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MLIR Specification

MLIR 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.

MLIR stands for one of "Multi-Level IR" or "Multi-dimensional Loop IR" or "Machine Learning IR" - the MLIR team prefers the first interpretation. This document defines and describes the key concepts in MLIR, and is intended to be a dry reference document - rationale documentation, system overview documentation and other content is 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

The top-level unit of code in MLIR is a Module. A module contains a list of Functions. Functions are represented as a composition of operations and contain a Control Flow Graph (CFG) of Blocks, which contain instructions and end with terminator operations (like branches).

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.

MLIR incorporates polyhedral compiler concepts, including for and if instructions, which model affine loops and affine conditionals. It also includes affine maps integrated into the type system - they are key to the representation of data and MemRefs, which are the representation for tensors in addressable memory. MLIR also supports a first-class Tensor type allowing it to concisely represent operations on N-dimensional arrays.

Finally, MLIR supports operations for allocating buffers, producing views to transform them, represent target-independent arithmetic, target-specific instructions, and even supports arbitrary user-defined high-level tensor operations.

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 memref<100x?xf32>(%n)
  tensor_store %A to %A_m : memref<100x?xf32>

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

  // Call ML 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.
  for  %i = 0 to 100 {
     for %j = 0 to 50 {
        store 0 to %C[%i, %j] : memref<100x50xf32>
        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. understanding the state of code as it is being transformed and for 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 ::= digit+ | `0x` hex_digit+
float-literal   ::= TODO
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)*
suffix-id ::= digit+ | ((letter|id-punct) (letter|id-punct|digit)*)

function-id ::= `@` bare-id
ssa-id ::= `%` suffix-id
ssa-id-list ::= ssa-id (`,` ssa-id)*

// Uses of an SSA value, e.g. in an operand list to an instruction.
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.

Polyhedral Structures

MLIR uses techniques from polyhedral compilation to make dependence analysis and loop transformations efficient and reliable. This section introduces some of the core concepts that are used throughout the document.

Dimensions and Symbols

Dimensions and symbols are the two kinds of identifiers that can appear in the polyhedral structures, and are always of 'index' type. Dimensions are declared in parentheses and symbols are declared in square brackets.

Examples:

// A 2d to 3d affine mapping.
// d0/d1 are dimensions, s0 is a symbol
#affine_map2to3 = (d0, d1)[s0] -> (d0, d1 + s0, d1 - s0) size (10, 20, 30)

Dimensional identifiers correspond to the dimensions of the underlying structure being represented (a map, set, or more concretely a loop nest or a tensor); for example, a three-dimensional loop nest has three dimensional identifiers. Symbol identifiers represent an unknown quantity that can be treated as constant for a region of interest.

Dimensions and symbols are bound to SSA values by various operations in MLIR and use the same parenthesized vs square bracket list to distinguish the two.

Syntax:

// Uses of SSA values that are passed to dimensional identifiers.
dim-use-list ::= `(` ssa-use-list? `)`

// Uses of SSA values that are used to bind symbols.
symbol-use-list ::= `[` ssa-use-list? `]`

// Most things that bind SSA values bind dimensions and symbols.
dim-and-symbol-use-list ::= dim-use-list symbol-use-list?

SSA values bound to dimensions and symbols must always have 'index' type.

A symbolic identifier can be bound to an SSA value that is either an argument to the function, a value defined at the top level of that function (outside of all loops and if instructions), the result of a constant operation, or the result of an affine_apply operation that recursively takes as arguments any symbolic identifiers. Dimensions may be bound not only to anything that a symbol is bound to, but also to induction variables of enclosing for instructions, and the results of an affine_apply operation (which recursively may use other dimensions and symbols).

Example:

#affine_map2to3 = (d0, d1)[s0] -> (d0, d1 + s0, d1 - s0) size (10,20,30)
// Binds %N to the s0 symbol in affine_map2to3.
%x = alloc()[%N] : memref<40x50xf32, #affine_map2to3>

Affine Expressions

Syntax:

affine-expr ::= `(` affine-expr `)`
              | affine-expr `+` affine-expr
              | affine-expr `-` affine-expr
              | `-`? integer-literal `*` affine-expr
              | affine-expr `ceildiv` integer-literal
              | affine-expr `floordiv` integer-literal
              | affine-expr `mod` integer-literal
              | `-`affine-expr
              | bare-id
              | `-`? integer-literal

multi-dim-affine-expr ::= `(` affine-expr (`,` affine-expr)* `)`

ceildiv is the ceiling function which maps the result of the division of its first argument by its second argument to the smallest integer greater than or equal to that result. floordiv is a function which maps the result of the division of its first argument by its second argument to the largest integer less than or equal to that result. mod is the modulo operation: since its second argument is always positive, its results are always positive in our usage. The integer-literal operand for ceildiv, floordiv, and mod is always expected to be positive. bare-id is an identifier which must have type index. The precedence of operations in an affine expression are ordered from highest to lowest in the order: (1) parenthesization, (2) negation, (3) modulo, multiplication, floordiv, and ceildiv, and (4) addition and subtraction. All of these operators associate from left to right.

A multi-dimensional affine expression is a comma separated list of one-dimensional affine expressions, with the entire list enclosed in parentheses.

Context: An affine function, informally, is a linear function plus a constant. More formally, a function f defined on a vector \vec{v} \in \mathbb{Z}^n$is a multidimensional affine function of\vec{v}$ if

f(\vec{v})

is a constant matrix from \mathbb{Z}^{m \times n} and \vec{c} is a constant vector from \mathbb{Z}. $m$ is the dimensionality of such an affine function. MLIR further extends the definition of an affine function to allow 'floordiv', 'ceildiv', and 'mod' with respect to positive integer constants. Such extensions to affine functions have often been referred to as quasi-affine functions by the polyhedral compiler community. MLIR uses the term 'affine map' to refer to these multi-dimensional quasi-affine functions. As examples, (i+j+1, j), (i \mod 2, j+i), (j, i/4, i \mod 4), (2i+1, j) are two-dimensional affine functions of (i, j), but (i \cdot j, i^2), (i \mod j, i/j) are not affine functions of (i, j).

Affine Maps

Syntax:

affine-map-inline
   ::= dim-and-symbol-id-lists `->` multi-dim-affine-expr
       ( `size` `(` dim-size (`,` dim-size)* `)` )?

dim-size ::= affine-expr
           | `min` `(` affine-expr ( `,` affine-expr)+ `)`

The identifiers in the dimensions and symbols lists must be unique. These are the only identifiers that may appear in 'multi-dim-affine-expr'. In addition, only symbolic identifiers and constants can appear in 'dim-size'. Affine maps with one or more symbols in its specification are known as "symbolic affine maps", and those with no symbols as "non-symbolic affine maps". An affine map has an optional "size" tuple which provides the size for each corresponding dimension. Affine maps with a size in their specification are known as "bounded affine maps", and those without a size are "unbounded affine maps".

