MLIR Specification

MLIR (Multi-Level IR) is a compiler intermediate representation with similarities to traditional three-address SSA representations (like LLVM IR or SIL), but which introduces notions from polyhedral loop optimization as first-class concepts. This hybrid design is optimized to represent, analyze, and transform high level dataflow graphs as well as target-specific code generated for high performance data parallel systems. Beyond its representational capabilities, its single continuous design provides a framework to lower from dataflow graphs to high-performance target-specific code.

This document defines and describes the key concepts in MLIR, and is intended to be a dry reference document - the rationale documentation, glossary, and other content are hosted elsewhere.

MLIR is designed to be used in three different forms: a human-readable textual form suitable for debugging, an in-memory form suitable for programmatic transformations and analysis, and a compact serialized form suitable for storage and transport. The different forms all describe the same semantic content. This document describes the human-readable textual form.

[TOC]

High-Level Structure

MLIR is an SSA-based IR, which means that values are defined before use and have scope defined by their dominance relations. Operations may produce zero or more results, and each is a distinct SSA value with its own type defined by the type system.

The unit of code in MLIR is an Operation. Operations allow for representing many different concepts: allocating buffers, producing views to transform them, target-independent arithmetic, target-specific operations, and even arbitrary user-defined high-level operations including the Module and Function operations. Operations may contain Regions that represent a Control Flow Graph (CFG) of Blocks, that contain operations and end with a terminator operation (like branches).

Here's an example of an MLIR module:

// Compute A*B using an implementation of multiply kernel and print the
// result using a TensorFlow op. The dimensions of A and B are partially
// known. The shapes are assumed to match.
func @mul(%A: tensor<100x?xf32>, %B: tensor<?x50xf32>) -> (tensor<100x50xf32>) {
  // Compute the inner dimension of %A using the dim operation.
  %n = dim %A, 1 : tensor<100x?xf32>

  // Allocate addressable "buffers" and copy tensors %A and %B into them.
  %A_m = alloc(%n) : memref<100x?xf32>
  tensor_store %A to %A_m : memref<100x?xf32>

  %B_m = alloc(%n) : memref<?x50xf32>
  tensor_store %B to %B_m : memref<?x50xf32>

  // Call function @multiply passing memrefs as arguments,
  // and getting returned the result of the multiplication.
  %C_m = call @multiply(%A_m, %B_m)
          : (memref<100x?xf32>, memref<?x50xf32>) -> (memref<100x50xf32>)

  dealloc %A_m : memref<100x?xf32>
  dealloc %B_m : memref<?x50xf32>

  // Load the buffer data into a higher level "tensor" value.
  %C = tensor_load %C_m : memref<100x50xf32>
  dealloc %C_m : memref<100x50xf32>

  // Call TensorFlow built-in function to print the result tensor.
  "tf.Print"(%C){message: "mul result"}
                  : (tensor<100x50xf32) -> (tensor<100x50xf32>)

  return %C : tensor<100x50xf32>
}

// A function that multiplies two memrefs and returns the result.
func @multiply(%A: memref<100x?xf32>, %B: memref<?x50xf32>)
          -> (memref<100x50xf32>)  {
  // Compute the inner dimension of %A.
  %n = dim %A, 1 : memref<100x?xf32>

  // Allocate memory for the multiplication result.
  %C = alloc() : memref<100x50xf32>

  // Multiplication loop nest.
  affine.for %i = 0 to 100 {
     affine.for %j = 0 to 50 {
        store 0 to %C[%i, %j] : memref<100x50xf32>
        affine.for %k = 0 to %n {
           %a_v  = load %A[%i, %k] : memref<100x?xf32>
           %b_v  = load %B[%k, %j] : memref<?x50xf32>
           %prod = mulf %a_v, %b_v : f32
           %c_v  = load %C[%i, %j] : memref<100x50xf32>
           %sum  = addf %c_v, %prod : f32
           store %sum, %C[%i, %j] : memref<100x50xf32>
        }
     }
  }
  return %C : memref<100x50xf32>
}

Notation

MLIR has a simple and unambiguous grammar, allowing it to reliably round-trip through a textual form. This is important for development of the compiler - e.g. for understanding the state of code as it is being transformed and writing test cases.

This document describes the grammar using Extended Backus-Naur Form (EBNF).

This is the EBNF grammar used in this document, presented in yellow boxes.

alternation ::= expr0 | expr1 | expr2  // Either expr0 or expr1 or expr2.
sequence    ::= expr0 expr1 expr2      // Sequence of expr0 expr1 expr2.
repetition0 ::= expr*  // 0 or more occurrences.
repetition1 ::= expr+  // 1 or more occurrences.
optionality ::= expr?  // 0 or 1 occurrence.
grouping    ::= (expr) // Everything inside parens is grouped together.
literal     ::= `abcd` // Matches the literal `abcd`.

Code examples are presented in blue boxes.

// This is an example use of the grammar above:
// This matches things like: ba, bana, boma, banana, banoma, bomana...
example ::= `b` (`an` | `om`)* `a`

Common syntax

The following core grammar productions are used in this document:

// TODO: Clarify the split between lexing (tokens) and parsing (grammar).
digit     ::= [0-9]
hex_digit ::= [0-9a-fA-F]
letter    ::= [a-zA-Z]
id-punct  ::= [$._-]

integer-literal ::= decimal-literal | hexadecimal-literal
decimal-literal ::= digit+
hexadecimal-literal ::= `0x` hex_digit+
float-literal ::= [-+]?[0-9]+[.][0-9]*([eE][-+]?[0-9]+)?
string-literal  ::= `"` [^"\n\f\v\r]* `"`   TODO define escaping rules

Not listed here, but MLIR does support comments. They use standard BCPL syntax, starting with a // and going until the end of the line.

