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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 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 operations 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 affine.for and affine.if operations defined by the affine dialect, 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 operations, 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(%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. 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 ::= decimal-literal | hexadecimal-literal
decimal-literal ::= digit+
hexadecimal-literal ::= `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 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.

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.

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 may be called through an operation, such as 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 operation, binding the
// SSA value %N to the symbol s0.
%a = alloc()[%N] : memref<4x4xf32, #affine_map42>

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

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 ::= 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 a Function
affine.if #set42(%i, %j)[%M, %N] {
  ...
}

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

Affine Dialect

MLIR provides a first class set of polyhedral operations and analyses within the affine dialect.

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 (there is no fixed list of types), and types may have application-specific semantics. For example, MLIR supports a set of standard types as well as dialect-specific types.

type ::= non-function-type
       | function-type

non-function-type ::= integer-type
                    | index-type
                    | float-type
                    | vector-type
                    | tensor-type
                    | memref-type
                    | dialect-type
                    | type-alias
                    | complex-type
                    | tuple-type
                    | none-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-specific 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 -> ()

Builtin Types

Builtin types consist of only the types needed for the validity of the IR.

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: 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.

Standard Types

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).

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.

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.

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 ::= `?` | 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>

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. 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.

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
#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, 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.
memref<16x4x?xf32, #imapB, memspace0>

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 = alloc (%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, 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.

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 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.

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>

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>

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.

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 in MLIR 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. It is possible to attach attributes to operations, functions, and function arguments. 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.

Function and Argument Attributes

Functions and function arguments in MLIR may have optional attributes attached to them. The sole constraint for these attributes is that they must be dialect specific attributes. This is because functions, and function arguments, are a generic entities and thus cannot apply any meaningful context necessary for dependent attributes. This has the added benefit of avoiding collisions between common attribute names, such as noalias.

Operation Attributes

Operations, unlike functions and function arguments, may include both dialect specific and dependent attributes. This is because an operation represents a distinct semantic context, and can thus provide a single source of meaning to dependent attributes.

Attribute Values

Attributes values are represented by the following forms:

attribute-value ::= affine-map-attribute
                  | array-attribute
                  | bool-attribute
                  | elements-attribute
                  | integer-attribute
                  | integer-set-attribute
                  | float-attribute
                  | function-attribute
                  | string-attribute
                  | type-attribute
                  | unit-attribute

AffineMap Attribute

Syntax:

affine-map-attribute ::= 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.

Elements Attributes

Syntax:

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

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  `,`
                              ( tensor-type | vector-type ) `,`
                              hex-string-literal `>`

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` `<` ( tensor-type | vector-type ) `,`
                              attribute-value `,` attribute-value `>`

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

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

Splat Elements Attribute

Syntax:

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

A splat elements attribute is an elements attribute that represents a tensor or vector constant where all elements have the same value.

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-map

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

Float Attribute

Syntax:

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

A float attribute is a literal attribute that represents a floating point value of the specified float type.

Function Attribute

Syntax:

function-attribute ::= function-id `:` function-type

A function attribute is a literal attribute that represents a reference to the given function object.

String Attribute

Syntax:

string-attribute ::= string-literal

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

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 {unitAttr : unit})

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

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 `)` (`->` function-result-type)?
argument-list ::= named-argument (`,` named-argument)* | /*empty*/
argument-list ::= type attribute-dict? (`,` type attribute-dict?)* | /*empty*/
named-argument ::= ssa-id `:` 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. A function definition contains a region made up of one or more blocks forming the function 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.

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
}

Regions

A region is a CFG of MLIR Blocks. Regions serve as a generalization of a function body that can be nested under arbitrary operations. A region semantics is defined by the containing entity (operation or function). Regions do not have a name or an address, only the blocks contained in a region do.

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

region ::= region-signature? region-body
region-signature ::= `(` argument-list `)` (`->` function-result-type)?
region-body ::= `{` block+ `}`

The function body is an example of a region, the body of an affine.for operation is another example, this time of an single-block region.

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 referenced values defined outside of the region, whenever it would have been legal to use them as operands to the enclosing operation. This can be further restricted using custom verifier.

Example:

func $@accelerator_compute(i64, i1) -> i64 {
^bb0(%a: i64, %cond: i1): // Code dominated by ^bb0 may refer to %a
  br_cond %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
...
}

Regions are Single-Entry-Multiple-Exit (SEME). It 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 blocks it contains. (This behavior is similar to that of functions in most programming languages). Nonetheless, when exiting the region from any of its multiple exit points, the control flows to the same successor.

Regions present in an operation can be executed any number of times. The IR does not guarantee if a region passed as an argument to an operation will be executed; if so, how many times and when. In particular, a region can be executed zero, one or multiple times, in no particular order with respect to other regions or operations. It may be executed as a part of an operation, or by some later operation using any values produced by the operation that contains the region. The successor to a region’s exit points may not necessarily exist: regions enclosing non-terminating code such as infinite loops are possible, as well as an operation implementing an infinite loop over a region. Concurrent or asynchronous execution of regions is unspecified. Operations may define pecific rules of execution, e.g. sequential loops or switch-like blocks.