Context: Affine maps are mathematical functions that transform a list of dimension indices and symbols into a list of results, with affine expressions combining the indices and symbols. Affine maps distinguish between indices and symbols because indices are inputs to the affine map when the latter is called through an affine_apply operation, whereas symbols are bound when an affine mapping is established (e.g. when a memref is formed, establishing a memory layout map).

Affine maps are used for various core structures in MLIR. The restrictions we impose on their form allows powerful analysis and transformation, while keeping the representation closed with respect to several operations of interest.

Named affine mappings

Syntax:

affine-map-id ::= `#` suffix-id

// Definitions of affine maps are at the top of the file.
affine-map-def    ::= affine-map-id `=` affine-map-inline
module-header-def ::= affine-map-def

// Uses of affine maps may use the inline form or the named form.
affine-map ::= affine-map-id | affine-map-inline

Affine mappings may be defined inline at the point of use, or may be hoisted to the top of the file and given a name with an affine map definition, and used by name.

Examples:

// Affine map out-of-line definition and usage example.
#affine_map42 =
  (d0, d1)[s0] -> (d0, d0 + d1 + floordiv(s0,2)) size (10, s0)

// Use an affine mapping definition in an alloc instruction, binding the
// SSA value %N to the symbol s0.
%a = alloc memref<4x4xf32, #affine_map42> () [%N]

// Same thing with an inline affine mapping definition.
%b = alloc memref<4x4xf32, (d0, d1)[s0] -> (d0, d0 + d1 + floordiv(s0,2))
                                           size (10, s0)> () [%N]

Semi-affine maps

Semi-affine maps are extensions of affine maps to allow multiplication, floordiv, ceildiv, and mod with respect to symbolic identifiers. Semi-affine maps are thus a strict superset of affine maps.

Syntax of semi-affine expressions:

semi-affine-expr ::= `(` semi-affine-expr `)`
                   | semi-affine-expr `+` semi-affine-expr
                   | semi-affine-expr `-` semi-affine-expr
                   | symbol-or-const `*` semi-affine-expr
                   | `ceildiv` `(` semi-affine-expr `,` symbol-or-const `)`
                   | `floordiv` `(` semi-affine-expr `,` symbol-or-const `)`
                   | semi-affine-expr `mod` symbol-or-const
                   | bare-id
                   | `-`? integer-literal

symbol-or-const ::= `-`? integer-literal | symbol-id

multi-dim-semi-affine-expr ::= `(` semi-affine-expr (`,` semi-affine-expr)* `)`

The precedence and associativity of operations in the syntax above is the same as that for affine expressions.

Syntax of semi-affine maps:

semi-affine-map-inline
   ::= dim-and-symbol-id-lists `->` multi-dim-semi-affine-expr
       ( `size` `(` dim-size (`,` dim-size)* `)` )?

Semi-affine maps may be defined inline at the point of use, or may be hoisted to the top of the file and given a name with a semi-affine map definition, and used by name.

semi-affine-map-id ::= `#` suffix-id

// Definitions of semi-affine maps are at the top of file.
semi-affine-map-def ::= semi-affine-map-id `=` semi-affine-map-inline
module-header-def ::= semi-affine-map-def

// Uses of semi-affine maps may use the inline form or the named form.
semi-affine-map ::= semi-affine-map-id | semi-affine-map-inline

Integer Sets

An integer set is a conjunction of affine constraints on a list of identifiers. The identifiers associated with the integer set are separated out into two classes: the set's dimension identifiers, and the set's symbolic identifiers. The set is viewed as being parametric on its symbolic identifiers. In the syntax, the list of set's dimension identifiers are enclosed in parentheses while its symbols are enclosed in square brackets.

Syntax of affine constraints:

affine-constraint ::= affine-expr `>=` `0`
                    | affine-expr `==` `0`
affine-constraint-conjunction ::= /*empty*/
                                | affine-constraint (`,` affine-constraint)*

Integer sets may be defined inline at the point of use, or may be hoisted to the top of the file and given a name with an integer set definition, and used by name.

integer-set-id ::= `#` suffix-id

integer-set-inline
   ::= dim-and-symbol-id-lists `:` affine-constraint-conjunction

// Declarations of integer sets are at the top of the file.
integer-set-decl ::= integer-set-id `=` integer-set-inline

// Uses of integer sets may use the inline form or the named form.
integer-set ::= integer-set-id | integer-set-inline

The dimensionality of an integer set is the number of identifiers appearing in dimension list of the set. The affine-constraint non-terminals appearing in the syntax above are only allowed to contain identifiers from dims and symbols. A set with no constraints is a set that is unbounded along all of the set's dimensions.

Example:

// A example two-dimensional integer set with two symbols.
#set42 = (d0, d1)[s0, s1]
   : d0 >= 0, -d0 + s0 - 1 >= 0, d1 >= 0, -d1 + s1 - 1 >= 0

// Inside an ML Function
if #set42(%i, %j)[%M, %N] {
  ...
}

d0 and d1 correspond to dimensional identifiers of the set, while s0 and s1 are symbol identifiers.

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 does not include complex numbers, tuples, structures, arrays, or dictionaries.

TODO: Revisit this in light of dialect extensible type systems.

type ::= integer-type
       | index-type
       | float-type
       | other-type
       | vector-type
       | tensor-type
       | memref-type
       | function-type

// MLIR doesn't have a tuple type but functions can return multiple values.
type-list ::= type-list-parens | 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)*

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).

Index Type

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).

Syntax:

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

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

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.

Other Types

In addition to the primary integer and floating point types, and derived types, MLIR supports some special purpose types:

// TensorFlow specific types (TODO: the rest ref data types)
other-type ::= `tf_control` | `tf_resource` | `tf_variant` | `tf_string`
               `tf_complex64` | `tf_complex128` | `tf_f32ref`

tf_control is used in TensorFlow graphs to represent control dependence edges.

Function Type

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

MLIR supports first-class functions: 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 instruction.

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

Vector Type

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

const-dimension-list ::= (integer-literal `x`)+

The vector type represents a SIMD style vector, used by target-specific instruction 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).