Identifiers and keywords

Syntax:

// Identifiers
bare-id ::= (letter|[_]) (letter|digit|[_$.])*
bare-id-list ::= bare-id (`,` bare-id)*
ssa-id ::= `%` suffix-id
suffix-id ::= (digit+ | ((letter|id-punct) (letter|id-punct|digit)*))

symbol-ref-id ::= `@` (suffix-id | string-literal)
ssa-id-list ::= ssa-id (`,` ssa-id)*

// Uses of an SSA value, e.g. in an operand list to an operation.
ssa-use ::= ssa-id
ssa-use-list ::= ssa-use (`,` ssa-use)*

Identifiers name entities such as SSA values, types and functions, and are chosen by the writer of MLIR code. Identifiers may be descriptive (e.g. %batch_size, @matmul), or may be non-descriptive when they are auto-generated (e.g. %23, @func42). Identifier names for SSA values may be used in an MLIR text file but are not persisted as part of the IR - the printer will give them anonymous names like %42.

MLIR guarantees identifiers never collide with keywords by prefixing identifiers with a sigil (e.g. %, #, @, ^, !). In certain unambiguous contexts (e.g. affine expressions), identifiers are not prefixed, for brevity. New keywords may be added to future versions of MLIR without danger of collision with existing identifiers.

The scope of SSA values is defined based on the standard definition of dominance. Argument identifiers in mapping functions are in scope for the mapping body. Function identifiers and mapping identifiers are visible across the entire module.

Dialects

Dialects are the mechanism by which to engage with and extend the MLIR ecosystem. They allow for defining new operations, as well as attributes and types. Each dialect is given a unique namespace that is prefixed to each defined attribute/operation/type. For example, the Affine dialect defines the namespace: affine.

MLIR allows for multiple dialects, even those outside of the main tree, to co-exist together within one module. Dialects are produced and consumed by certain passes. MLIR provides a framework to convert between, and within, different dialects.

A few of the dialects supported by MLIR:

• Affine dialect
• GPU dialect
• LLVM dialect
• SPIR-V dialect
• Standard dialect
• Vector dialect

Target specific operations

Dialects provide a modular way in which targets can expose target-specific operations directly through to MLIR. As an example, some targets go through LLVM. LLVM has a rich set of intrinsics for certain target-independent operations (e.g. addition with overflow check) as well as providing access to target-specific operations for the targets it supports (e.g. vector permutation operations). LLVM intrinsics in MLIR are represented via operations that start with an "llvm." name.

Example:

// LLVM: %x = call {i16, i1} @llvm.sadd.with.overflow.i16(i16 %a, i16 %b)
%x:2 = "llvm.sadd.with.overflow.i16"(%a, %b) : (i16, i16) -> (i16, i1)

These operations only work when targeting LLVM as a backend (e.g. for CPUs and GPUs), and are required to align with the LLVM definition of these intrinsics.

Operations

Syntax:

operation         ::= op-result-list? (generic-operation | custom-operation)
                      trailing-location?
generic-operation ::= string-literal '(' ssa-use-list? ')' attribute-dict?
                      `:` function-type
custom-operation  ::= bare-id custom-operation-format
op-result-list    ::= op-result (`,` op-result)* `=`
op-result         ::= ssa-id (`:` integer-literal)
successor-list    ::= successor (`,` successor)*
successor         ::= caret-id (`:` bb-arg-list)?
region-list       ::= region (`,` region)*
trailing-location ::= (`loc` `(` location `)`)?

MLIR introduces a uniform concept called operations to enable describing many different levels of abstractions and computations. Operations in MLIR are fully extensible (there is no fixed list of operations) and have application-specific semantics. For example, MLIR supports target-independent operations, affine operations, and target-specific machine operations.

The internal representation of an operation is simple: an operation is identified by a unique string (e.g. dim, tf.Conv2d, x86.repmovsb, ppc.eieio, etc), can return zero or more results, take zero or more SSA operands, may have zero or more attributes, may have zero or more successors, and zero or more enclosed regions. The generic printing form includes all these elements literally, with a function type to indicate the types of the results and operands.

Example:

// An operation that produces two results.
// The results of %result can be accessed via the <name> `#` <opNo> syntax.
%result:2 = "foo_div"() : () -> (f32, i32)

// Pretty form that defines a unique name for each result.
%foo, %bar = "foo_div"() : () -> (f32, i32)

// Invoke a TensorFlow function called tf.scramble with two inputs
// and an attribute "fruit".
%2 = "tf.scramble"(%result#0, %bar) {fruit: "banana"} : (f32, i32) -> f32

In addition to the basic syntax above, dialects may register known operations. This allows those dialects to support custom assembly form for parsing and printing operations. In the operation sets listed below, we show both forms.

Terminator Operations

These are a special category of operations that must terminate a block, e.g. branches. These operations may also have a list of successors (blocks and their arguments).

Example:

// Branch to ^bb1 or ^bb2 depending on the condition %cond.
// Pass value %v to ^bb2, but not to ^bb1.
"cond_br"(%cond)[^bb1, ^bb2(%v : index)] : (i1) -> ()

Module

module ::= `module` symbol-ref-id? (`attributes` attribute-dict)? region

An MLIR module represents an opaque top-level container operation. It contains a single region containing a single block that is comprised of any operations. Operations within this region must not implicitly capture values defined above it. Modules have an optional symbol name that can be used to refer to them in operations.

