Upstream MLIR PyTACO implementation.

[bixia]

Summary

Add TACO tests to test/Integration/Dialect/SparseTensor/taco. Add the MLIR
PyTACO implementation as tools under the directory.

[mehdi_amini]

mlir/test/Integration/Dialect/SparseTensor/taco/tools/mlir_pytaco.py

151

Could this be implemented as a registered C++ pipeline instead of listing all the passes here?

[aartbik]

mlir/test/Integration/Dialect/SparseTensor/taco/tools/mlir_pytaco.py

151

Yeah, agreed, this is very related to https://github.com/llvm/llvm-project/issues/51751 where we express the need to avoid proliferating these veyr specific passes all over the place.

[aartbik]

mlir/test/Integration/Dialect/SparseTensor/taco/MTTKRP_test.py

12 ↗

(On Diff #399828)

can you make this a line break comment, as in

This PyTACO part is taken from the TACO open-source project.

and then close it at line 41 with

just so that the part that we used from TACO jumps more out

17 ↗

(On Diff #399828)

maybe spell out "formats"

mlir/test/Integration/Dialect/SparseTensor/taco/SpMV_test.py

12 ↗

(On Diff #399828)

same request

mlir/test/Integration/Dialect/SparseTensor/taco/tools/mlir_pytaco.py

5

empty line after copyright header

81

np.float16

151

Bixia, for now, I would suggest to add a TODO and link to bug

[aartbik]

mlir/test/Integration/Dialect/SparseTensor/taco/MTTKRP_test.py

12 ↗

(On Diff #399828)

Ah, now I see what went wrong last time, my suggestion is being formatted by phacricator.
I really want something as

#### This PyTACO part is taken from the TACO open-source project. ####

and then end it as

#######################################

[mehdi_amini]

mlir/test/Integration/Dialect/SparseTensor/taco/tools/mlir_pytaco.py

151

This seems like a trivial thing to do isn't it? It probably won't take much longer that the TODO and filing the bug :)

[bixia]

Address review comments.
Rename test files so that module names start with lower cases to confirm to python style guide.

[bixia]

mlir/test/Integration/Dialect/SparseTensor/taco/tools/mlir_pytaco.py

151

Replaced this with a TODO as the ticket already has a owner and I can't just take over it.

[mehdi_amini]

mlir/test/Integration/Dialect/SparseTensor/taco/tools/mlir_pytaco.py

151

Alright, LG, thanks :)

[aartbik]

Well done Bixia!

mlir/python/requirements.txt

 numpy
 pybind11>=2.8.0
 PyYAML
+dataclasses

mlir/test/Integration/Dialect/SparseTensor/taco/README.md

+# MLIR-PyTACO: Implementing PyTACO with MLIR
+
+TACO (http://tensor-compiler.org/) is a tensor algebra compiler. TACO defines
+PyTACO, a domain specific language in Python, for writing tensor algebra
+applications.
+
+This directory contains the implementation of PyTACO using MLIR. In particular,
+we implement a Python layer that accepts the PyTACO language, generates MLIR
+linalg.generic OPs with sparse tensor annotation to represent the tensor
+computation, and invokes the MLIR sparse tensor code generator
+(https://mlir.llvm.org/docs/Dialects/SparseTensorOps/) as well as other MLIR
+compilation passes to generate an executable. Then, we invoke the MLIR execution
+engine to execute the program and pass the result back to the Python layer.
+
+As can be seen from the tests in this directory, in order to port a PyTACO
+program to MLIR-PyTACO, we basically only need to replace this line that imports
+PyTACO:
+
+```python
+import pytaco as pt
+```
+
+with this line to import MLIR-PyTACO:
+
+```python
+from tools import mlir_pytaco_api as pt
+```

mlir/test/Integration/Dialect/SparseTensor/taco/data/gold_A.tns

+1 1 12
+1 2 12
+1 3 12
+1 4 12
+1 5 12
+1 6 12
+1 7 12
+1 8 12
+1 9 12
+1 10 12
+1 11 12
+1 12 12
+1 13 12
+1 14 12
+1 15 12
+1 16 12
+1 17 12
+1 18 12
+1 19 12
+1 20 12
+1 21 12
+1 22 12
+1 23 12
+1 24 12
+1 25 12
+2 1 6
+2 2 6
+2 3 6
+2 4 6
+2 5 6
+2 6 6
+2 7 6
+2 8 6
+2 9 6
+2 10 6
+2 11 6
+2 12 6
+2 13 6
+2 14 6
+2 15 6
+2 16 6
+2 17 6
+2 18 6
+2 19 6
+2 20 6
+2 21 6
+2 22 6
+2 23 6
+2 24 6
+2 25 6