Tensor Type

Syntax:

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

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

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>

Memref Type

Syntax:

memref-type ::= `memref` `<` dimension-list-ranked tensor-memref-element-type
                (`,` semi-affine-map-composition)? (`,` memory-space)? `>`

semi-affine-map-composition ::= (semi-affine-map `,` )* semi-affine-map
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, but do not allow unknown rank.

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. HBM, VMEM, ...) 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
#imapA = (d0, d1) -> (d0, d1) size (16, 32)

// Column major layout.
#imapB = (d0, d1, d2) [s0] -> (d2, d1, d0) size (s0, 4, 16)

// The dimension list "16x32" defines the following 2D index space:
//
//   { (i, j) : 0 <= i < 16, 0 <= j < 32 }
//
memref<16x32xf32, #imapA, hbm>
// 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.
memref<16x4x?xf32, #imapB, hbm>

Symbol capture example:

// Affine map with symbol 's0' used as offset for first dimension.
#imapA = (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 = <strong>alloc</strong> (%n)[%o] : <16x?xf32, #imapA>

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, hbm>

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

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.

Index map examples:

// Index map from [MS, NS] slice index space to larger [M, N]
// matrix index space at slice offset symbols OI, OJ:
// Maps from [MS, NS] -> [M, N]
#imap_slice = (i, j) [M, N, OI, OJ] -> (i + OI , j + OJ) size (M, N)

// Index map from 4-dimensional tiled index space to
// 2-dimensional index space.
// Maps from [M/128, N/128, 128, 128] -> [M, N]
#imap_tiled = (d0, d1, d2, d3) [M, N] -> (128 * d0 + d2, 128 * d1 + d3)
                                         size (M, N)

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.

Layout map examples:

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

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

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 instructions 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.

Attributes

Syntax:

attribute-dict ::= `{` `}`
                 | `{` attribute-entry (`,` attribute-entry)* `}`
attribute-entry ::= bare-id `:` attribute-value

Attributes are the mechanism for specifying constant data in MLIR in places where a variable is never allowed - e.g. the index of a dim operation, or the stride of a convolution.

Attributes have a name, and their values are represented by the following forms:

attribute-value ::= bool-literal
                  | integer-literal ( `:` (index-type | integer-type) )?
                  | float-literal ( `:` float-type )?
                  | string-literal
                  | affine-map
                  | type
                  | `[` (attribute-value (`,` attribute-value)*)? `]`
                  | function-id `:` function-type

It is possible to attach attributes to instructions and functions, and the set of expected attributes, their structure, and the definition of that meaning is contextually dependent on the operation they are attached to.

Module

module ::= module-header-def* function*

An MLIR module may optionally have a list of header definitions (e.g. affine mappings) at the top of the file, but is principally made up of a list of functions.

TODO: We should allow specifying a "dialect" in the module header. This will prepopulate a symbol table with known named types and mappings (e.g. for TPU) and will define the set of operations that are allowed (allowing the verifier to detect common errors).

Functions

MLIR functions have a signature (including argument and result types) and associated attributes according to the following grammar:

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

function-signature ::= function-id `(` argument-list `)` (`->` type-list)?
argument-list ::= named-argument (`,` named-argument)* | /*empty*/
argument-list ::= type (`,` type)* | /*empty*/ named-argument ::= ssa-id `:`
type

function-attributes ::= `attributes` attribute-dict
function-body ::= `{` block+ `}`

An external function declaration (used when referring to a function declared in some other module) has no body. A function definition contains a control flow graph made up of one or more blocks. 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 function.

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
}

Blocks

Syntax:

block           ::= bb-label instruction+
bb-label        ::= bb-id bb-arg-list? `:`
bb-id           ::= caret-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)*

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

A basic block is a sequential list of instructions without control flow (calls are not considered control flow for this purpose) that are executed from top to bottom. The last instruction 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
  br_cond %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 instructions" 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].

Instruction Kinds

MLIR has three kinds of instructions: dialect-defined operations, an affine for instruction, and an affine if instruction.

inst ::= operation | for-inst | if-inst

'for' instruction {#'for'-instruction}

Syntax:

for-inst ::= `for` ssa-id `=` lower-bound `to` upper-bound
              (`step` integer-literal)? `{` inst* `}`

lower-bound ::= `max`? affine-map dim-and-symbol-use-list | shorthand-bound
upper-bound ::= `min`? affine-map dim-and-symbol-use-list | shorthand-bound
shorthand-bound ::= ssa-id | `-`? integer-literal

The for instruction represents an affine loop nest, defining an SSA value for its induction variable. This SSA value always has type index, which is the size of the machine word.

The for instruction executes its body a number of times iterating from a lower bound to an upper bound by a stride. The stride, represented by step, is a positive constant integer which defaults to "1" if not present. The lower and upper bounds specify a half-open range: the range includes the lower bound but does not include the upper bound.

The lower and upper bounds of a for instruction are represented as an application of an affine mapping to a list of SSA values passed to the map. The same restrictions hold for these SSA values as for all bindings of SSA values to dimensions and symbols.

The affine mappings for the bounds may return multiple results, in which case the max/min keywords are required (for the lower/upper bound respectively), and the bound is the maximum/minimum of the returned values. There is no semantic ambiguity, but MLIR syntax requires the use of these keywords to make things more obvious to human readers.

Many upper and lower bounds are simple, so MLIR accepts two shorthand syntaxes: the form that accepts a single 'ssa-id' (e.g. %N) is shorthand for applying that SSA value to a function that maps a single symbol to itself, e.g., ()[s]->(s)()[%N]. The integer literal form (e.g. -42) is shorthand for a nullary mapping function that returns the constant value (e.g. ()->(-42)()).

Example showing reverse iteration of the inner loop:

#map57 = (d0, d1)[s0] -> (d0, s0 - d1 - 1)

func @simple_example(%A: memref<?x?xf32>, %B: memref<?x?xf32>) {
  %N = dim %A, 0 : memref<?x?xf32>
  for %i = 0 to %N step 1 {
    for %j = 0 to %N {   // implicitly steps by 1
      %0 = affine_apply #map57(%i, %j)[%N]
      %tmp = call @F1(%A, %0#0, %0#1) : (memref<?x?xf32>, index, index)->(f32)
      call @F2(%tmp, %B, %0#0, %0#1) : (f32, memref<?x?xf32>, index, index)->()
    }
  }
  return
}

'if' instruction {#'if'-instruction}

Syntax:

if-inst-head ::= `if` if-inst-cond `{` inst* `}`
               | if-inst-head `else` `if` if-inst-cond `{` inst* `}`
if-inst-cond ::= integer-set dim-and-symbol-use-list

if-inst ::= if-inst-head
          | if-inst-head `else` `{` inst* `}`

The if instruction restricts execution to a subset of the loop iteration space defined by an integer set (a conjunction of affine constraints). A single if may have a number of optional else if clauses, and may end with an optional else clause.