Functions

An MLIR Function is an operation with a name containing one region. The region of a function is not allowed to implicitly capture values defined outside of the function, and all external references must use function arguments or attributes that establish a symbolic connection (e.g. symbols referenced by name via a string attribute like SymbolRefAttr):

function ::= `func` function-signature function-attributes? function-body?

function-signature ::= symbol-ref-id `(` argument-list `)`
                       (`->` function-result-list)?

argument-list ::= (named-argument (`,` named-argument)*) | /*empty*/
argument-list ::= (type attribute-dict? (`,` type attribute-dict?)*) | /*empty*/
named-argument ::= ssa-id `:` type attribute-dict?

function-result-list ::= function-result-list-parens
                       | non-function-type
function-result-list-parens ::= `(` `)`
                              | `(` function-result-list-no-parens `)`
function-result-list-no-parens ::= function-result (`,` function-result)*
function-result ::= type attribute-dict?

function-attributes ::= `attributes` attribute-dict
function-body ::= region

An external function declaration (used when referring to a function declared in some other module) has no body. While the MLIR textual form provides a nice inline syntax for function arguments, they are internally represented as "block arguments" to the first block in the region.

Only dialect attribute names may be specified in the attribute dictionaries for function arguments, results, or the function itself.

Examples:

// External function definitions.
func @abort()
func @scribble(i32, i64, memref<? x 128 x f32, #layout_map0>) -> f64

// A function that returns its argument twice:
func @count(%x: i64) -> (i64, i64)
  attributes {fruit: "banana"} {
  return %x, %x: i64, i64
}

// A function with an argument attribute
func @example_fn_arg(%x: i32 {swift.self = unit})

// A function with a result attribute
func @example_fn_result() -> (f64 {dialectName.attrName = 0 : i64})

// A function with an attribute
func @example_fn_attr() attributes {dialectName.attrName = false}

Blocks

Syntax:

block           ::= block-label operation+
block-label     ::= block-id block-arg-list? `:`
block-id        ::= caret-id
caret-id        ::= `^` suffix-id
ssa-id-and-type ::= ssa-id `:` type

// Non-empty list of names and types.
ssa-id-and-type-list ::= ssa-id-and-type (`,` ssa-id-and-type)*

block-arg-list ::= `(` ssa-id-and-type-list? `)`

A block is a sequential list of operations without control flow (calls are not considered control flow for this purpose) that are executed from top to bottom. The last operation in a block is a terminator operation, which ends the block.

Blocks in MLIR take a list of block arguments, which represent SSA PHI nodes in a functional notation. The arguments are defined by the block, and values are provided for these block arguments by branches that go to the block.

Here is a simple example function showing branches, returns, and block arguments:

func @simple(i64, i1) -> i64 {
^bb0(%a: i64, %cond: i1): // Code dominated by ^bb0 may refer to %a
  cond_br %cond, ^bb1, ^bb2

^bb1:
  br ^bb3(%a: i64)    // Branch passes %a as the argument

^bb2:
  %b = addi %a, %a : i64
  br ^bb3(%b: i64)    // Branch passes %b as the argument

// ^bb3 receives an argument, named %c, from predecessors
// and passes it on to bb4 twice.
^bb3(%c: i64):
  br ^bb4(%c, %c : i64, i64)

^bb4(%d : i64, %e : i64):
  %0 = addi %d, %e : i64
  return %0 : i64
}

Context: The "block argument" representation eliminates a number of special cases from the IR compared to traditional "PHI nodes are operations" SSA IRs (like LLVM). For example, the parallel copy semantics of SSA is immediately apparent, and function arguments are no longer a special case: they become arguments to the entry block [more rationale].

Regions

Definition

A region is a CFG of MLIR Blocks. Regions serve to group semantically connected blocks, where the semantics is not imposed by the IR. Instead, the containing operation defines the semantics of the regions it contains. Regions do not have a name or an address, only the blocks contained in a region do. Regions are meaningless outside of the containing entity and have no type or attributes.

The first block in the region cannot be a successor of any other block. The syntax for the region is as follows:

region ::= `{` block* `}`

The function body is an example of a region: it consists of a CFG of blocks and has additional semantic restrictions that other types of regions may not have (block terminators must either branch to a different block, or return from a function where the types of the return arguments must match the result types of the function signature).

Control and Value Scoping

Regions provide nested control isolation: it is impossible to branch to a block within a region from outside it or to branch from within a region to a block outside it. Similarly, it provides a natural scoping for value visibility: SSA values defined in a region don't escape to the enclosing region, if any. By default, a region can reference values defined outside of the region whenever it would have been legal to use them as operands to the enclosing operation.

Example:

func @accelerator_compute(i64, i1) -> i64 {
^bb0(%a: i64, %cond: i1): // Code dominated by ^bb0 may refer to %a
  cond_br %cond, ^bb1, ^bb2

^bb1:
  // This def for %value does not dominate ^bb2
  %value = "op.convert"(%a) : (i64) -> i64
  br ^bb3(%a: i64)    // Branch passes %a as the argument

^bb2:
  "accelerator.launch"() {
    ^bb0:
      // Region of code nested under "accelerator.launch", it can reference %a but
      // not %value.
      %new_value = "accelerator.do_something"(%a) : (i64) -> ()
  }
  // %new_value cannot be referenced outside of the region

^bb3:
  ...
}

This can be further restricted using the custom verifier associated with the enclosing operation, for example, disallowing references to values defined outside the region completely.

Control Flow

Regions are Single-Entry-Multiple-Exit (SEME). This means that control can only flow into the first block of the region, but can flow out of the region at the end of any of the contained blocks (This behavior is similar to that of a function body in most programming languages). When exiting a Region, control is returned to the enclosing operation.

The enclosing operation determines the way in which control is transmitted into the entry block of a Region. The successor to a region’s exit points may not necessarily exist: for example a call to a function that does not return. Concurrent or asynchronous execution of regions is unspecified. Operations may define specific rules of execution, e.g. sequential loops or switch cases.