mlir/test/Integration/Dialect/SparseTensor/taco/data/gold_y.tns

+# See http://frostt.io/tensors/file-formats.html for FROSTT (.tns) format
+1 37102
+2 -20.4138
+3 804927

mlir/test/Integration/Dialect/SparseTensor/taco/data/nell-2.tns

+1 1 1 1.0
+1 2 2 2.0
+1 3 4 3.0
+2 1 1 1.0
+2 4 3 2.0

mlir/test/Integration/Dialect/SparseTensor/taco/data/pwtk.mtx

+%%MatrixMarket matrix coordinate real symmetric
+%-------------------------------------------------------------------------------
+% To download a matrix for a real world application
+% https://math.nist.gov/MatrixMarket/
+%-------------------------------------------------------------------------------
+3 3 5
+1 1 37423.0879671
+2 1 -22.4050781162
+3 1 -300.654980157
+3 2 -.00869762944058
+3 3 805225.750212

mlir/test/Integration/Dialect/SparseTensor/taco/test_MTTKRP.py

+# RUN: SUPPORTLIB=%mlir_runner_utils_dir/libmlir_c_runner_utils%shlibext %PYTHON %s | FileCheck %s
+
+import numpy as np
+import os
+import sys
+import tempfile
+
+_SCRIPT_PATH = os.path.dirname(os.path.abspath(__file__))
+sys.path.append(_SCRIPT_PATH)
+from tools import mlir_pytaco_api as pt
+
+###### This PyTACO part is taken from the TACO open-source project. ######
+# See http://tensor-compiler.org/docs/data_analytics/index.html.
+
+compressed = pt.compressed
+dense = pt.dense
+
+# Define formats for storing the sparse tensor and dense matrices.
+csf = pt.format([compressed, compressed, compressed])
+rm = pt.format([dense, dense])
+
+# Load a sparse three-dimensional tensor from file (stored in the FROSTT
+# format) and store it as a compressed sparse fiber tensor. We use a small
+# tensor for the purpose of testing. To run the program using the data from
+# the real application, please download the data from:
+# http://frostt.io/tensors/nell-2/
+B = pt.read(os.path.join(_SCRIPT_PATH, "data/nell-2.tns"), csf)
+
+# These two lines have been modified from the original program to use static
+# data to support result comparison.
+C = pt.from_array(np.full((B.shape[1], 25), 1, dtype=np.float64))
+D = pt.from_array(np.full((B.shape[2], 25), 2, dtype=np.float64))
+
+# Declare the result to be a dense matrix.
+A = pt.tensor([B.shape[0], 25], rm)
+
+# Declare index vars.
+i, j, k, l = pt.get_index_vars(4)
+
+# Define the MTTKRP computation.
+A[i, j] = B[i, k, l] * D[l, j] * C[k, j]
+
+##########################################################################
+
+# CHECK: Compare result True
+# Perform the MTTKRP computation and write the result to file.
+with tempfile.TemporaryDirectory() as test_dir:
+  actual_file = os.path.join(test_dir, "A.tns")
+  pt.write(actual_file, A)
+  actual = np.loadtxt(actual_file, np.float64)
+  expected = np.loadtxt(
+      os.path.join(_SCRIPT_PATH, "data/gold_A.tns"), np.float64)
+  print(f"Compare result {np.allclose(actual, expected, rtol=0.01)}")

