow-unregistered-dialect -convert-linalg-to-loops
// This converted representation is in the `scf` dialect.
// It's syntax can be found here: https://mlir.llvm.org/docs/Dialects/SCFDialect/
#map0 = affine_map<(d0) -> (d0 * 2 + 1)>

func @example(%arg0: memref<?xf32>, %arg1: memref<?xvector<4xf32>, #map0>) {
  %c0 = constant 0 : index
  %c1 = constant 1 : index
  %0 = dim %arg0, %c0 : memref<?xf32>
  scf.for %arg2 = %c0 to %0 step %c1 {
    %1 = load %arg0[%arg2] : memref<?xf32>
    %2 = load %arg1[%arg2] : memref<?xvector<4xf32>, #map0>
    %3 = "some_compute"(%1, %2) : (f32, vector<4xf32>) -> vector<4xf32>
    store %3, %arg1[%arg2] : memref<?xvector<4xf32>, #map0>
  }
  return
}
```

The property participates in simplifying analyses and transformations. For
instance, it guarantees no out-of bounds access can occur by construction
(assuming dynamic operand dimensions agree with each other, which is the purpose
of the `assert` runtime check).

Before lowering to loop form, loop induction variables and iterators are
implicit (i.e. *not yet materialized*).

The main implications are that:

1.  The semantics of the ops are *restricted to operate on structured data
    types*, on which we can define an iterator.

2.  This does not model arbitrary code with side-effects.

We do not think these are serious limitations in practice because MLIR is all
about mixing different levels of abstractions in the same IR. As long as Linalg
can progressively lower to the next level of abstraction, it can also be just
bypassed for things that do not fit.

At the same time, conditioning op semantics on structured data types is a very
promising path towards extensibility to non-dense tensors as experience with
LIFT abstractions for
[sparse](https://www.lift-project.org/publications/2016/harries16sparse.pdf) and
[position-dependent arrays](https://www.lift-project.org/publications/2019/pizzuti19positiondependentarrays.pdf),
as well as [TACO](http://tensor-compiler.org/), has shown.

#### Property 2: Reversible Mappings Between Control and Data Structures<a name="prop2"></a>

A `linalg.generic` *defines* the mapping between the iteration space (i.e. the
loops) and the data.

Consider the following fully specified `linalg.generic` example. Here, the first
`memref` is a 2-strided one on both of its dimensions, and the second `memref`
uses an identity layout.

```mlir
// File name: example2.mlir
#indexing_maps = [
  affine_map<(i, j) -> (j, i)>,
  affine_map<(i, j) -> (j)>
]

#attrs = {
  indexing_maps = #indexing_maps,
  iterator_types = ["parallel", "parallel"]
}

func @example(%A: memref<8x?xf32, offset: 0, strides: [2, 2]>,
              %B: memref<?xvector<4xf32>>) {
  linalg.generic #attrs
  ins(%A: memref<8x?xf32, offset: 0, strides: [2, 2]>)
  outs(%B: memref<?xvector<4xf32>>) {
  ^bb0(%a: f32, %b: vector<4xf32>):
    %c = "some_compute"(%a, %b): (f32, vector<4xf32>) -> (vector<4xf32>)
    linalg.yield %c: vector<4xf32>
  }
  return
}
```

The property "*Reversible Mappings Between Control and Data Structures*" is
materialized by a lowering into a form that will resemble:

```mlir
// Run: mlir-opt example2.mlir -allow-unregistered-dialect -convert-linalg-to-loops
#map0 = affine_map<(d0, d1) -> (d0 * 2 + d1 * 2)>