A Region may also enter another region within the enclosing operation. If an operation has multiple regions, the semantics of the operation defines into which regions the control flows and in which order, if any. An operation may transmit control into regions that were specified in other operations, in particular those that defined the values the given operation uses. Thus such operations can be treated opaquely in the enclosing control flow graph, providing a level of control flow isolation similar to that of the call operation.

Closure

Regions allow defining an operation that creates a closure, for example by “boxing” the body of the region into a value they produce. It remains up to the operation to define its semantics. Note that if an operation triggers asynchronous execution of the region, it is under the responsibility of the operation caller to wait for the region to be executed guaranteeing that any directly used values remain live.

Arguments and Results

The arguments of the first block of a region are treated as arguments of the region. The source of these arguments is defined by the semantics of the parent operation. They may correspond to some of the values the operation itself uses.

Regions produce a (possibly empty) list of values. The operation semantics defines the relation between the region results and the operation results.

Type System

Each SSA value in MLIR has a type defined by the type system below. There are a number of primitive types (like integers) and also aggregate types for tensors and memory buffers. MLIR standard types do not include structures, arrays, or dictionaries.

MLIR has an open type system (i.e. there is no fixed list of types), and types may have application-specific semantics. For example, MLIR supports a set of dialect types.

type ::= type-alias | dialect-type | standard-type

type-list-no-parens ::=  type (`,` type)*
type-list-parens ::= `(` `)`
                   | `(` type-list-no-parens `)`

// This is a common way to refer to an SSA value with a specified type.
ssa-use-and-type ::= ssa-use `:` type

// Non-empty list of names and types.
ssa-use-and-type-list ::= ssa-use-and-type (`,` ssa-use-and-type)*

Type Aliases

type-alias-def ::= '!' alias-name '=' 'type' type
type-alias ::= '!' alias-name

MLIR supports defining named aliases for types. A type alias is an identifier that can be used in the place of the type that it defines. These aliases must be defined before their uses. Alias names may not contain a '.', since those names are reserved for dialect types.

Example:

!avx_m128 = type vector<4 x f32>

// Using the original type.
"foo"(%x) : vector<4 x f32> -> ()

// Using the type alias.
"foo"(%x) : !avx_m128 -> ()

Dialect Types

Similarly to operations, dialects may define custom extensions to the type system.

dialect-namespace ::= bare-id

opaque-dialect-item ::= dialect-namespace '<' string-literal '>'

pretty-dialect-item ::= dialect-namespace '.' pretty-dialect-item-lead-ident
                                              pretty-dialect-item-body?

pretty-dialect-item-lead-ident ::= '[A-Za-z][A-Za-z0-9._]*'
pretty-dialect-item-body ::= '<' pretty-dialect-item-contents+ '>'
pretty-dialect-item-contents ::= pretty-dialect-item-body
                              | '(' pretty-dialect-item-contents+ ')'
                              | '[' pretty-dialect-item-contents+ ']'
                              | '{' pretty-dialect-item-contents+ '}'
                              | '[^[<({>\])}\0]+'

dialect-type ::= '!' opaque-dialect-item
dialect-type ::= '!' pretty-dialect-item

Dialect types can be specified in a verbose form, e.g. like this:

// LLVM type that wraps around llvm IR types.
!llvm<"i32*">

// Tensor flow string type.
!tf.string

// Complex type
!foo<"something<abcd>">

// Even more complex type
!foo<"something<a%%123^^^>>>">

Dialect types that are simple enough can use the pretty format, which is a lighter weight syntax that is equivalent to the above forms:

// Tensor flow string type.
!tf.string

// Complex type
!foo.something<abcd>

Sufficiently complex dialect types are required to use the verbose form for generality. For example, the more complex type shown above wouldn't be valid in the lighter syntax: !foo.something<a%%123^^^>>> because it contains characters that are not allowed in the lighter syntax, as well as unbalanced <> characters.

See here to learn how to define dialect types.

Standard Types

Standard types are a core set of dialect types that are defined in a builtin dialect and thus available to all users of MLIR.

standard-type ::=     complex-type
                    | float-type
                    | function-type
                    | index-type
                    | integer-type
                    | memref-type
                    | none-type
                    | tensor-type
                    | tuple-type
                    | vector-type

Complex Type

Syntax:

complex-type ::= `complex` `<` type `>`

The value of complex type represents a complex number with a parameterized element type, which is composed of a real and imaginary value of that element type. The element must be a floating point or integer scalar type.

Examples:

complex<f32>
complex<i32>

Floating Point Types

Syntax:

// Floating point.
float-type ::= `f16` | `bf16` | `f32` | `f64`

MLIR supports float types of certain widths that are widely used as indicated above.

Function Type

Syntax:

// MLIR functions can return multiple values.
function-result-type ::= type-list-parens
                       | non-function-type

function-type ::= type-list-parens `->` function-result-type

MLIR supports first-class functions: for example, the constant operation produces the address of a function as an SSA value. This SSA value may be passed to and returned from functions, merged across control flow boundaries with block arguments, and called with the call_indirect operation.

Function types are also used to indicate the arguments and results of operations.

Index Type

Syntax:

// Target word-sized integer.
index-type ::= `index`

The index type is a signless integer whose size is equal to the natural machine word of the target (rationale) and is used by the affine constructs in MLIR. Unlike fixed-size integers, it cannot be used as an element of vector, tensor or memref type (rationale).

Rationale: integers of platform-specific bit widths are practical to express sizes, dimensionalities and subscripts.

Integer Type

Syntax:

// Sized integers like i1, i4, i8, i16, i32.
integer-type ::= `i` [1-9][0-9]*

MLIR supports arbitrary precision integer types. Integer types are signless, but have a designated width.