The condition of the if is represented by an integer set (a conjunction of affine constraints), and the SSA values bound to the dimensions and symbols in the integer set. The same restrictions hold for these SSA values as for all bindings of SSA values to dimensions and symbols.

Example:

#set = (d0, d1)[s0]: (d0 - 10 >= 0, s0 - d0 - 9 >= 0,
                      d1 - 10 >= 0, s0 - d1 - 9 >= 0)
func @reduced_domain_example(%A, %X, %N) : (memref<10xi32>, i32, i32) {
  for %i = 0 to %N {
     for %j = 0 to %N {
       %0 = affine_apply #map42(%i, %j)
       %tmp = call @S1(%X, %0#0, %0#1)
       if #set(%i, %j)[%N] {
          %1 = affine_apply #map43(%i, %j)
          call @S2(%tmp, %A, %1#0, %1#1)
       }
    }
  }
  return
}

Operations

Syntax:

operation ::= (ssa-id `=`)? string-literal `(` ssa-use-list? `)`
              attribute-dict? `:` function-type

MLIR represents computation within functions with a uniform concept called operations. 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, high-level TensorFlow ops, and target-specific machine instructions.

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, and may have zero or more attributes. When parsed or printed in the raw form, these are all printed literally, and a function type is used to indicate the types of the results and operands.

Example:

// Invoke a TensorFlow function called tf.scramble with two inputs
// and an attribute "fruit".
%2 = "tf.scramble"(%42, %12){fruit: "banana"} : (f32, i32) -> f32

// Invoke the TPU specific add instruction that takes two vector register
// as input and produces a vector register.
%7 = "tpu.add"(%42, %12)
             : (vector<8x128xf32>, vector<8x128xf32>) -> vector<8x128xf32>

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

Context: TensorFlow has an open "op" ecosystem, and we directly apply these ideas to the design of MLIR, but generalize it much further. To make it easy to reason about IR dumps and manipulate operations in C++, the MLIR compiler infrastructure uses C++ templates to make working with them convenient and safe. The details of this are not described in this document.

Standard Operations

TODO: shape, which returns a 1D tensor, and can take an unknown rank tensor as input.

TODO: rank, which returns an index.

Terminator operations

'br' terminator operation {#'br'-terminator-operation}

Syntax:

operation ::= `br` bb-id branch-use-list?
branch-use-list ::= `(` ssa-use-list `:` type-list-no-parens `)`

The br terminator instruction represents an unconditional jump to a target block. The count and types of operands to the branch must align with the arguments in the target block.

The MLIR branch instruction is not allowed to target the entry block for a function.

'cond_br' terminator operation {#'cond_br'-terminator-operation}

Syntax:

operation ::=
  `cond_br` ssa-use `,` bb-id branch-use-list? `,` bb-id branch-use-list?

The cond_br terminator instruction represents a conditional branch on a boolean (1-bit integer) value. If the bit is set, then the first destination is jumped to; if it is false, the second destination is chosen. The count and types of operands must align with the arguments in the corresponding target blocks.

The MLIR conditional branch instruction is not allowed to target the entry block for a function. The two destinations of the conditional branch instruction are allowed to be the same.

The following example illustrates a CFG function with a conditional branch instruction that targets the same block:

func @select(%a : i32, %b :i32, %flag : i1) -> i32 {
    // Both targets are the same, operands differ
    cond_br %flag, ^bb1(%a : i32), ^bb1(%b : i32)

^bb1(%x : i32) :
    return %x : i32
}

'return' terminator operation {#'return'-terminator-operation}

Syntax:

operation ::= `return` (ssa-use-list `:` type-list-no-parens)?

The return terminator instruction represents the completion of a function, and produces the result values. The count and types of the operands must match the result types of the enclosing function. It is legal for multiple blocks in a single function to return.

Core Operations

'affine_apply' operation {#'affine_apply'-operation}

Syntax:

operation ::= ssa-id `=` `affine_apply` affine-map dim-and-symbol-use-list

The affine_apply instruction applies an affine mapping to a list of SSA values, yielding another list of SSA values. The number of dimension and symbol arguments to affine_apply must be equal to the respective number of dimensional and symbolic inputs to the affine mapping, and the number of results is the dimensionality of the range of the affine mapping. The input SSA values must all have 'index' type, and the results are all of 'index' type.

Example:

#map10 = (d0, d1) -> (floordiv(d0,8), floordiv(d1,128),
                      d0 mod 8, d1 mod 128)
...
%1 = affine_apply #map10 (%s, %t)

// Inline example.
%2 = affine_apply (i)[s0] -> (i+s0) (%42)[%n]

'call' operation {#'call'-operation}

Syntax:

operation ::=
    ssa-id `=` `call` function-id `(` ssa-use-list? `)` `:` function-type

The call operation represents a direct call to a function. The operands and result types of the call must match the specified function type. The callee is encoded as a function attribute named "callee".

Example:

// Calling the CFG function my_add.
%31 = call @my_add(%0, %1) : (tensor<16xf32>, tensor<16xf32>) -> tensor<16xf32>

'call_indirect' operation {#'call_indirect'-operation}

Syntax:

operation ::= ssa-id `=` `call_indirect` ssa-use
                `(` ssa-use-list? `)` `:` function-type

The call_indirect operation represents an indirect call to a value of function type. Functions are first class types in MLIR, and may be passed as arguments and merged together with block arguments. The operands and result types of the call must match the specified function type.

Function values can be created with the constant operation.

Example:

%31 = call_indirect %15(%0, %1)
        : (tensor<16xf32>, tensor<16xf32>) -> tensor<16xf32>

'dim' operation {#'dim'-operation}

Syntax:

operation ::= ssa-id `=` `dim` ssa-id `,` integer-literal `:` type

The dim operation takes a memref or tensor operand and a dimension index, and returns an 'index' that is the size of that dimension.

The dim operation is represented with a single integer attribute named index, and the type specifies the type of the memref or tensor operand.