func @example(%arg0: memref<8x?xf32, #map0>, %arg1: memref<?xvector<4xf32>>) {
  %c8 = constant 8 : index
  %c0 = constant 0 : index
  %c1 = constant 1 : index
  %0 = dim %arg0, %c1 : memref<8x?xf32, #map0>
  scf.for %arg2 = %c0 to %0 step %c1 {
    scf.for %arg3 = %c0 to %c8 step %c1 {
      %1 = load %arg0[%arg3, %arg2] : memref<8x?xf32, #map0>
      %2 = load %arg1[%arg3] : memref<?xvector<4xf32>>
      %3 = "some_compute"(%1, %2) : (f32, vector<4xf32>) -> vector<4xf32>
      store %3, %arg1[%arg3] : memref<?xvector<4xf32>>
    }
  }
  return
}
```

This mapping needs to be reversible because we want to be able to go back and
forth between the two and answer questions such as:

-   Given a subset of the iteration space, what subset of data does it read and
    write?
-   Given a subset of data read or written, what subset of the iteration space
    is responsible for this read or write?

Answering these `2` questions is one of the main analyses that Linalg uses to
implement transformations such as tiling, tiled producer-consumer fusion, and
promotion to temporary buffers in fast memory.

In the current implementation, `linalg.generic` uses a list of
[AffineMaps](https://mlir.llvm.org/docs/LangRef/#affinemap-attribute) (see the
`#indexing_maps` attribute in the previous examples). This is a pragmatic
short-term solution, but in the longer term note that this property could be
even evaluated dynamically, similarly to inspector-executor algorithms.

#### Property 3: The Type Of Iterators is Defined Explicitly<a name="prop3"></a>

A `linalg.generic` op fully *declares* the type of its iterators. This
information is used in transformations.

These properties are derived from established practice in the field and mirror
the properties from Ken Kennedy's
[Optimizing Compilers for Modern Architectures](https://www.elsevier.com/books/optimizing-compilers-for-modern-architectures/allen/978-0-08-051324-9).
The key idea of legality of loop transformations expressed by Kennedy is that
***the lexicographic order of all dependence vectors must be preserved***.

This can be better captured directly at the loop level thanks to specific
iterator types, among which: *parallel*, *reduction*, *partition*,
*permutable/monotonic*, *sequential*, *dependence distance*, ...

These types are traditionally the result of complex dependence analyses and have
been referred to as "*bands*" in the polyhedral community (e.g. *parallel
bands*, *permutable bands*, etc, in
[ISL](https://en.wikipedia.org/wiki/Integer_set_library) schedule tree
parlance).

Specifying the information declaratively in a `linalg.generic` allows conveying
properties that may be hard (or even impossible) to derive from lower-level
information. These properties can be brought all the way to the moment when they
are useful for transformations, used and then discarded.

Additionally, these properties may also be viewed as a contract that the
frontend/user guarantees and that the compiler may take advantage of. The common
example is the use of data-dependent reduction semantics for specifying
histogram computations. If the frontend has additional knowledge that proper
atomic operations are available, it may be better to specify parallel semantics
and use the special atomic in the computation region.

At this time, Linalg only has an explicit use for *parallel* and *reduction*
loops but previous experience shows that the abstraction generalizes.

#### Property 4: The Compute Payload is Specified With a Region<a name="prop4"></a>

A `linalg.generic` op has a compute payload that is fully generic thanks to the
use of
[Regions](https://github.com/llvm/llvm-project/blob/58265ad42a90ae8905be6a447cb42e53529a54a0/mlir/docs/LangRef.md#regions).

The region takes as arguments the scalar elemental types of the tensor or buffer
operands of the `linalg.generic`. For flexibility and ability to match library
calls, additional special values may be passed. For instance, a `linalg.fill`
operation takes a buffer and an additional scalar value.

At this time there are no additional restrictions to the region semantics. This
is meant to allow the exploration of various design tradeoffs at the
intersection of regions and iterator types. In particular, the frontend is
responsible for the semantics of iterator types to correspond to the operations
inside the region: the region can capture buffers arbitrarily and write into
them. If this conflicts with some parallel iterator requirement, this is
undefined behavior.

Previous examples already elaborate compute payloads with an unregistered
function `"some_compute"`. The following code snippet shows what the result will
be when using a concrete operation `addf`:

```mlir
// File name: example3.mlir
#map = affine_map<(i, j) -> (i, j)>

#attrs = {
  indexing_maps = [#map, #map, #map],
  iterator_types = ["parallel", "parallel"]
}

func @example(%A: memref<?x?xf32>, %B: memref<?x?xf32>, %C: memref<?x?xf32>) {
  linalg.generic #attrs
  ins(%A, %B: memref<?x?xf32>, memref<?x?xf32>)
  outs(%C: memref<?x?xf32>) {
    ^bb0(%a: f32, %b: f32, %c: f32):
      %d = addf %a, %b : f32
      linalg.yield %d : f32
  }

  return
}
```

This function basically element-wise adds up two matrices (`%A` and `%B`) and
stores the result into another one (`%C`).

The property "*The Compute Payload is Specified With a Region*" is materialized
by a lowering into a form that will resemble:

```mlir
func @example(%arg0: memref<?x?xf32>, %arg1: memref<?x?xf32>, %arg2: memref<?x?xf32>) {
  %c0 = constant 0 : index
  %c1 = constant 1 : index
  %0 = dim %arg0, %c0 : memref<?x?xf32>
  %1 = dim %arg0, %c1 : memref<?x?xf32>
  scf.for %arg3 = %c0 to %0 step %c1 {
    scf.for %arg4 = %c0 to %1 step %c1 {
      %2 = load %arg0[%arg3, %arg4] : memref<?x?xf32>
      %3 = load %arg1[%arg3, %arg4] : memref<?x?xf32>
      %4 = addf %2, %3 : f32
      store %4, %arg2[%arg3, %arg4] : memref<?x?xf32>
    }
  }
  return
}
```

In the process of lowering to loops and lower-level constructs, similar
requirements are encountered, as are discussed in the
[inlined call op proposal](https://llvm.discourse.group/t/introduce-std-inlined-call-op-proposal/282/2).
We expect to be able to reuse the common lower-level infrastructure provided it
evolves to support both region arguments and captures.

#### Property 5: May Map To an External Library Call<a name="prop5"></a>

A `linalg.generic` op may map to an external library call by specifying a
`SymbolAttr`. At this level of abstraction, the important glue is the ability to
perform transformations that preserve the structure necessary to ***call the
external library after different transformations have been applied***.

This involves considerations related to preservation of op semantics and
integration at the ABI level. Regardless of whether one wants to use external
library calls or a custom ISA, the problem for codegen is similar: preservation
of a fixed granularity.

Consider the following example that adds an additional attribute
`library_call="pointwise_add"` that specifies the name of an external library
call we intend to use:

```mlir
// File name: example4.mlir
#indexing_maps = [
  affine_map<(i, j) -> (i, j)>,
  affine_map<(i, j) -> (i, j)>,
  affine_map<(i, j) -> (i, j)>
]

#attrs = {
  indexing_maps = #indexing_maps,
  iterator_types = ["parallel", "parallel"],
  library_call = "pointwise_add"
}

func @example(%A: memref<?x?xf32>, %B: memref<?x?xf32>, %C: memref<?x?xf32>) {
  linalg.generic #attrs
  ins(%A, %B: memref<?x?xf32>, memref<?x?xf32>)
  outs(%C: memref<?x?xf32>) {
  ^bb0(%a: f32, %b: f32, %c: f32):
    %d = addf %a, %b : f32
    linalg.yield %d : f32
  }
  return
}
```

The property "*Map To an External Library Call*" is materialized by a lowering
into a form that will resemble:

```mlir
// Run: mlir-opt example4.mlir -convert-linalg-to-std
// Note that we lower the Linalg dialect directly to the Standard dialect.
// See this doc: https://mlir.llvm.org/docs/Dialects/Standard/