Rationale: low precision integers (like i2, i4 etc) are useful for low-precision inference chips, and arbitrary precision integers are useful for hardware synthesis (where a 13 bit multiplier is a lot cheaper/smaller than a 16 bit one).

TODO: Need to decide on a representation for quantized integers (initial thoughts).

Memref Type

Syntax:

memref-type ::= ranked-memref-type | unranked-memref-type

ranked-memref-type ::= `memref` `<` dimension-list-ranked tensor-memref-element-type
                      (`,` layout-specification)? |
                      (`,` memory-space)? `>`

unranked-memref-type ::= `memref` `<*x` tensor-memref-element-type
                         (`,` memory-space)? `>`

stride-list ::= `[` (dimension (`,` dimension)*)? `]`
strided-layout ::= `offset:` dimension `,` `strides: ` stride-list
layout-specification ::= semi-affine-map | strided-layout
memory-space ::= integer-literal /* | TODO: address-space-id */

A memref type is a reference to a region of memory (similar to a buffer pointer, but more powerful). The buffer pointed to by a memref can be allocated, aliased and deallocated. A memref can be used to read and write data from/to the memory region which it references. Memref types use the same shape specifier as tensor types. Note that memref<f32>, memref<0 x f32>, memref<1 x 0 x f32>, and memref<0 x 1 x f32> are all different types.

A memref is allowed to have an unknown rank (e.g. memref<*xf32>). The purpose of unranked memrefs is to allow external library functions to receive memref arguments of any rank without versioning the functions based on the rank. Other uses of this type are disallowed or will have undefined behavior.

Codegen of Unranked Memref

Using unranked memref in codegen besides the case mentioned above is highly discouraged. Codegen is concerned with generating loop nests and specialized instructions for high-performance, unranked memref is concerned with hiding the rank and thus, the number of enclosing loops required to iterate over the data. However, if there is a need to code-gen unranked memref, one possible path is to cast into a static ranked type based on the dynamic rank. Another possible path is to emit a single while loop conditioned on a linear index and perform delinearization of the linear index to a dynamic array containing the (unranked) indices. While this is possible, it is expected to not be a good idea to perform this during codegen as the cost of the translations is expected to be prohibitive and optimizations at this level are not expected to be worthwhile. If expressiveness is the main concern, irrespective of performance, passing unranked memrefs to an external C++ library and implementing rank-agnostic logic there is expected to be significantly simpler.

Unranked memrefs may provide expressiveness gains in the future and help bridge the gap with unranked tensors. Unranked memrefs will not be expected to be exposed to codegen but one may query the rank of an unranked memref (a special op will be needed for this purpose) and perform a switch and cast to a ranked memref as a prerequisite to codegen.

Example:

// With static ranks, we need a function for each possible argument type
%A = alloc() : memref<16x32xf32> %B = alloc() :
memref<16x32x64xf32> call @helper_2D(%A) : (memref<16x32xf32>)->() call
@helper_3D(%B) : (memref<16x32x64xf32>)->()

// With unknown rank, the functions can be unified under one unranked type
%A = alloc() : memref<16x32xf32>
%B = alloc() : memref<16x32x64xf32>
// Remove rank info
%A_u = memref_cast %A : memref<16x32xf32> -> memref<*xf32>
%B_u = memref_cast %B : memref<16x32x64xf32> -> memref<*xf32>
// call same function with dynamic ranks
call @helper(%A_u) : (memref<*xf32>)->()
call @helper(%B_u) : (memref<*xf32>)->()

The core syntax and representation of a layout specification is a semi-affine map. Additionally, syntactic sugar is supported to make certain layout specifications more intuitive to read. For the moment, a memref supports parsing a strided form which is converted to a semi-affine map automatically.

The memory space of a memref is specified by a target-specific integer index. If no memory space is specified, then the default memory space (0) is used. The default space is target specific but always at index 0.

TODO: MLIR will eventually have target-dialects which allow symbolic use of memory hierarchy names (e.g. L3, L2, L1, ...) but we have not spec'd the details of that mechanism yet. Until then, this document pretends that it is valid to refer to these memories by bare-id.

The notionally dynamic value of a memref value includes the address of the buffer allocated, as well as the symbols referred to by the shape, layout map, and index maps.

Examples of memref static type

// Identity index/layout map
#identity = affine_map<(d0, d1) -> (d0, d1)>

// Column major layout.
#col_major = affine_map<(d0, d1, d2) -> (d2, d1, d0)>

// A 2-d tiled layout with tiles of size 128 x 256.
#tiled_2d_128x256 = affine_map<(d0, d1) -> (d0 div 128, d1 div 256, d0 mod 128, d1 mod 256)>

// A tiled data layout with non-constant tile sizes.
#tiled_dynamic = affine_map<(d0, d1)[s0, s1] -> (d0 floordiv s0, d1 floordiv s1,
                             d0 mod s0, d1 mod s1)>

// A layout that yields a padding on two at either end of the minor dimension.
#padded = affine_map<(d0, d1) -> (d0, (d1 + 2) floordiv 2, (d1 + 2) mod 2)>


// The dimension list "16x32" defines the following 2D index space:
//
//   { (i, j) : 0 <= i < 16, 0 <= j < 32 }
//
memref<16x32xf32, #identity, memspace0>

// The dimension list "16x4x?" defines the following 3D index space:
//
//   { (i, j, k) : 0 <= i < 16, 0 <= j < 4, 0 <= k < N }
//
// where N is a symbol which represents the runtime value of the size of
// the third dimension.
//
// %N here binds to the size of the third dimension.
%A = alloc(%N) : memref<16x4x?xf32, #col_major, memspace0>