In case of zero executions, control does not flow into the region. In case of multiple executions, the control may exit the region from any of the region exit points and enter it again at its entry point. It may also enter another region. If an operation has multiple region arguments, the semantics of the operation defines into which regions the control flows and in which order, if any. An operation may trigger execution of regions that were specified in other operations, in particular those that defined the values the given operation uses. When all argument regions were executed the number of times required by the operation semantics, the control flows from any of the region exit points to the original control-successor of the operation that triggered the execution. Thus operations with region arguments can be treated opaquely in the enclosing control flow graph, providing a level of control flow isolation similar to that of the call operation.

Regions allow to define 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. In this situation, the value “containing” the region may be passed to or returned from a function/region, at which point the values defined in dominating blocks are no longer accessible. If this region directly uses such values, passing a value “containing” it across function boundaries or using it in operations leads to undefined behavior. This is similar to returning a lambda capturing a reference to a local variable in C++. 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.

Regions produce a (possibly empty) list of values. For function body regions, return is the standard region-exiting terminator, but dialects can provide their own. For regions passed as operation arguments, the operation semantics defines the relation between the region results and the operation results.

Blocks

Syntax:

block           ::= bb-label operation+
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 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
  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 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].

Operations

Syntax:

operation ::= op-result? string-literal `(` ssa-use-list? `)`
              (`[` successor-list `]`)? (`(` region-list `)`)?
              attribute-dict? `:` function-type
op-result ::= ssa-id ((`:` integer-literal) | (`,` ssa-id)*) `=`
successor-list ::= successor (`,` successor)*
region-list    ::= region (`,` region)*

MLIR represents computations 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, 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. When parsed or printed in the generic assembly form, these are all printed literally, and a function type is used 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

Terminator 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.
"br_cond"(%cond)[^bb1, ^bb2(%v : index)] : (i1) -> ()

In addition to the basic syntax above, dialects may register tables of 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.

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

Terminator operations are required at the end of each block. They may contain a list of successors, i.e. other blocks to which the control flow will proceed. Currently, all terminator operations must be registered in some known dialect, unlike regular operations.

'br' terminator operation

Syntax:

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

The br terminator operation 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 operation is not allowed to target the entry block for a function.

'cond_br' terminator operation

Syntax:

operation ::=
  `cond_br` ssa-use `,` successor `,` successor

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

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

func @select(i32, i32, i1) -> i32 {
^bb0(%a : i32, %b :i32, %flag : i1) :
    // 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

Syntax:

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

The return terminator operation 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

'call' operation

Syntax:

operation ::= `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 function my_add.
%31 = call @my_add(%0, %1) : (tensor<16xf32>, tensor<16xf32>) -> tensor<16xf32>

'call_indirect' operation

Syntax:

operation ::= `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

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 generic form:
%x = "std.dim"(%A){index: 0} : (tensor<4 x ? x f32>) -> index
%y = "std.dim"(%A){index: 1} : (tensor<4 x ? x f32>) -> index

Memory Operations

'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 operation, and destroyed by the dealloc operation.

Example:

// Allocating memref for a fully static shape.
%A = alloc() : memref<1024x64xf32, #layout_map0, memspace0>

// %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, memspace1>

'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 operation in those cases).

Example:

%A = alloc_static(0x1232a00) : memref<1024 x 64 x f32, #layout_map0, memspace0>

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

'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 operation.

Example:

dealloc %A : memref<128 x f32, #layout, memspace0>

'dma_start' operation

Syntax:

operation ::= `dma_start` ssa-use`[`ssa-use-list`]` `,`
               ssa-use`[`ssa-use-list`]` `,` ssa-use `,`
               ssa-use`[`ssa-use-list`]` (`,` ssa-use, ssa-use)?
              `:` 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), a tag memref with its indices, and optionally two additional arguments corresponding to the stride (in terms of number of elements) and the number of elements to transfer per stride. 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 operation in a affine context (whenever DMA operations appear in an affine context). See restrictions on dimensions and symbols in affine contexts. 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

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

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 affine.if or affine.for body, the indices of a load are restricted to SSA values bound to surrounding loop induction variables, symbols, results of a constant operation, or the result of an affine.apply operation that can in turn take as arguments all of the aforementioned SSA values or the recursively result of such an affine.apply operation.

Example:

%1 = affine.apply (d0, d1) -> (3*d0) (%i, %j)
%2 = affine.apply (d0, d1) -> (d1+1) (%i, %j)
%12 = load %A[%1, %2] : memref<8x?xi32, #layout, memspace0>

// Example of an indirect load (treated as non-affine)
%3 = affine.apply (d0) -> (2*d0 + 1)(%12)
%13 = load %A[%3, %2] : memref<4x?xi32, #layout, memspace0>

Context: The load and store operations are specifically crafted to fully resolve a reference to an element of a memref, and (in affine affine.if and affine.for operations) 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 these contexts.