Examples:

// Always returns 4, can be constant folded:
%x = dim %A, 0 : tensor<4 x ? x f32>

// Returns the dynamic dimension of %A.
%y = dim %A, 1 : tensor<4 x ? x f32>

// Equivalent longhand form:
%x = "dim"(%A){index: 0} : (tensor<4 x ? x f32>) -> index
%y = "dim"(%A){index: 1} : (tensor<4 x ? x f32>) -> index

'reshape' operation {#'reshape'-operation}

Syntax:

operation ::= ssa_id `=` `reshape` ssa-use `:` memref-type
    dim-and-symbol-use-list `:` memref-type

Reshapes the input to the requested shape. The reshape instruction creates a new memref, but changes how the total dimension size is factored into individual dimensions sizes as long as the products of the dimension sizes of both shapes are the same. For example, a 16x16xf32 memref can be reshaped into a 16x8x2xf32 one, but not to a 16x4xf32 one.

Example:

// Allocate base memref with dynamic 16x?xf32.
#lmapD = (i, j)[S0] -> (i, j) size (16, S0)
%D = alloc <16x?xf32, #lmapD, hbm>(%N)[%N]

// Create memref which reshapes from 16x?xf32 to 16x4x?xf32.
#imapDR = (i, j, k)[S0] -> (i, j * S0 + k) size (16, 4 * S0)
%N4 = affine_apply (S -> floordiv(S,4)) (%N)
%DR = reshape %D : memref<16x?xf32, #lmapD, hbm> (%N4)[%N4] to
      (memref<16x?xf32, #lmapD, hbm> -> memref<16x4x?xf32, #imapDR, hbm>)

'view' operation {#'view'-operation}

Syntax:

operation ::=
  `view` memref-type dim-use-list symbol-use-list? ssa-id `:` memref-type

Creates a view of a base memref with a potentially different index space. A view is only defined when its index map maps to a range that is contained in the base memref's index space. The element type and memory space of a view matches those of the operand memref.

The view instruction defines a new memref which aliases the buffer of its operand memref, but in a new index system, specified by the index map in its type (and any captured symbols). See the figure below for an example.

Example:

#map_b = (i,j)[s0, s1] -> (i + s0, j) size (16, s1)

// %B is a view of %A with a window of size 4 with offset %0 along the
// first dimension. The SSA value %0 is bound to the offset symbol of
// its index map (#map_b)
%n1 = dim %A, 1 : memref<16x?xf32, #map_a, hbm>
%B = view memref<16x?xf32, #map_a, hbm> -> memref<4x?xf32, #map_b, hbm>
          (%n1) [%0, %n1] %A : memref<16x?xf32, #map_a, hbm>

// A view memref that is a dynamic sized slice along an already dynamic
// sized base memref with the slice size being half the base memref's
// dynamic size and with an offset of %0
#map_c = (i,j)[s0, s1] -> (i + s0, j) size (4, s1)
%s1 = "divi"(%n1, 2) : (i32, i32) -> i32
%C = view memref<16x?xf32, #map_a, hbm> -> memref<4x?xf32, #map_c, hbm>
          (%s1) [%0, %n1] %A : memref<16x?xf32, #map_a, hbm>

Memory Operations

'alloc' operation {#'alloc'-operation}

Syntax:

operation ::= ssa-id `=` `alloc` dim-and-symbol-use-list `:` memref-type

Allocates a new memref of specified type. Values required for dynamic dimension sizes are passed as arguments in parentheses (in the same order in which they appear in the shape signature of the memref) while the symbols required by the layout map are passed in the square brackets in lexicographical order. If no layout maps are specified in the memref, then an identity mapping is used.

The buffer referenced by a memref type is created by the alloc instruction, and destroyed by the dealloc instruction.

Example:

// Allocating memref for a fully static shape.
%A = alloc() : memref<1024x64xf32, #layout_map0, hbm>

// %M, %N, %x, %y are SSA values of integer type.  M and N are bound to the
// two unknown dimensions of the type and x/y are bound to symbols in
// #layout_map1.
%B = alloc(%M, %N)[%x, %y] : memref<?x?xf32, #layout_map1, vmem>

'alloc_static' operation {#'alloc_static'-operation}

Syntax:

operation ::=
    ssa-id `=` `alloc_static` `(` integer-literal `)` :  memref-type

Allocates a new memref of specified type with a fixed base pointer location in memory. 'alloc_static' does not support types that have dynamic shapes or that require dynamic symbols in their layout function (use the alloc'instruction in those cases).

Example:

%A = alloc_static(0x1232a00) : memref<1024 x 64 x f32, #layout_map0, hbm>

The alloc_static instruction is used to represent code after buffer allocation has been performed.

'dealloc' operation {#'dealloc'-operation}

Syntax:

operation ::= `dealloc` ssa-use `:` memref-type

Delineates the end of the lifetime of the memory corresponding to a memref allocation. It is paired with an alloc or alloc_static instruction.

Example:

dealloc %A : memref<128 x f32, #layout, hbm>

'dma_start' operation

Syntax:

operation ::= `dma_start` ssa-use`[`ssa-use-list`]`,
               ssa-use`[`ssa-use-list`]`, ssa-use,
               ssa-use`[`ssa-use-list`]`
              `:` memref-type, memref-type, memref-type

Starts a non-blocking DMA operation that transfers data from a source memref to a destination memref. The operands include the source and destination memref's each followed by its indices, size of the data transfer in terms of the number of elements (of the elemental type of the memref), and a tag memref with its indices. The tag location is used by a dma_wait operation to check for completion. The indices of the source memref, destination memref, and the tag memref have the same restrictions as any load/store instruction in an ML Function (whenever DMA operations appear in ML Functions). This allows powerful static analysis and transformations in the presence of such DMAs including rescheduling, pipelining / overlap with computation, and checking for matching start/end operations. The source and destination memref need not be of the same dimensionality, but need to have the same elemental type.

For example, a dma_start operation that transfers 32 vector elements from a memref %src at location [%i, %j] to memref %dst at [%k, %l] would be specified as shown below.

Example:

%size = constant 32 : index
%tag = alloc() : memref<1 x i32, (d0) -> (d0), 4>
%idx = constant 0 : index
dma_start %src[%i, %j], %dst[%k, %l], %size, %tag[%idx] :
     memref<40 x 8 x vector<16xf32>, (d0) -> (d0), 0>,
     memref<2 x 4 x vector<16xf32>, (d0) -> (d0), 2>,
     memref<1 x i32>, (d0) -> (d0), 4>

'dma_wait' operation

Syntax:

operation ::= `dma_wait` ssa-use`[`ssa-use-list`]`, ssa-use `:` memref-type

Blocks until the completion of a DMA operation associated with the tag element specified with a tag memref and its indices. The operands include the tag memref followed by its indices and the number of elements associated with the DMA being waited on. The indices of the tag memref have the same restrictions as load/store indices.