mlir/test/Integration/Dialect/SparseTensor/taco/test_SpMV.py

+# RUN: SUPPORTLIB=%mlir_runner_utils_dir/libmlir_c_runner_utils%shlibext %PYTHON %s | FileCheck %s
+
+import numpy as np
+import os
+import sys
+import tempfile
+
+_SCRIPT_PATH = os.path.dirname(os.path.abspath(__file__))
+sys.path.append(_SCRIPT_PATH)
+from tools import mlir_pytaco_api as pt
+
+###### This PyTACO part is taken from the TACO open-source project. ######
+# See http://tensor-compiler.org/docs/scientific_computing/index.html.
+
+compressed = pt.compressed
+dense = pt.dense
+
+# Define formats for storing the sparse matrix and dense vectors.
+csr = pt.format([dense, compressed])
+dv = pt.format([dense])
+
+# Load a sparse matrix stored in the matrix market format) and store it
+# as a CSR matrix.  The matrix in this test is a reduced version of the data
+# downloaded from here:
+# https://www.cise.ufl.edu/research/sparse/MM/Boeing/pwtk.tar.gz
+# In order to run the program using the matrix above, you can download the
+# matrix and replace this path to the actual path to the file.
+A = pt.read(os.path.join(_SCRIPT_PATH, "data/pwtk.mtx"), csr)
+
+# These two lines have been modified from the original program to use static
+# data to support result comparison.
+x = pt.from_array(np.full((A.shape[1],), 1, dtype=np.float64))
+z = pt.from_array(np.full((A.shape[0],), 2, dtype=np.float64))
+
+# Declare the result to be a dense vector
+y = pt.tensor([A.shape[0]], dv)
+
+# Declare index vars
+i, j = pt.get_index_vars(2)
+
+# Define the SpMV computation
+y[i] = A[i, j] * x[j] + z[i]
+
+##########################################################################
+
+# CHECK: Compare result True
+# Perform the SpMV computation and write the result to file
+with tempfile.TemporaryDirectory() as test_dir:
+  actual_file = os.path.join(test_dir, "y.tns")
+  pt.write(actual_file, y)
+  actual = np.loadtxt(actual_file, np.float64)
+  expected = np.loadtxt(
+      os.path.join(_SCRIPT_PATH, "data/gold_y.tns"), np.float64)
+  print(f"Compare result {np.allclose(actual, expected, rtol=0.01)}")

mlir/test/Integration/Dialect/SparseTensor/taco/test_simple_tensor_algebra.py

+# RUN: SUPPORTLIB=%mlir_runner_utils_dir/libmlir_c_runner_utils%shlibext %PYTHON %s | FileCheck %s
+
+import os
+import sys
+
+_SCRIPT_PATH = os.path.dirname(os.path.abspath(__file__))
+sys.path.append(_SCRIPT_PATH)
+from tools import mlir_pytaco_api as pt
+
+compressed = pt.compressed
+dense = pt.dense
+
+# Ensure that we can run an unmodified PyTACO program with a simple tensor
+# algebra expression using tensor index notation, and produce the expected
+# result.
+i, j = pt.get_index_vars(2)
+A = pt.tensor([2, 3])
+B = pt.tensor([2, 3])
+C = pt.tensor([2, 3])
+D = pt.tensor([2, 3], dense)
+A.insert([0, 1], 10)
+A.insert([1, 2], 40)
+B.insert([0, 0], 20)
+B.insert([1, 2], 30)
+C.insert([0, 1], 5)
+C.insert([1, 2], 7)
+D[i, j] = A[i, j] + B[i, j] - C[i, j]
+
+# CHECK: [20. 5. 0. 0. 0. 63.]
+print(D.to_array().reshape(6))

