Defining Custom MLIR Dialects

Creating a Custom Dialect

When bridging two different systems, like Lean4 and JAX, you need a common language. MLIR provides this through dialects, which are collections of operations, types, and attributes tailored to a specific domain. A custom dialect lets us represent Lean4's high-level concepts, like function application and lambda abstractions, directly in MLIR. This preserves the program's original meaning before we lower it into something more generic, like numerical computations for JAX.

Instead of writing boilerplate C++ code for every new operation, MLIR uses a tool called TableGen to automate the process. You describe your dialect, its operations, and its types in a declarative format using .td files (Table Definition files). The MLIR build system then uses these descriptions to generate the necessary C++ classes. This approach keeps the definitions clean and separate from the implementation logic.

Defining the Dialect Itself

The first step is to formally declare the dialect. This involves creating a .td file and defining a Dialect class. This definition gives our dialect a name (which will be used as a prefix for operations, like lean.app) and a C++ namespace to avoid naming conflicts with other dialects.

// In lean/LeanDialect.td

#ifndef LEAN_DIALECT
#define LEAN_DIALECT

include "mlir/IR/OpBase.td"

def Lean_Dialect : Dialect {
  let name = "lean";
  let cppNamespace = "::mlir::lean";
  let summary = "A dialect for representing Lean 4 expressions.";

  // C++ header for dialect declarations
  let cppHeader = "Lean/LeanDialect.h";
}

#endif // LEAN_DIALECT

This definition is the anchor for everything else. All operations, types, and attributes in our dialect will be associated with Lean_Dialect.

Operations, Types, and Attributes

With the dialect defined, we can start adding its core components. The most important of these are the operations, which represent the actual computational steps or structural elements of a Lean4 program.

Operation

noun

The fundamental unit of code in MLIR. An operation (or 'Op') can represent anything from a machine instruction (like add) to a high-level language construct (like a function call) or even a structural element (like a module).

Let's define a few operations that mirror core Lean4 constructs. We'll start with Lean.App for function application, Lean.Lambda for lambda abstractions, and Lean.Const for constants. Each operation is defined in our .td file with a summary, a description, its arguments (inputs), and its results (outputs).

This is the crucial step in bridging the gap between Lean4's functional world and the computational graphs expected by frameworks like . We are creating a formal, intermediate representation that understands Lean's structure before it's transformed into something JAX can execute.

// In lean/LeanOps.td

include "Lean/LeanDialect.td"

// Base class for all our Lean operations
class Lean_Op<string mnemonic, list<Trait> traits = []> :
    Op<Lean_Dialect, mnemonic, traits>;

// lean.const operation
def Lean_ConstOp : Lean_Op<"const"> {
  let summary = "A constant value operation";
  let arguments = (ins StrAttr:$name);
  let results = (outs AnyType:$result);
}

// lean.app operation
def Lean_AppOp : Lean_Op<"app"> {
  let summary = "Function application operation";
  let arguments = (ins AnyType:$fn, Variadic<AnyType>:$args);
  let results = (outs AnyType:$result);
}

// lean.lambda operation
def Lean_LambdaOp : Lean_Op<"lambda"> {
  let summary = "Lambda abstraction operation";
  let arguments = (ins Variadic<AnyType>:$args);
  let results = (outs AnyType:$result);

  // A lambda has a body, which is a region of other operations
  let regions = (region SizedRegion<1>:$body);
}

Notice the Lean.LambdaOp has a region. A region is how MLIR represents nested structures, like the body of a function or a loop. This allows us to embed a graph of other operations inside the lambda, perfectly capturing its scope.

Integration with the Build System

Defining .td files is only half the battle. You also need to integrate them into the C++ build system. This typically involves using CMake to tell MLIR where to find your dialect definitions and where to put the generated C++ files.

# In your project's CMakeLists.txt

# Find the MLIR package
find_package(MLIR REQUIRED CONFIG)

# Define a library for our dialect
add_mlir_dialect(LeanDialect
  LeanDialect.td
  LeanOps.td
)

The add_mlir_dialect function is a CMake helper provided by MLIR. It automatically sets up the build rules to invoke mlir-tblgen (the TableGen executable for MLIR) and compile the resulting C++ source files into a library. You then link this library into your main compiler tool.

With this setup, you have a self-contained dialect. It's ready to be used as a target for a Lean4 frontend and can then be progressively lowered through other dialects until it becomes executable code.

Quiz Questions 1/5

What is the primary role of MLIR when bridging a high-level language like Lean4 with a numerical computation framework like JAX?

Quiz Questions 2/5

What is the main purpose of using TableGen (.td files) when creating a custom MLIR dialect?

Building Lean4-JAX with MLIR