Example:

dma_wait %tag[%index], %num_elements : memref<1 x i32, (d0) -> (d0), 4>

'extract_element' operation {#'extract_element'-operation}

Syntax:

operation ::= ssa-id `=` `extract_element` ssa-use `[` ssa-use-list `]` `:` type

The extract_element op reads a tensor or vector and returns one element from it specified by an index list. The output of the 'extract_element' is a new value with the same type as the elements of the tensor or vector. The arity of indices matches the rank of the accessed value (i.e., if a tensor is of rank 3, then 3 indices are required for the extract. The indices should all be of affine_int type.

Examples:

%3 = extract_element %v[%1, %2] : vector<4x4xi32>
%4 = extract_element %t[%1, %2] : tensor<4x4xi32>
%5 = extract_element %ut[%1, %2] : tensor<*xi32>

'load' operation {#'load'-operation}

Syntax:

operation ::= ssa-id `=` `load` ssa-use `[` ssa-use-list `]` `:` memref-type

The load op reads an element from a memref specified by an index list. The output of load is a new value with the same type as the elements of the memref. The arity of indices is the rank of the memref (i.e., if the memref loaded from is of rank 3, then 3 indices are required for the load following the memref identifier).

In an ML Function, the indices of a load are restricted to SSA values bound to surrounding loop induction variables, symbols, results of a constant operation, or the results of an affine_apply operation that can in turn take as arguments all of the aforementioned SSA values or the recursively results of such an affine_apply operation.

Example:

#remap1 = (d0, d1) -> (3*d0, d1+1)
#remap2 = (d0) -> (2*d0 + 1)
 ...
%1 = affine_apply #remap1(%i, %j)
%12 = load %A[%1#0, %1#1] : memref<8x?xi32, #layout, hbm>

// Example of an indirect load (treated as non-affine)
%2 = affine_apply #remap2(%12)
%13 = load %A[%2, %1#1] : memref<4x?xi32, #layout, hbm>

Context: The load and store instructions are specifically crafted to fully resolve a reference to an element of a memref, and (in an ML function) the compiler can follow use-def chains (e.g. through affine_apply operations) to precisely analyze references at compile-time using polyhedral techniques. This is possible because of the restrictions on dimensions and symbols in ML functions.

'store' operation {#'store'-operation}

Syntax:

operation ::= `store` ssa-use `,` ssa-use
    `[` ssa-use-list `]` `:` memref-type

Store value to memref location given by indices. The value stored should have the same type as the elemental type of the memref. The number of arguments provided within brackets need to match the rank of the memref.

In an ML Function, the indices of a store are restricted to SSA values bound to surrounding loop induction variables, symbols, results of a constant operation, or the results of an affine_apply operation that can in turn take as arguments all of the aforementioned SSA values or the recursively results of such an affine_apply operation.

Example:

store %100, %A[%1, 1023] : memref<4x?xf32, #layout, hbm>

Context: The load and store instructions are specifically crafted to fully resolve a reference to a scalar member of a memref, and (in an ML function) the compiler can follow use-def chains (e.g. through affine_apply operations) to precisely analyze references at compile-time using polyhedral techniques. This is possible because of the restrictions on dimensions and symbols in ML functions.

'tensor_load' operation {#'tensor_load'-operation}

Syntax:

operation ::= ssa-id `=` `tensor_load` ssa-use-and-type

Create a tensor from a memref, making an independent copy of the element data. The result value is a tensor whose shape and element type match the memref operand.

Example:

// Produces a value of tensor<4x?xf32> type.
%12 = tensor_load %10 : memref<4x?xf32, #layout, hbm>

'tensor_store' operation {#'tensor_store'-operation}

Syntax:

operation ::= `tensor_store` ssa-use `,` ssa-use `:` memref-type

Stores the contents of a tensor into a memref. The first operand is a value of tensor type, the second operand is a value of memref type. The shapes and element types of these must match, and are specified by the memref type.

Example:

%9 = dim %8, 1 : tensor<4x?xf32>
%10 = alloc(%9) : memref<4x?xf32, #layout, hbm>
tensor_store %8, %10 : memref<4x?xf32, #layout, hbm>

Arithmetic Operations

Basic arithmetic in MLIR is specified by standard operations described in this section.

TODO: "sub" etc. Let's not get excited about filling this out yet, we can define these on demand. We should be highly informed by and learn from the operations supported by HLO and LLVM.

'addi' operation {#'addi'-operation}

Examples:

// Scalar addition.
%a = addi %b, %c : i64

// SIMD vector element-wise addition, e.g. for Intel SSE.
%f = addi %g, %h : vector<4xi32>

// Tensor element-wise addition, analogous to HLO's add operation.
%x = addi %y, %z : tensor<4x?xi8>

The addi operation takes two operands and returns one result, each of these is required to be the same type. This type may be an integer scalar type, a vector whose element type is integer, or a tensor of integers. It has no standard attributes.

'addf' operation {#'addf'-operation}

Examples:

// Scalar addition.
%a = addf %b, %c : f64

// SIMD vector addition, e.g. for Intel SSE.
%f = addf %g, %h : vector<4xf32>

// Tensor addition, analogous to HLO's add operation.
%x = addf %y, %z : tensor<4x?xbf16>

The addf operation takes two operands and returns one result, each of these is required to be the same type. This type may be a floating point scalar type, a vector whose element type is a floating point type, or a floating point tensor.

It has no standard attributes.

TODO: In the distant future, this will accept optional attributes for fast math, contraction, rounding mode, and other controls.

'cmpi' operation {#'cmpi'-operation}

Examples:

// Scalar "signed less than" comparison.
%x = cmpi "slt", %lhs, %rhs : i32

// Long-hand notation of the same operation.
%x = "cmpi"(%lhs, %rhs){predicate: 2} : (i32, i32) -> i1

// Vector equality comparison.
%x = cmpi "eq", %lhs, %rhs : vector<4xi64>

// Long-hand notation of the same operation.
%x = "cmpi"(%lhs, %rhs){predicate: 0}
    : (vector<4xi64>, vector<4xi64> -> vector<4xi1>

The cmpi operation is a generic comparison for integer-like types. Its two arguments can be integers, vectors or tensors thereof as long as their types match. The operation produces an i1 for the former case, a vector or a tensor of i1 with the same shape as inputs in the other cases.