mlir/test/Integration/Dialect/SparseTensor/taco/tools/lit.local.cfg

+# Files in this directory are tools, not tests.
+config.unsupported = True

mlir/test/Integration/Dialect/SparseTensor/taco/tools/mlir_pytaco.py

+#  Part of the LLVM Project, under the Apache License v2.0 with LLVM Exceptions.
+#  See https://llvm.org/LICENSE.txt for license information.
+#  SPDX-License-Identifier: Apache-2.0 WITH LLVM-exception
+
+"""Experimental MLIR-PyTACO with sparse tensor support.
+
+See http://tensor-compiler.org/ for TACO tensor compiler.
+
+This module implements the Python classes for PyTACO index notation. These
+include classes for data types, tensor dimension formats (aka mode formats),
+tensor dimension orderings (aka mode ordering), tensor storage formats, and
+tensors.
+
+The PyTACO API doesn't follow the naming conversion required by the style guide
+for this module. As such, we first implement the supporting classes and routines
+following the style guide, and then define the type aliases and constants to
+support the PyTACO API in the pytaco_api module.
+"""
+
+from typing import Any, Callable, Dict, Iterable, List, Optional, Set, Tuple, Union
+
+import abc
+import ctypes
+import dataclasses
+import enum
+import numpy as np
+import functools
+import operator
+import os
+import threading
+
+# Import MLIR related modules.
+from mlir import all_passes_registration  # Register MLIR compiler passes.
+from mlir import execution_engine
+from mlir import ir
+from mlir import runtime
+from mlir.dialects import arith
+from mlir.dialects import builtin
+from mlir.dialects import linalg
+from mlir.dialects import std
+from mlir.dialects import sparse_tensor
+from mlir.dialects.linalg.opdsl import lang
+from mlir.passmanager import PassManager
+
+from . import mlir_pytaco_utils as utils
+
+# TACO naming prefixes.
+_TACO_INDEX_PREFIX = "i"
+_TACO_TENSOR_PREFIX = "A"
+
+# Bitwidths for pointers and indices.
+_POINTER_BIT_WIDTH = 0
+_INDEX_BIT_WIDTH = 0
+# The name for the environment variable that provides the full path for the
+# supporting library.
+_SUPPORTLIB_ENV_VAR = "SUPPORTLIB"
+# The default supporting library if the environment variable is not provided.
+_DEFAULT_SUPPORTLIB = "libmlir_c_runner_utils.so"
+# The JIT compiler optimization level.
+_OPT_LEVEL = 2
+# The entry point to the JIT compiled program.
+_ENTRY_NAME = "main"
+
+# Type aliases for type annotation.
+_BinaryOp = Callable[[Any, Any], Any]
+_ExprVisitor = Callable[..., None]
+_ExprInfoDict = Dict["IndexExpr", "_ExprInfo"]
+_LogicalOp = Callable[[bool, bool], bool]
+_ModeFormatOp = Callable[["ModeFormat", "ModeFormat"], "ModeFormat"]
+_SubtreeLeafChecker = Optional[Callable[..., bool]]
+
+
+class Type(enum.Enum):
+  """The data types supported by TACO.
+
+  We use numpy data types to implement the enum data types.
+  """
+  INT16 = np.int16
+  INT32 = np.int32
+  INT64 = np.int64
+  # numpy _ctype_from_dtype_scalar can't handle np.float16 yet.
+  FLOAT32 = np.float32
+  FLOAT64 = np.float64
+
+
+# All floating point type enums.
+_FLOAT_TYPES = (Type.FLOAT32, Type.FLOAT64)
+# All integral type enums.
+_INT_TYPES = (Type.INT16, Type.INT32, Type.INT64)
+# Type alias for any numpy type used to implement the runtime support for the
+# enum data types.
+_AnyRuntimeType = Union[np.int16, np.int32, np.int64, np.float32, np.float64]
+
+
[e-mail](frozen=True)
+class DType:
+  """The data type class.
+
+  We support the TACO API dtype class with an alias of this class.
+
+  The following methods are defined by the TACO API:
+    is_float: Returns whether the data type represents a floating point value.
+    is_int:   Returns whether the data type represents an integral value.
+
+  Attributes:
+    kind: A Type enum representing the data type.
+    value: The numpy data type for the TACO data type.
+  """
+  kind: Type = Type.FLOAT64
+
+  def is_float(self) -> bool:
+    """Returns whether the data type represents a floating point value."""
+    return self.kind in _FLOAT_TYPES
+
+  def is_int(self) -> bool:
+    """Returns whether the data type represents an integral value."""
+    return self.kind in _INT_TYPES
+
+  @property
+  def value(self) -> _AnyRuntimeType:
+    """Returns the numpy dtype for the data type."""
+    return self.kind.value
+
+
+def _mlir_type_from_taco_type(dtype: DType) -> ir.Type:
+  """Returns the MLIR type corresponding to the given TACO type."""
+  dtype_to_irtype = {