#map0 = affine_map<(d0, d1)[s0, s1, s2] -> (d0 * s1 + s0 + d1 * s2)>

func @example(%arg0: memref<?x?xf32>, %arg1: memref<?x?xf32>, %arg2: memref<?x?xf32>) {
  %0 = memref_cast %arg0 : memref<?x?xf32> to memref<?x?xf32, #map0>
  %1 = memref_cast %arg1 : memref<?x?xf32> to memref<?x?xf32, #map0>
  %2 = memref_cast %arg2 : memref<?x?xf32> to memref<?x?xf32, #map0>
  call @pointwise_add(%0, %1, %2) : (memref<?x?xf32, #map0>, memref<?x?xf32, #map0>, memref<?x?xf32, #map0>) -> ()
  return
}
func @pointwise_add(memref<?x?xf32, #map0>, memref<?x?xf32, #map0>, memref<?x?xf32, #map0>) attributes {llvm.emit_c_interface}
```

Which, after lowering to LLVM resembles:

```mlir
// Run: mlir-opt example4.mlir -convert-linalg-to-std | mlir-opt -convert-std-to-llvm
// Some generated code are omitted here.
func @example(%arg0: !llvm<"float*">, ...) {
  ...
  llvm.call @pointwise_add(...) : (!llvm<"float*">, ...) -> ()
  return
}

llvm.func @pointwise_add(%arg0: !llvm<"float*">, ...) attributes {llvm.emit_c_interface} {
  ...
  llvm.call @_mlir_ciface_pointwise_add(%9, %19, %29) : (!llvm."{ float*, float*, i64, [2 x i64], [2 x i64] }*">, !llvm<"{ f32*, f32*, i64, [2 x i64], [2 x i64] }*">, !llvm<"{ float*, float*, i64, [2 x i64], [2 x i64] }
*">) -> ()
  llvm.return
}
llvm.func @_mlir_ciface_pointwise_add(!llvm."{ float*, float*, i64, [2 x i64], [2 x i64] }*">, !llvm<"{ f32*, f32*, i64, [2 x i64], [2 x i64] }*">, !llvm<"{ f32*, f32*, i64, [2 x i64], [2 x i64] }*">) attributes {llvm.emit_c_interface}
```

##### Convention For External Library Interoperability

The `linalg` dialect adopts a convention that is similar to `BLAS` when
offloading operations to fast library implementations: pass a non-owning pointer
to input and output data with additional metadata. This convention is also found
in libraries such as `MKL`, `OpenBLAS`, `BLIS`, `cuBLAS`, `cuDNN`, etc.. and
more generally at interface points across language boundaries (e.g. C++ /
Python).

Generally, `linalg` passes non-owning pointers to View data structures to
pre-compiled library calls linked externally.

There is an
[ongoing discussion](https://llvm.discourse.group/t/lowering-optional-attributes-in-linalg-structuredops-to-standard-dialect/333/3)
on the topic of extending interoperability in the presence of key attributes.

#### Property 6: Perfectly Nested Writes To The Whole Output Operands<a name="prop6"></a>

Perfectly nested loops form a particularly important class of structure that
enables key loop transformations such as tiling and mapping to library calls.
Unfortunately, this type of structure is easily broken by transformations such
as partial loop fusion. Tiling and mapping to library calls become more
challenging, or even infeasible. Linalg ops adopt perfect-nestedness as a
first-class property: the structure cannot be broken and is transported in the
IR by construction.

A `linalg.generic` op represents a perfectly nested loop nest that writes the
entire memory region. This is a structural constraint across regions and loops
that has proven to be key in simplifying transformations.

One particular point to mention is that converting imperfectly nested code into
perfectly nested code can often be done with enough loop distribution and
embedding of conditionals down to the innermost loop level.

Previous experience with Tensor Comprehensions gave us the intuition that
forcing innermost control-flow nesting is a lot like writing data-parallel code
with arrays of boolean values and predication. This type of trick has also been
used before in polyhedral compilers to convert non-affine control into affine
compute dependencies.

While it may be possible to automate such rewrites from generic IR,
`linalg.generic` just forces the semantics for now.

The key implication is that this conversion to deep predication needs to be
undone once we are done with Linalg transformations. After iterators and
induction variables are materialized (i.e. after lowering out of
`linalg.generic` occurred), the overall performance will be greatly influenced
by the quality of canonicalizations, foldings and *Loop Independent Code Motion*
(LICM).

In the grander scheme, the reliance on late LICM was deemed a necessary risk.

#### Putting it Together<a name="summary"></a>

As it stands, the six properties above define the semantics of a
`linalg.generic` op. It is an open question whether all of these semantics are
strictly necessary in practice and whether some should or could be derived
automatically while still maintaining the
[core guiding principles](#guiding_principles).

For the time being, we have settled on the combination of these properties
because of empirical evidence building and working on multiple high-level
compilers. As we lay those down and engage more with the community, we expect
multiple rounds of discussions and design changes to the original architecture.

### Data Representation: Views<a name="views"></a>

The current implementation uses the
[Strided MemRef (a.k.a View)](https://groups.google.com/a/tensorflow.org/forum/#!topic/mlir/MaL8m2nXuio)
abstraction. The name *View* is used interchangeably in `linalg` to signify
*Strided MemRef*. In the future we expect to use other structured data types and
support ragged, mixed-sparse and other types. We expect to draw on the
experience from existing LIFT abstractions for
[sparse](https://www.lift-project.org/publications/2016/harries16sparse.pdf) and
[position-dependent arrays](https://www.lift-project.org/publications/2019/pizzuti19positiondependentarrays.pdf).

### Metadata Ops<a name="metadata_ops"></a>

A set of ops that manipulate metadata but do not move memory. These ops take
`view` operands + extra attributes and return new `view`s. The returned `view`s
generally alias the operand `view`. At the moment the existing ops are:

```
* `std.view`,
* `std.subview`,
* `std.transpose`.
* `linalg.range`,
* `linalg.slice`,
* `linalg.reshape`,
```

Future ops are added on a per-need basis but should include:

```
* `linalg.tile`,
* `linalg.intersection`,
* `linalg.convex_union`,
* `linalg.difference` (would need to work on a list of views).
```

These additional operations correspond to abstractions that have been known to
work in the field of large-scale distributed stencil computations.

In a longer-term future, the abstractions from
[Legion data-centric programming model](https://legion.stanford.edu/overview/)
seem generally appealing.

### Named Payload-Carrying Ops<a name="named_ops"></a>

Additionally, `linalg` provides a small subset of commonly named operations:

```
* `linalg.copy`,
* `linalg.fill`,
* `linalg.dot`,
* `linalg.matmul`,
* `linalg.conv`.
```

These named operations adhere to the `linalg.generic` op interface. Work is in
progress to define declarative mechanisms to automatically generate named ops
from a description in terms of only the generic op interface.

This is the main reason there are only a small number of ops today: we expect
them to be auto-generated from Tablegen soon.

### Named Payload Ops Specification

Linalg provides a declarative specification and a generation tool
(`mlir-linalg-ods-gen`) to automatically produce named ops from a notation that
is inspired by Einstein notation.

The syntax and semantics used in `mlir-linalg-ods-gen` are very much in flight
and borrow from Tensor Comprehensions (TC) but differ in a few dimensions, to
better adapt to Linalg:

1.  The input and output tensor parameters are specified as `id :
    type(symbolic-affine-expression-list)` (e.g. `A : f32(M, N + M)`) and each
    new symbol is discovered eagerly. TC on the other hand does not allow
    general symbolic affine expressions.
1.  The output shapes are specified explicitly, in TC they are always derived
    from the input shapes.
1.  The operations used to specify computations use EDSC intrinsics so that they
    can easily be parsed and emitted into a simple region builder without
    resorting to more general MLIR parsing.
1.  Reduction dimensions are specified with angle bracket notation on the
    operation they apply to (e.g. `std_add<k>` specifies that `k` is a reduction
    dimension). In TC, a reduction is specified with `op=` operator and the
    reduction dimensions are inferred.
1.  The parallel and reduction dimension are ordered by the textual program
    order. For instance, in the comprehension `O(i, j) = std_add<k, l>(...)`,
    `i` (resp. `j`) is a parallel iterator encoded by affine dimension of
    position `0` (resp. `1`); `k` (resp. `l`) is a reduction iterator encoded by
    an affine dimension of position `2` (resp. `3`).
1.  A list of attributes can be defined for the op with the format of `attr(
    strides: 2xi32)` and referenced in comprehension like `strides[0]`. These
    attribute uses will be parsed as affine symbols to generate op definition
    and implementation. For a concrete op instance, the runtime constant values
    from the attributes will be used to replace the affine symbols and simplify
    the indexing maps.

These decisions and syntax are subject to evolution and change. In particular,
op-specific attributes, dynamic ranks, some form of templating, shape
calculation function specification, etc. may be added in the future.

At this time, the following restrictions are imposed on the syntax and
semantics:

1.  Each def may only contain a single comprehension but each comprehension may
    perform multiple updates.
2.  Each tensor may only be used with a single indexing expression.

A `"""`-wrapped doc string can be attached to the named op. It should contain a
oneliner for summary first, followed by lengthy description.

The following specification may be used to define a named `batchmatmul` op:

```
def batchmatmul(A: f32(Batch, M, K), B: f32(K, N)) -> (C: f32(Batch, M, N))
"""Batch matrix-multiply operation.

This operation performs batch matrix-multiply over ...
"""
{
  C(b, m, n) = std_addf<k>(std_mulf(A(b, m, k), B(k, n)));
}
```

When `mlir-linalg-ods-gen -gen-ods-decl=1` is called, the following ODS is
produced:

```
def batchmatmulOp : LinalgNamedStructured_Op<"batchmatmul", [
  NInputs<2>,
  NOutputs<1>,
  NamedStructuredOpTrait]> { ... }
```

When `mlir-linalg-ods-gen -gen-impl=1` is called, the following C++ is produced:

```
llvm::Optional<SmallVector<StringRef, 8>> batchmatmul::referenceIterators() {
  return SmallVector<StringRef, 8>{
    getParallelIteratorTypeName(),
    getParallelIteratorTypeName(),
    getParallelIteratorTypeName(),
    getReductionIteratorTypeName() };
}
llvm::Optional<SmallVector<AffineMap, 8>> batchmatmul::referenceIndexingMaps() {
  MLIRContext *context = getContext();
  AffineExpr d0, d1, d2, d3;
  bindDims(context, d0, d1, d2, d3);
  return SmallVector<AffineMap, 8>{
      AffineMap::get(4, 0, {d0, d1, d3}),
      AffineMap::get(4, 0, {d3, d2}),
      AffineMap::get(4, 0, {d0, d1, d2}) };
}
void batchmatmul::regionBuilder(ArrayRef<BlockArgument> args) {
  using namespace edsc;
  using namespace intrinsics;
  Value _0(args[0]), _1(args[1]), _2(args[2]);
  Value _4 = std_mulf(_0, _1);
  Value _5 = std_addf(_2, _4);
  (linalg_yield(ValueRange{ _5 }));
}
```

## Open Issues and Design Alternatives<a name="open_issues"></a>

Multiple open issues and design alternatives are in flight and it is time to lay
them out for the community to discuss and pick apart:

1.  Should `linalg.generic` support nesting?
1.  Should `linalg.generic` regions take views or only scalars?
1.  Should we try to solve automatic differentiation at this level of
    abstraction?
1.  Are all the six properties really necessary?
1.  Is this relying too much on declarative specification and would we be better
    off relying more on analyses?
1.  Is this general enough for the community's needs? If not how should this be
    extended, if at all? ...

These key questions (and much more) should be really thought of in the general
context of MLIR in which different levels of IR interoperate seamlessly. In
practice, it is not necessary (or beneficial) to try and solve all problems in
the same IR.

## Operations

[include "Dialects/LinalgOps.md"]