// A 2-d dynamic shaped memref that also has a dynamically sized tiled layout.
// The memref index space is of size %M x %N, while %B1 and %B2 bind to the
// symbols s0, s1 respectively of the layout map #tiled_dynamic. Data tiles of
// size %B1 x %B2 in the logical space will be stored contiguously in memory.
// The allocation size will be (%M ceildiv %B1) * %B1 * (%N ceildiv %B2) * %B2
// f32 elements.
%T = alloc(%M, %N) [%B1, %B2] : memref<?x?xf32, #tiled_dynamic>

// A memref that has a two-element padding at either end. The allocation size
// will fit 16 * 68 float elements of data.
%P = alloc() : memref<16x64xf32, #padded>

// Affine map with symbol 's0' used as offset for the first dimension.
#imapS = affine_map<(d0, d1) [s0] -> (d0 + s0, d1)>
// Allocate memref and bind the following symbols:
// '%n' is bound to the dynamic second dimension of the memref type.
// '%o' is bound to the symbol 's0' in the affine map of the memref type.
%n = ...
%o = ...
%A = alloc (%n)[%o] : <16x?xf32, #imapS>

Index Space

A memref dimension list defines an index space within which the memref can be indexed to access data.

Index

Data is accessed through a memref type using a multidimensional index into the multidimensional index space defined by the memref's dimension list.

Examples

// Allocates a memref with 2D index space:
//   { (i, j) : 0 <= i < 16, 0 <= j < 32 }
%A = alloc() : memref<16x32xf32, #imapA, memspace0>

// Loads data from memref '%A' using a 2D index: (%i, %j)
%v = load %A[%i, %j] : memref<16x32xf32, #imapA, memspace0>

Index Map

An index map is a one-to-one semi-affine map that transforms a multidimensional index from one index space to another. For example, the following figure shows an index map which maps a 2-dimensional index from a 2x2 index space to a 3x3 index space, using symbols S0 and S1 as offsets.

The number of domain dimensions and range dimensions of an index map can be different, but must match the number of dimensions of the input and output index spaces on which the map operates. The index space is always non-negative and integral. In addition, an index map must specify the size of each of its range dimensions onto which it maps. Index map symbols must be listed in order with symbols for dynamic dimension sizes first, followed by other required symbols.

Layout Map

A layout map is a semi-affine map which encodes logical to physical index space mapping, by mapping input dimensions to their ordering from most-major (slowest varying) to most-minor (fastest varying). Therefore, an identity layout map corresponds to a row-major layout. Identity layout maps do not contribute to the MemRef type identification and are discarded on construction. That is, a type with an explicit identity map is memref<?x?xf32, (i,j)->(i,j)> is strictly the same as the one without layout maps, memref<?x?xf32>.

Layout map examples:

// MxN matrix stored in row major layout in memory:
#layout_map_row_major = (i, j) -> (i, j)

// MxN matrix stored in column major layout in memory:
#layout_map_col_major = (i, j) -> (j, i)

// MxN matrix stored in a 2-d blocked/tiled layout with 64x64 tiles.
#layout_tiled = (i, j) -> (i floordiv 64, j floordiv 64, i mod 64, j mod 64)

Affine Map Composition

A memref specifies a semi-affine map composition as part of its type. A semi-affine map composition is a composition of semi-affine maps beginning with zero or more index maps, and ending with a layout map. The composition must be conformant: the number of dimensions of the range of one map, must match the number of dimensions of the domain of the next map in the composition.

The semi-affine map composition specified in the memref type, maps from accesses used to index the memref in load/store operations to other index spaces (i.e. logical to physical index mapping). Each of the semi-affine maps and thus its composition is required to be one-to-one.

The semi-affine map composition can be used in dependence analysis, memory access pattern analysis, and for performance optimizations like vectorization, copy elision and in-place updates. If an affine map composition is not specified for the memref, the identity affine map is assumed.

Strided MemRef

A memref may specify strides as part of its type. A stride specification is a list of integer values that are either static or ? (dynamic case). Strides encode the distance, in number of elements, in (linear) memory between successive entries along a particular dimension. A stride specification is syntactic sugar for an equivalent strided memref representation using semi-affine maps. For example, memref<42x16xf32, offset: 33 strides: [1, 64]> specifies a non-contiguous memory region of 42 by 16 f32 elements such that:

1. the minimal size of the enclosing memory region must be 33 + 42 * 1 + 16 * 64 = 1066 elements;
2. the address calculation for accessing element (i, j) computes 33 + i + 64 * j
3. the distance between two consecutive elements along the outer dimension is 1 element and the distance between two consecutive elements along the outer dimension is 64 elements.

This corresponds to a column major view of the memory region and is internally represented as the type memref<42x16xf32, (i, j) -> (33 + i + 64 * j)>.

The specification of strides must not alias: given an n-D strided memref, indices (i1, ..., in) and (j1, ..., jn) may not refer to the same memory address unless i1 == j1, ..., in == jn.

Strided memrefs represent a view abstraction over preallocated data. They are constructed with special ops, yet to be introduced. Strided memrefs are a special subclass of memrefs with generic semi-affine map and correspond to a normalized memref descriptor when lowering to LLVM.

None Type

Syntax:

none-type ::= `none`

The none type is a unit type, i.e. a type with exactly one possible value, where its value does not have a defined dynamic representation.

Tensor Type

Syntax:

tensor-type ::= `tensor` `<` dimension-list tensor-memref-element-type `>`
tensor-memref-element-type ::= vector-element-type | vector-type | complex-type

// memref requires a known rank, but tensor does not.
dimension-list ::= dimension-list-ranked | (`*` `x`)
dimension-list-ranked ::= (dimension `x`)*
dimension ::= `?` | decimal-literal

SSA values of tensor type represents aggregate N-dimensional data values, and have a known element type. It may have an unknown rank (indicated by *) or may have a fixed rank with a list of dimensions. Each dimension may be a static non-negative decimal constant or be dynamically determined (indicated by ?).