+      Type.INT16: ir.IntegerType.get_signless(16),
+      Type.INT32: ir.IntegerType.get_signless(32),
+      Type.INT64: ir.IntegerType.get_signless(64),
+      Type.FLOAT32: ir.F32Type.get(),
+      Type.FLOAT64: ir.F64Type.get()
+  }
+  return dtype_to_irtype[dtype.kind]
+
+
+def _compile_mlir(module: ir.Module) -> ir.Module:
+  """Compiles an MLIR module and returns the compiled module."""
+  # TODO: Replace this with a pipeline implemented for 
+  #   https://github.com/llvm/llvm-project/issues/51751.
+  pipeline = (
+      f"sparsification,"
+      f"sparse-tensor-conversion,"
+      f"builtin.func(linalg-bufferize,convert-linalg-to-loops,convert-vector-to-scf),"
+      f"convert-scf-to-std,"
+      f"func-bufferize,"
+      f"tensor-constant-bufferize,"
+      f"builtin.func(tensor-bufferize,std-bufferize,finalizing-bufferize),"
+      f"convert-vector-to-llvm{{reassociate-fp-reductions=1 enable-index-optimizations=1}},"
+      f"lower-affine,"
+      f"convert-memref-to-llvm,"
+      f"convert-std-to-llvm,"
+      f"reconcile-unrealized-casts")
+  PassManager.parse(pipeline).run(module)
+  return module
+
+
[e-mail]_cache()
+def _get_support_lib_name() -> str:
+  """Returns the string for the supporting C shared library."""
+  return os.getenv(_SUPPORTLIB_ENV_VAR, _DEFAULT_SUPPORTLIB)
+
+
+def _ctype_pointer_from_array(array: np.ndarray) -> ctypes.pointer:
+  """Returns the ctype pointer for the given numpy array."""
+  return ctypes.pointer(
+      ctypes.pointer(runtime.get_ranked_memref_descriptor(array)))
+
+
+class ModeFormat(enum.Enum):
+  """The tensor dimension storage format class.
+
+  We support the TACO API mode_format class with an alias of this class.
+
+  In TACO, a tensor dimension is called a mode and the storage format for a
+  tensor dimension is called a mode format.
+  """
+  DENSE = sparse_tensor.DimLevelType.dense
+  COMPRESSED = sparse_tensor.DimLevelType.compressed
+
+
+def _mode_format_operation(a: ModeFormat, b: ModeFormat,
+                           op: _LogicalOp) -> ModeFormat:
+  """Implements the given operator on ModeFormat."""
+  return (ModeFormat.COMPRESSED
+          if op(a == ModeFormat.COMPRESSED, b == ModeFormat.COMPRESSED) else
+          ModeFormat.DENSE)
+
+
+def _mode_format_estimator(op: _BinaryOp) -> _ModeFormatOp:
+  """Produces a ModeFormat operator for the given binary operator.
+
+  The ModeFormat operator is used as a heuristic to derive the destination
+  dimension sparsity from the source dimension sparsity. In particular, if the
+  binary operator produces a disjunction of the zero values from its source
+  operands, such as the MUL operator, we return a ModeFormat operator that
+  uses operator.or_. That is, we estimate that a dimension for the MUL
+  operation result to be sparse if either of its source operands is sparse.
+
+  On the other hand, if the binary operator produces a conjunction of the
+  zero values from its source operands, such as the ADD operator, we return
+  a ModeFormat operator that uses operator.and_. In this case, we estimate
+  that a dimension for the ADD operation result to be sparse if both of its
+  source operands are sparse.
+
+  Args:
+    op: A _BinaryOp object representing a supporting operator on tensors.
+
+  Returns:
+    A ModeFormatOp for estimating the destination dimension sparsity from
+    the source dimension sparsity.
+  """
+  conjunction = functools.partial(_mode_format_operation, op=operator.and_)
+  disjunction = functools.partial(_mode_format_operation, op=operator.or_)
+  return conjunction if op(0, 1) != 0 else disjunction
+
+
+def _all_instance_of(collection: Iterable, cls: Any) -> bool:
+  """Returns true if all elements of the iterable is an instance of cls."""
+  return all(isinstance(e, cls) for e in collection)
+
+
+def _identity_ordering(rank: int) -> List[int]:
+  """Returns the identity ordering for tensor of given rank."""
+  return list(range(rank))
+
+
[e-mail](frozen=True)
+class ModeOrdering:
+  """The tensor dimension ordering class.
+
+  We support the TACO API mode_ordering class with an alias of this class.
+
+  Attributes:
+    ordering: A list of integers representing the ordering of the tensor
+      dimensions.
+  """
+  ordering: List[int]
+
+  def __post_init__(self) -> None:
+    """Verifies the value in ordering.
+
+    Raises:
+       ValueError: If ordering is not a list of integers.
+    """