Its first argument is an attribute that defines which type of comparison is performed. The following comparisons are supported:

equal (mnemonic: "eq"; integer value: 0)
not equal (mnemonic: "ne"; integer value: 1)
signed less than (mnemonic: "slt"; integer value: 2)
signed less than or equal (mnemonic: "slt"; integer value: 3)
signed greater than (mnemonic: "sgt"; integer value: 4)
signed greater than or equal (mnemonic: "sge"; integer value: 5)
unsigned less than (mnemonic: "ult"; integer value: 6)
unsigned less than or equal (mnemonic: "ult"; integer value: 7)
unsigned greater than (mnemonic: "ugt"; integer value: 8)
unsigned greater than or equal (mnemonic: "uge"; integer value: 9)

The result is 1 if the comparison is true and 0 otherwise. For vector or tensor operands, the comparison is performed elementwise and the element of the result indicates whether the comparison is true for the operand elements with the same indices as those of the result.

Note: while the short-hand notation uses strings, the actual underlying attribute has integer type (or rather enum class in C++ code) as seen from the long-hand notation. String literals are used to improve readability of the IR by humans.

This operation only applies to integer-like operands, but not floats. The main reason being that comparison operations have diverging sets of attributes: integers require sign specification while floats require various floating point-related particularities, e.g., -ffast-math behavior, IEEE754 compliance, etc (rationale). The type of comparison is specified as attribute to avoid introducing ten similar operations, taking into account that they are often implemented using the same operation downstream (rationale). The separation between signed and unsigned order comparisons is necessary because of integers being signless. The comparison operation must know how to interpret values with the foremost bit being set: negatives in two's complement or large positives (rationale).

'constant' operation {#'constant'-operation}

Syntax:

operation ::= ssa-id `=` `constant` attribute-value `:` type

The constant operation produces an SSA value equal to some constant specified by an attribute. This is the way that MLIR uses to form simple integer and floating point constants, as well as more exotic things like references to functions and (TODO!) tensor/vector constants.

The constant operation is represented with a single attribute named "value". The type specifies the result type of the operation.

Examples:

// Integer constant
%1 = constant 42 : i32

// Reference to function @myfn.
%3 = constant @myfn : (tensor<16xf32>, f32) -> tensor<16xf32>

// Equivalent longhand forms
%1 = "constant"(){value: 42} : i32
%3 = "constant"(){value: @myfn}
   : () -> (tensor<16xf32>, f32) -> tensor<16xf32>

MLIR does not allow direct references to functions in SSA operands because we anticipate the desire to multithread the compiler, and disallowing SSA values to directly reference a function simplifies this (rationale).

'divis' operation {#'divis'-operation}

Signed integer division. Rounds towards zero. Treats the leading bit as sign, i.e. 6 / -2 = -3.

Note: the semantics of division by zero or signed division overflow (minimum value divided by -1) is TBD; do NOT assume any specific behavior.

Syntax:

operation ::= ssa-id `=` `divis` ssa-use, ssa-use `:` type

Examples:

// Scalar signed integer division.
%a = divis %b, %c : i64

// SIMD vector element-wise division.
%f = divis %g, %h : vector<4xi32>

// Tensor element-wise integer division.
%x = divis %y, %z : tensor<4x?xi8>

The divis operation takes two operands and returns one result, each of these is required to be the same type. This type may be an integer scalar type, a vector whose element type is integer, or a tensor of integers. It has no standard attributes.

'diviu' operation {#'diviu'-operation}

Unsigned integer division. Rounds towards zero. Treats the leading bit as the most significant, i.e. for i16 given two's complement representation, 6 / -2 = 6 / (2^16 - 2) = 0.

Note: the semantics of division by zero is TBD; do NOT assume any specific behavior.

Syntax:

operation ::= ssa-id `=` `diviu` ssa-use, ssa-use `:` type

Examples:

// Scalar unsigned integer division.
%a = diviu %b, %c : i64

// SIMD vector element-wise division.
%f = diviu %g, %h : vector<4xi32>

// Tensor element-wise integer division.
%x = diviu %y, %z : tensor<4x?xi8>

The diviu operation takes two operands and returns one result, each of these is required to be the same type. This type may be an integer scalar type, a vector whose element type is integer, or a tensor of integers. It has no standard attributes.

'memref_cast' operation

Syntax:

operation ::= ssa-id `=` `memref_cast` ssa-use `:` type `to` type

Examples:

// Discard static dimension information.
%3 = memref_cast %2 : memref<4x?xf32> to memref<?x?xf32>

// Convert to a type with more known dimensions.
%4 = memref_cast %3 : memref<?x?xf32> to memref<4x?xf32>

Convert a memref from one type to an equivalent type without changing any data elements. The source and destination types must both be memref types with the same element type, same mappings, same address space, and same rank, yet the source and destination types may not be the same. The operation is invalid if converting to a mismatching constant dimension.

'mulf' operation {#'mulf'-operation}

Examples:

// Scalar multiplication.
%a = mulf %b, %c : f64

// SIMD pointwise vector multiplication, e.g. for Intel SSE.
%f = mulf %g, %h : vector<4xf32>

// Tensor pointwise multiplication, analogous to HLO's pointwise multiply operation.
%x = mulf %y, %z : tensor<4x?xbf16>

The mulf operation takes two operands and returns one result, each of these is required to be the same type. This type may be a floating point scalar type, a vector whose element type is a floating point type, or a floating point tensor.

It has no standard attributes.

TODO: In the distant future, this will accept optional attributes for fast math, contraction, rounding mode, and other controls.

'remis' operation {#'remis'-operation}

Signed integer division remainder. Treats the leading bit as sign, i.e. 6 % -2 = 0.

Note: the semantics of division by zero is TBD; do NOT assume any specific behavior.

Syntax:

operation ::= ssa-id `=` `remis` ssa-use, ssa-use `:` type

Examples:

// Scalar signed integer division remainder.
%a = remis %b, %c : i64

// SIMD vector element-wise division remainder.
%f = remis %g, %h : vector<4xi32>

// Tensor element-wise integer division remainder.
%x = remis %y, %z : tensor<4x?xi8>

The remis operation takes two operands and returns one result, each of these is required to be the same type. This type may be an integer scalar type, a vector whose element type is integer, or a tensor of integers. It has no standard attributes.

'remiu' operation {#'remiu'-operation}

Unsigned integer division remainder. Treats the leading bit as the most significant, i.e. for i16, 6 % -2 = 6 % (2^16 - 2) = 6.

Note: the semantics of division by zero is TBD; do NOT assume any specific behavior.