'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 affine context, the indices of a store are restricted to SSA values bound to surrounding loop induction variables, symbols, results of a constant operation, or the result of an affine.apply operation that can in turn take as arguments all of the aforementioned SSA values or the recursively result of such an affine.apply operation.

Example:

store %100, %A[%1, 1023] : memref<4x?xf32, #layout, memspace0>

Context: The load and store operations are specifically crafted to fully resolve a reference to an element of a memref, and (in polyhedral affine.if and affine.for operations) 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 these contexts.

'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, memspace0>

'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, memspace0>
tensor_store %8, %10 : memref<4x?xf32, #layout, memspace0>

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

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

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.

'and' operation

Bitwise integer and.

Syntax:

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

Examples:

// Scalar integer bitwise and.
%a = and %b, %c : i64

// SIMD vector element-wise bitwise integer and.
%f = and %g, %h : vector<4xi32>

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

The and 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.

'cmpi' operation

Examples:

// Custom form of scalar "signed less than" comparison.
%x = cmpi "slt", %lhs, %rhs : i32

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

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

// Generic form of the same operation.
%x = "std.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 custom assembly form uses strings, the actual underlying attribute has integer type (or rather enum class in C++ code) as seen from the generic assembly form. 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

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

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

'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

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. The operation is invalid if converting to a mismatching constant dimension.

'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.

'or' operation

Bitwise integer or.

Syntax:

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

Examples:

// Scalar integer bitwise or.
%a = or %b, %c : i64

// SIMD vector element-wise bitwise integer or.
%f = or %g, %h : vector<4xi32>

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

The or 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.

'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

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

Syntax:

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

Examples:

// Custom form of scalar selection.
%x = select %cond, %true, %false : i32

// Generic form of the same operation.
%x = "std.select"(%cond, %true, %false) : (i1, i32, i32) -> i32

// Vector selection is element-wise
%vx = "std.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

Syntax:

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

Examples:

// Convert from unknown rank to rank 2 with unknown dimension sizes.
%2 = "std.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 = "std.tensor_cast"(%2) : (tensor<?x?xf32>) -> tensor<4x?xf32>

// Discard static dimension and rank information.
%4 = "std.tensor_cast"(%3) : (tensor<4x?xf32>) -> tensor<?x?xf32>
%5 = "std.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. If both are ranked, then the rank should be the same and static dimensions should match. The operation is invalid if converting to a mismatching constant dimension.

'xor' operation

Bitwise integer xor.

Syntax:

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

Examples:

// Scalar integer bitwise xor.
%a = xor %b, %c : i64

// SIMD vector element-wise bitwise integer xor.
%f = xor %g, %h : vector<4xi32>

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

The xor 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.

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
Vector 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.

In addition to the TPU 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 operations). LLVM intrinsics 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.

Dialect specific types

Similarly to operations, dialects may define custom extensions to the type system. These extensions fit within the same type system as described in the type system overview.

dialect-type ::= '!' dialect-namespace '<' '"' type-specific-data '"' '>'
dialect-type ::= '!' alias-name pretty-dialect-type-body?

pretty-dialect-type-body ::= '<' pretty-dialect-type-contents+ '>'
pretty-dialect-type-contents ::= pretty-dialect-type-body
                              | '(' pretty-dialect-type-contents+ ')'
                              | '[' pretty-dialect-type-contents+ ']'
                              | '{' pretty-dialect-type-contents+ '}'
                              | '[^[<({>\])}\0]+'

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.

TensorFlow types

The TensorFlow dialect in MLIR defines several extended types:

// TensorFlow specific types (TODO: the rest ref data types)
type-specific-data ::= `control` | `resource` | `variant` | `string`
               `complex64` | `complex128` | `f32ref`

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

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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

Affine Dialect

Type System

Type Aliases

Builtin Types

Function Type

Standard Types

Index Type

Integer Type

Floating Point Types

Vector Type

Tensor Type

Memref Type

Index Space

Index

Index Map

Layout Map

Affine Map Composition

Complex Type

Tuple Type

None Type

Attributes

Function and Argument Attributes

Operation Attributes

Attribute Values

AffineMap Attribute

Array Attribute

Boolean Attribute

Elements Attributes

Dense Elements Attribute

Opaque Elements Attribute

Sparse Elements Attribute

Splat Elements Attribute

Integer Attribute

Integer Set Attribute

Float Attribute

Function Attribute

String Attribute

Type Attribute

Unit Attribute

Module

Functions

Regions

Blocks

Operations

Standard Operations

Terminator operations

'br' terminator operation

'cond_br' terminator operation

'return' terminator operation

Core Operations

'call' operation

'call_indirect' operation

'dim' operation

Memory Operations

'alloc' operation

'alloc_static' operation

'dealloc' operation

'dma_start' operation

'dma_wait' operation

'extract_element' operation

'load' operation

'store' operation

'tensor_load' operation

'tensor_store' operation

Arithmetic Operations

'addi' operation

'addf' operation

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