+    if (not isinstance(self.ordering, list) or
+        not _all_instance_of(self.ordering, int)):
+      raise ValueError("Ordering must be a list of integers: "
+                       f"{self.ordering}")
+    # Check that ordering is a permutation of the dimension numbers.
+    if sorted(self.ordering) != _identity_ordering(self.rank()):
+      raise ValueError(f"Invalid ordering: {self.ordering} != "
+                       f"permutation{_identity_ordering(self.rank())}.")
+
+  def rank(self) -> int:
+    """Returns the number of dimensions represented by the ordering."""
+    return len(self.ordering)
+
+
[e-mail](frozen=True)
+class ModeFormatPack:
+  """The tensor dimension format class.
+
+  We support the TACO API mode_format_pack class with an alias of this class.
+
+  The storage format of a tensor contains one mode_format for each tensor
+  dimension.
+
+  Attributes:
+    formats: A list of ModeFormat representing the storage format for each of
+      the tensor dimension.
+  """
+  formats: List[ModeFormat]
+
+  def __post_init__(self) -> None:
+    """Verifies the value in formats.
+
+    Raises:
+       ValueError: If formats is not a list of ModeFormats.
+    """
+    if (not isinstance(self.formats, list) or
+        not _all_instance_of(self.formats, ModeFormat)):
+      raise ValueError("Formats must be a list of ModeFormat: "
+                       f"{self.formats}")
+
+  def rank(self) -> int:
+    """Returns the number of dimensions represented by the format pack."""
+    return len(self.formats)
+
+
[e-mail]
+class Format:
+  """The tensor format class defined by the TACO API.
+
+  Attributes:
+    format_pack: A ModeFormatPack representing the storage format for the tensor
+      dimensions.
+    ordering: A ModeOrdering representing the tensor dimension ordering in the
+      storage.
+  """
+  format_pack: ModeFormatPack
+  ordering: Optional[ModeOrdering] = None
+
+  def __post_init__(self) -> None:
+    """Verifies and fixes up the values in format_pack and ordering.
+
+    Verifies and fixes up the values in format_pack and ordering to supports the
+    initializer syntax defined by the TACO API. If format_pack is a list of
+    ModeFormat, replaces it with ModeFormatPack constructed from the list. If
+    ordering is not provided, set ordering to the natural ordering for the rank
+    corresponding to format_pack.
+
+    Raises:
+       ValueError: If format_pack is not an instance of ModeFormatPack or if
+         ordering is not an instance of ModeOrdering.
+    """
+    if isinstance(self.format_pack, list):
+      if not _all_instance_of(self.format_pack, ModeFormat):
+        raise ValueError(f"Expected a list of ModeFormat: {self.format_pack}")
+      self.format_pack = ModeFormatPack(self.format_pack)
+    if not isinstance(self.format_pack, ModeFormatPack):
+      raise ValueError(f"Expected ModeFormatpack: {self.format_pack}")
+
+    if self.ordering is None:
+      self.ordering = ModeOrdering(list(range(self.rank())))
+    if not isinstance(self.ordering, ModeOrdering):
+      raise ValueError(f"Expected ModeOrdering: {self.ordering}")
+
+    if self.format_pack.rank() != self.ordering.rank():
+      raise ValueError("Inconsistent ModeFormatPack and ModeOrdering: "
+                       f"len({self.format_pack}) != "
+                       f"len({self.ordering})")
+
+  def is_dense(self) -> bool:
+    """Returns true if all the Tensor dimensions have a dense format."""
+    return all([f == ModeFormat.DENSE for f in self.format_pack.formats])
+
+  def rank(self) -> int:
+    """Returns the number of dimensions represented by the format."""
+    return self.format_pack.rank()
+
+  def mlir_tensor_attr(self) -> Optional[sparse_tensor.EncodingAttr]:
+    """Constructs the MLIR attributes for the tensor format."""
+    if self.is_dense():
+      return None
+
+    order = (
+        range(self.rank()) if
+        (self.ordering is None) else self.ordering.ordering)
+    mlir_storage_format = [f.value for f in self.format_pack.formats]
+    return sparse_tensor.EncodingAttr.get(mlir_storage_format,
+                                          ir.AffineMap.get_permutation(order),
+                                          _POINTER_BIT_WIDTH, _INDEX_BIT_WIDTH)
+
+
+def _make_format(formats: List[ModeFormat],
+                 ordering: Optional[List[int]] = None) -> Format:
+  """Constructs a format from a list of ModeFormat and an optional ordering.