Syntax:

operation ::= ssa-id `=` `remiu` ssa-use, ssa-use `:` type

Examples:

// Scalar unsigned integer division remainder.
%a = remiu %b, %c : i64

// SIMD vector element-wise division remainder.
%f = remiu %g, %h : vector<4xi32>

// Tensor element-wise integer division remainder.
%x = remiu %y, %z : tensor<4x?xi8>

The remiu operation takes two operands and returns one result, each of these is required to be the same type. This type may be an integer scalar type, a vector whose element type is integer, or a tensor of integers. It has no standard attributes.

'select' operation {#'select'-operation}

Syntax:

operation ::= ssa-id `=` `select` ssa-use, ssa-use, ssa-use `:` type

Examples:

// Short-hand notation of scalar selection.
%x = select %cond, %true, %false : i32

// Long-hand notation of the same operation.
%x = "select"(%cond, %true, %false) : (i1, i32, i32) -> i32

// Vector selection is element-wise
%vx = "select"(%vcond, %vtrue, %vfalse)
    : (vector<42xi1>, vector<42xf32>, vector<42xf32>) -> vector<42xf32>

The select operation chooses one value based on a binary condition supplied as its first operand. If the value of the first operand is 1, the second operand is chosen, otherwise the third operand is chosen. The second and the third operand must have the same type.

The operation applies to vectors and tensors elementwise given the shape of all operands is identical. The choice is made for each element individually based on the value at the same position as the element in the condition operand.

The select operation combined with cmpi can be used to implement min and max with signed or unsigned comparison semantics.

'tensor_cast' operation {#'tensor_cast'-operation}

Syntax:

operation ::= ssa-id `=` `tensor_cast` ssa-use `:` type `to` type

Examples:

// Convert from unknown rank to rank 2 with unknown dimension sizes.
%2 = "tensor_cast"(%1) : (tensor<*xf32>) -> tensor<?x?xf32>
%2 = tensor_cast %1 : tensor<*xf32> to tensor<?x?xf32>

// Convert to a type with more known dimensions.
%3 = "tensor_cast"(%2) : (tensor<?x?xf32>) -> tensor<4x?xf32>

// Discard static dimension and rank information.
%4 = "tensor_cast"(%3) : (tensor<4x?xf32>) -> tensor<?x?xf32>
%5 = "tensor_cast"(%4) : (tensor<?x?xf32>) -> tensor<*xf32>

Convert a tensor from one type to an equivalent type without changing any data elements. The source and destination types must both be tensor types with the same element type, and the source and destination types may not be the same. They must either have the same rank, or one may be an unknown rank. The operation is invalid if converting to a mismatching constant dimension.

Dialects

MLIR supports multiple dialects containing a set of operations and types defined together, potentially outside of the main tree. Dialects are produced and consumed by certain passes. MLIR can be converted between different dialects by a conversion pass.

Currently, MLIR supports the following dialects:

Standard dialect
SuperVector dialect
TensorFlow dialect

TensorFlow operations

MLIR operations can represent arbitrary TensorFlow operations with a reversible mapping. Switch and merge nodes are represented with the MLIR control flow graph. TensorFlow dataflow operations are mapped over to MLIR operations whose name gets a "tf." prefix.

The normal dtypes supported by TensorFlow are mapped onto a Tensor type with an unknown rank. The resource and variant dtypes are mapped onto our resource and variant type specifically (TODO: Specify this). Attributes get mapped over directly, with type attributes represented as strings.

Examples:

// TensorFlow Add operation.
%a = "tf.Add"(%b, %c)
  : (tensor<*xf32>,tensor<*xf32>) -> tensor<*xf32>

// TensorFlow Add operation with partially inferred shapes.
%d = "tf.Add"(%e, %f)
  : (tensor<?x42x?xf32>,tensor<?x42x?xf32>) -> tensor<?x42x?xf32>

// TensorFlow Conv2d operation.
%y = "tf.Conv2d"(%input, %filter)
          {strides: [1,1,2,1], padding: "SAME", dilations: [2,1,1,1]}
   : (tensor<*xf32>, tensor<*xf32>) -> tensor<*xf32>

Target specific operations

We expect to expose many target-specific (such as TPU-specific) operations directly through to MLIR.

Example:

// TPU vector add instruction
%f = "tpu.vaddf32"(%a, %b)
             : (vector<8x128xf32>, vector<8x128xf32>) -> vector<8x128xf32>

In addition to the LLO backend, 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 instructions). LLVM intrinsics start with an "llvm." name.

Example:

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

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MLIR Specification

High-Level Structure

Notation

Common syntax

Identifiers and keywords

Polyhedral Structures

Dimensions and Symbols

Affine Expressions

Affine Maps

Named affine mappings

Semi-affine maps

Integer Sets

Type System

Integer Type

Index Type

Floating Point Types

Other Types

Function Type

Vector Type

Tensor Type

Memref Type

Index Space

Index

Index Map

Layout Map

Affine Map Composition

Attributes

Module

Functions

Blocks

Instruction Kinds

'for' instruction {#'for'-instruction}

'if' instruction {#'if'-instruction}

Operations

Standard Operations

Terminator operations

'br' terminator operation {#'br'-terminator-operation}

'cond_br' terminator operation {#'cond_br'-terminator-operation}

'return' terminator operation {#'return'-terminator-operation}

Core Operations

'affine_apply' operation {#'affine_apply'-operation}

'call' operation {#'call'-operation}

'call_indirect' operation {#'call_indirect'-operation}

'dim' operation {#'dim'-operation}

'reshape' operation {#'reshape'-operation}

'view' operation {#'view'-operation}

Memory Operations

'alloc' operation {#'alloc'-operation}

'alloc_static' operation {#'alloc_static'-operation}

'dealloc' operation {#'dealloc'-operation}

'dma_start' operation

'dma_wait' operation

'extract_element' operation {#'extract_element'-operation}

'load' operation {#'load'-operation}

'store' operation {#'store'-operation}

'tensor_load' operation {#'tensor_load'-operation}

'tensor_store' operation {#'tensor_store'-operation}

Arithmetic Operations

'addi' operation {#'addi'-operation}

'addf' operation {#'addf'-operation}

'cmpi' operation {#'cmpi'-operation}

'constant' operation {#'constant'-operation}

'divis' operation {#'divis'-operation}

'diviu' operation {#'diviu'-operation}

'memref_cast' operation

'mulf' operation {#'mulf'-operation}

'remis' operation {#'remis'-operation}

'remiu' operation {#'remiu'-operation}

'select' operation {#'select'-operation}

'tensor_cast' operation {#'tensor_cast'-operation}

Dialects

TensorFlow operations

Target specific operations

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