The runtime representation of the MLIR tensor type is intentionally abstracted - you cannot control layout or get a pointer to the data. For low level buffer access, MLIR has a memref type. This abstracted runtime representation holds both the tensor data values as well as information about the (potentially dynamic) shape of the tensor. The dim operation returns the size of a dimension from a value of tensor type.

Note: hexadecimal integer literals are not allowed in tensor type declarations to avoid confusion between 0xf32 and 0 x f32. Zero sizes are allowed in tensors and treated as other sizes, e.g., tensor<0 x 1 x i32> and tensor<1 x 0 x i32> are different types. Since zero sizes are not allowed in some other types, such tensors should be optimized away before lowering tensors to vectors.

Examples:

// Tensor with unknown rank.
tensor<* x f32>

// Known rank but unknown dimensions.
tensor<? x ? x ? x ? x f32>

// Partially known dimensions.
tensor<? x ? x 13 x ? x f32>

// Full static shape.
tensor<17 x 4 x 13 x 4 x f32>

// Tensor with rank zero. Represents a scalar.
tensor<f32>

// Zero-element dimensions are allowed.
tensor<0 x 42 x f32>

// Zero-element tensor of f32 type (hexadecimal literals not allowed here).
tensor<0xf32>

Tuple Type

Syntax:

tuple-type ::= `tuple` `<` (type ( `,` type)*)? `>`

The value of tuple type represents a fixed-size collection of elements, where each element may be of a different type.

Rationale: Though this type is first class in the type system, MLIR provides no standard operations for operating on tuple types (rationale).

Examples:

// Empty tuple.
tuple<>

// Single element
tuple<f32>

// Many elements.
tuple<i32, f32, tensor<i1>, i5>

Vector Type

Syntax:

vector-type ::= `vector` `<` static-dimension-list vector-element-type `>`
vector-element-type ::= float-type | integer-type

static-dimension-list ::= (decimal-literal `x`)+

The vector type represents a SIMD style vector, used by target-specific operation sets like AVX. While the most common use is for 1D vectors (e.g. vector<16 x f32>) we also support multidimensional registers on targets that support them (like TPUs).

Vector shapes must be positive decimal integers.

Note: hexadecimal integer literals are not allowed in vector type declarations, vector<0x42xi32> is invalid because it is interpreted as a 2D vector with shape (0, 42) and zero shapes are not allowed.

Attributes

Syntax:

attribute-dict ::= `{` `}`
                 | `{` attribute-entry (`,` attribute-entry)* `}`
attribute-entry ::= dialect-attribute-entry | dependent-attribute-entry
dialect-attribute-entry ::= dialect-namespace `.` bare-id `=` attribute-value
dependent-attribute-entry ::= dependent-attribute-name `=` attribute-value
dependent-attribute-name ::= (letter|[_]) (letter|digit|[_$])*

Attributes are the mechanism for specifying constant data on operations in places where a variable is never allowed - e.g. the index of a dim operation, or the stride of a convolution. They consist of a name and a concrete attribute value. The set of expected attributes, their structure, and their interpretation are all contextually dependent on what they are attached to.

There are two main classes of attributes: dependent and dialect. Dependent attributes derive their structure and meaning from what they are attached to; e.g., the meaning of the index attribute on a dim operation is defined by the dim operation. Dialect attributes, on the other hand, derive their context and meaning from a specific dialect. An example of a dialect attribute may be a swift.self function argument attribute that indicates an argument is the self/context parameter. The context of this attribute is defined by the swift dialect and not the function argument.

Attribute values are represented by the following forms:

attribute-value ::= attribute-alias | dialect-attribute | standard-attribute

Attribute Value Aliases

attribute-alias ::= '#' alias-name '=' attribute-value
attribute-alias ::= '#' alias-name

MLIR supports defining named aliases for attribute values. An attribute alias is an identifier that can be used in the place of the attribute that it defines. These aliases must be defined before their uses. Alias names may not contain a '.', since those names are reserved for dialect attributes.

Example:

#map = affine_map<(d0) -> (d0 + 10)>

// Using the original attribute.
%b = affine.apply affine_map<(d0) -> (d0 + 10)> (%a)

// Using the attribute alias.
%b = affine.apply #map(%a)

Dialect Attribute Values

Similarly to operations, dialects may define custom attribute values. The syntactic structure of these values is identical to custom dialect type values, except that dialect attributes values are distinguished with a leading '#', while dialect types are distinguished with a leading '!'.

dialect-attribute ::= '#' opaque-dialect-item
dialect-attribute ::= '#' pretty-dialect-item

Dialect attributes can be specified in a verbose form, e.g. like this:

// Complex attribute
#foo<"something<abcd>">

// Even more complex attribute
#foo<"something<a%%123^^^>>>">

Dialect attributes that are simple enough can use the pretty format, which is a lighter weight syntax that is equivalent to the above forms:

// Complex attribute
#foo.something<abcd>

Sufficiently complex dialect attributes are required to use the verbose form for generality. For example, the more complex type shown above wouldn't be valid in the lighter syntax: #foo.something<a%%123^^^>>> because it contains characters that are not allowed in the lighter syntax, as well as unbalanced <> characters.

See here to learn how to define dialect attribute values.

Standard Attribute Values

Standard attributes are a core set of dialect attributes that are defined in a builtin dialect and thus available to all users of MLIR.

standard-attribute ::=   affine-map-attribute
                       | array-attribute
                       | bool-attribute
                       | dictionary-attribute
                       | elements-attribute
                       | float-attribute
                       | integer-attribute
                       | integer-set-attribute
                       | string-attribute
                       | symbol-ref-attribute
                       | type-attribute
                       | unit-attribute

AffineMap Attribute

Syntax:

affine-map-attribute ::= `affine_map` `<` affine-map `>`

An affine-map attribute is an attribute that represents a affine-map object.