+
+  Args:
+    formats: A list of ModeFormat, one for each dimension of a tensor.
+    ordering: An optional list of integer, for the ordering of the tensor
+      dimensions. When an ordering is not given, the identity ordering is used.
+
+  Returns:
+    A tensor format object.
+
+  Raises:
+    ValueError: If formats is not a list of ModeFormat or the length of formats
+      is not consistent with the len of ordering.
+  """
+  ordering = ordering or _identity_ordering(len(formats))
+  return Format(ModeFormatPack(formats), ModeOrdering(ordering))
+
+
+class _AtomicCounter:
+  """An atomic counter."""
+
+  def __init__(self):
+    self._counter = 0
+    self._counter_lock = threading.Lock()
+
+  def increment(self) -> int:
+    """Increments the counter by one and returns the old value."""
+    old_value = self._counter
+    with self._counter_lock:
+      self._counter = self._counter + 1
+    return old_value
+
+
+class IndexVar:
+  """The tensor index class.
+
+  We support the TACO API index_var class with an alias of this class.
+
+  An IndexVar object represents an index variable in tensor index notation.
+
+  Attributes:
+    name: A unique string name of the IndexVar.
+  """
+  _counter = _AtomicCounter()
+
+  def __init__(self):
+    id = self._counter.increment()
+    self._name = f"{_TACO_INDEX_PREFIX}{id}"
+
+  def __repr__(self) -> str:
+    return f"IndexVar(name={repr(self._name)})"
+
+  @property
+  def name(self) -> str:
+    """Returns the name of the IndexVar."""
+    return self._name
+
+
+def get_index_vars(n: int) -> List[IndexVar]:
+  """Returns a list of n IndexVar.
+
+  This routine is defined by the TACO API.
+
+  Args:
+    n: An interger representing the number of IndexVar to get.
+
+  Returns:
+    A list of IndexVar.
+
+  Raises:
+    ValueError: if n is not a positive integer.
+  """
+  if not isinstance(n, int) or n <= 0:
+    raise ValueError(f"Expected an integer: {n}.")
+  # If lock contention ever becomes an issue, we could implement a bulk getter
+  # that returns a range by only claiming the lock once.
+  return [IndexVar() for i in range(n)]
+
+
+def _mlir_symbols_from_index_vars(
+    index_vars: Tuple[IndexVar, ...]) -> Tuple[lang.SymbolDef, ...]:
+  """Returns a tuple of MLIR symbols for the given tuple of index_var."""
+  return tuple(getattr(lang.S, i.name) for i in index_vars)
+
+
+def _mlir_dimensions_from_index_vars(
+    index_vars: Tuple[IndexVar, ...]) -> Tuple[lang.DimDef, ...]:
+  """Returns a tuple of MLIR dimensions for the given tuple of index_var."""
+  return tuple(getattr(lang.D, i.name) for i in index_vars)
+
+
+def _mlir_tensor_type(
+    dtype: DType, shape: Tuple[int, ...],
+    attr: Optional[sparse_tensor.EncodingAttr]) -> ir.RankedTensorType:
+  """Returns an MLIR tensor type.
+
+  Args:
+    dtype: An DType object for the element data type of the tensor.
+    shape: A tuple of integer for the shape of the tensor.
+    attr: An optional MLIR sparse tensor attribute, only provided if the tensor
+      is a sparse tensor.
+
+  Returns:
+    An MLIR ranked tensor type.
+  """
+  ir_type = _mlir_type_from_taco_type(dtype)
+  return ir.RankedTensorType.get(shape, ir_type, attr)
+
+
+def _verify_and_normalize_indices(indices) -> Tuple[IndexVar, ...]:
+  """Verifies and normalizes the indices for a tensor access.
+
+  Args:
+    indices: The index expression used to access a tensor, which could be any
+      Python object from user inputs.
+
+  Returns:
+    A tuple of IndexVar.
+
+  Raises:
+    ValueError: If indices is not an IndexVar or a tuple of IndexVar.
+  """
+  if isinstance(indices, IndexVar):
+    return (indices,)
+  elif isinstance(indices, tuple) and _all_instance_of(indices, IndexVar):
+    return indices
+
+  raise ValueError(f"Expected IndexVars: {indices}")
+
+
[e-mail](frozen=True)
+class _StructOpInfo:
+  """Information for generating a structured op in the linalg dialect.
+
+  This information is associated with an expression node that serves as the
+  root for an expression subtree implemented with a structured op.
+
+  Attributes:
+    dst_indices: A tuple of IndexVar, representing the result dimensions of the
+      structured op. This is used to construct the temporary variable for the
+      tensor to hold the structured op result.
+    dst_dims: A tuple of int, representing the result shape of the structured
+      op.