Array Attribute

Syntax:

array-attribute ::= `[` (attribute-value (`,` attribute-value)*)? `]`

An array attribute is an attribute that represents a collection of attribute values.

Boolean Attribute

Syntax:

bool-attribute ::= bool-literal

A boolean attribute is a literal attribute that represents a one-bit boolean value, true or false.

Dictionary Attribute

Syntax:

dictionary-attribute ::= `{` (attribute-entry (`,` attribute-entry)*)? `}`

A dictionary attribute is an attribute that represents a sorted collection of named attribute values. The elements are sorted by name, and each name must be unique within the collection.

Elements Attributes

Syntax:

elements-attribute ::= dense-elements-attribute
                     | opaque-elements-attribute
                     | sparse-elements-attribute

An elements attribute is a literal attribute that represents a constant vector or tensor value.

Dense Elements Attribute

Syntax:

dense-elements-attribute ::= `dense` `<` attribute-value `>` `:`
                             ( tensor-type | vector-type )

A dense elements attribute is an elements attribute where the storage for the constant vector or tensor value has been packed to the element bitwidth. The element type of the vector or tensor constant must be of integer, index, or floating point type.

Opaque Elements Attribute

Syntax:

opaque-elements-attribute ::= `opaque` `<` dialect-namespace  `,`
                              hex-string-literal `>` `:`
                              ( tensor-type | vector-type )

An opaque elements attribute is an elements attribute where the content of the value is opaque. The representation of the constant stored by this elements attribute is only understood, and thus decodable, by the dialect that created it.

Note: The parsed string literal must be in hexadecimal form.

Sparse Elements Attribute

Syntax:

sparse-elements-attribute ::= `sparse` `<` attribute-value `,` attribute-value
                              `>` `:` ( tensor-type | vector-type )

A sparse elements attribute is an elements attribute that represents a sparse vector or tensor object. This is where very few of the elements are non-zero.

The attribute uses COO (coordinate list) encoding to represent the sparse elements of the elements attribute. The indices are stored via a 2-D tensor of 64-bit integer elements with shape [N, ndims], which specifies the indices of the elements in the sparse tensor that contains non-zero values. The element values are stored via a 1-D tensor with shape [N], that supplies the corresponding values for the indices.

Example:

  sparse<[[0, 0], [1, 2]], [1, 5]> : tensor<3x4xi32>

// This represents the following tensor:
///  [[1, 0, 0, 0],
///   [0, 0, 5, 0],
///   [0, 0, 0, 0]]

Float Attribute

Syntax:

float-attribute ::= (float-literal (`:` float-type)?)
                  | (hexadecimal-literal `:` float-type)

A float attribute is a literal attribute that represents a floating point value of the specified float type. It can be represented in the hexadecimal form where the hexadecimal value is interpreted as bits of the underlying binary representation. This form is useful for representing infinity and NaN floating point values. To avoid confusion with integer attributes, hexadecimal literals must be followed by a float type to define a float attribute.

Examples:

42.0         // float attribute defaults to f64 type
42.0 : f32   // float attribute of f32 type
0x7C00 : f16 // positive infinity
0x7CFF : f16 // NaN (one of possible values)
42 : f32     // Error: expected integer type

Integer Attribute

Syntax:

integer-attribute ::= integer-literal ( `:` (index-type | integer-type) )?

An integer attribute is a literal attribute that represents an integral value of the specified integer or index type. The default type for this attribute, if one is not specified, is a 64-bit integer.

Integer Set Attribute

Syntax:

integer-set-attribute ::= `affine_set` `<` integer-set `>`

An integer-set attribute is an attribute that represents an integer-set object.

String Attribute

Syntax:

string-attribute ::= string-literal (`:` type)?

A string attribute is an attribute that represents a string literal value.

Symbol Reference Attribute

Syntax:

symbol-ref-attribute ::= symbol-ref-id (`::` symbol-ref-id)*

A symbol reference attribute is a literal attribute that represents a named reference to an operation that is nested within an operation with the OpTrait::SymbolTable trait. As such, this reference is given meaning by the nearest parent operation containing the OpTrait::SymbolTable trait. It may optionally contain a set of nested references that further resolve to a symbol nested within a different symbol table.

This attribute can only be held internally by array attributes and dictionary attributes(including the top-level operation attribute dictionary), i.e. no other attribute kinds such as Locations or extended attribute kinds. If a reference to a symbol is necessary from outside of the symbol table that the symbol is defined in, a string attribute can be used to refer to the symbol name.

Rationale: Given that MLIR models global accesses with symbol references, to enable efficient multi-threading, it becomes difficult to effectively reason about their uses. By restricting the places that can legally hold a symbol reference, we can always opaquely reason about a symbols usage characteristics.

Type Attribute

Syntax:

type-attribute ::= type

A type attribute is an attribute that represents a type object.

Unit Attribute

unit-attribute ::= `unit`

A unit attribute is an attribute that represents a value of unit type. The unit type allows only one value forming a singleton set. This attribute value is used to represent attributes that only have meaning from their existence.

One example of such an attribute could be the swift.self attribute. This attribute indicates that a function parameter is the self/context parameter. It could be represented as a boolean attribute(true or false), but a value of false doesn't really bring any value. The parameter either is the self/context or it isn't.

// A unit attribute defined with the `unit` value specifier.
func @verbose_form(i1) attributes {dialectName.unitAttr = unit}

// A unit attribute can also be defined without the value specifier.
func @simple_form(i1) attributes {dialectName.unitAttr}