+    dst_dtype: A DType representing the data type of the structured op result.
+    dst_name: A string representing the name of the structured op result.
+    dst_format: A Format object representing the destination tensor format.
+  """
+  dst_indices: Tuple[IndexVar, ...]
+  dst_dims: Tuple[int, ...]
+  dst_dtype: DType
+  dst_name: str
+  dst_format: Format
+
+  def __post_init__(self) -> None:
+    """Verifies the integrity of the attribute values."""
+    assert len(self.dst_indices) == len(self.dst_dims)
+    assert self.dst_format is not None
+
+  def emit_tensor_init(self) -> ir.RankedTensorType:
+    """Returns an initialization for the destination tensor."""
+    if self.dst_format.is_dense():
+      # Initialize the dense tensor.
+      ir_type = _mlir_type_from_taco_type(self.dst_dtype)
+      tensor = linalg.InitTensorOp(self.dst_dims, ir_type).result
+      zero = arith.ConstantOp(ir_type, 0.0)
+      return linalg.FillOp(output=tensor, value=zero).results[0]
+
+    # Initialize the sparse tensor.
+    mlir_type = _mlir_tensor_type(self.dst_dtype, self.dst_dims,
+                                  self.dst_format.mlir_tensor_attr())
+    index_type = ir.IndexType.get()
+    dims = [arith.ConstantOp(index_type, d).result for d in mlir_type.shape]
+    return sparse_tensor.InitOp(mlir_type, dims)
+
+
+class _Stats:
+  """Information to describe how a tensor expression is implemented.
+
+  Currently, we only record the temporary tensors introduced for splitting the
+  original expression.
+  """
+
+  def __init__(self):
+    self._temps = []
+
+  def __repr__(self) -> str:
+    return f"_Stats({repr(self._temps)})"
+
+  def add_element(self, structop: _StructOpInfo):
+    """Adds a temporary tensor."""
+    self._temps.append(structop)
+
+  def get_total(self) -> int:
+    """Gets the total number of temporary tensors."""
+    return len(self._temps)
+
+  def _get_element(self, idx: int) -> _StructOpInfo:
+    """Gets the ith temporary tensor."""
+    assert idx < self.get_total()
+    return self._temps[idx]
+
+  def get_dimensions(self, idx: int) -> Tuple[int]:
+    """Gets the dimensions for the ith temporary tensor."""
+    return self._get_element(idx).dst_dims
+
+  def get_formats(self, idx: int) -> Tuple[ModeFormat]:
+    """Gets the ModeFormats for the ith temporary tensor."""
+    return tuple(self._get_element(idx).dst_format.format_pack.formats)
+
+
+class Tensor:
+  """The tensor class.
+
+  We support the TACO API tensor class with an alias of this class.
+
+  This class is part of the TACO API with the following methods:
+    insert: Inserts a value to the given coordinate in the tensor.
+    to_array: Returns a numpy ndarray for the tensor.
+
+  TACO API also defines the following arrtibutes for the class:
+    dtype: A dtype object representing the data type of the tensor.
+    format: A format object representing the storage format of the tensor.
+    name: A string object representing the name of the tensor.
+    order: An integral rank of the tensor.
+    shape: A list of integers representing the shape of the tensor.
+
+  We currently ignore the tensor dimension ordering for dense tensor.
+  """
+  _counter = _AtomicCounter()
+
+  def _get_unique_name(self) -> str:
+    """Returns a unique name for creating a new Tensor."""
+    return f"{_TACO_TENSOR_PREFIX}{self._counter.increment()}"
+
+  def _init_format(self, fmt: Union[ModeFormat, List[ModeFormat],
+                                    Format]) -> None:
+    """Process the fmt argument for the Tensor constructor.
+
+    Args:
+      fmt: This argument can be a ModeFormat, List[ModeFormat], or format. If
+        this argument is a ModeFormat, uses this ModeFormat for all the tensor
+        dimensions. If this argument is a list of ModeFormat, the len of the
+        list should equal to the rank of the tensor. If this argument is a
+        format, uses it for the format of the tensor.
+
+    Raises:
+      ValueError: If fmt is not one of the expected type or is inconsistent
+        with the rank of the tensor. This is because fmt could be an users
+        input.
+    """
+    if isinstance(fmt, ModeFormat):
+      self._format = _make_format([fmt] * self.order)
+    elif isinstance(fmt, list):
+      if len(fmt) == self.order and isinstance(fmt[0], ModeFormat):
+        self._format = _make_format(fmt)
+      else:
+        raise ValueError("Inconsistent shape and format: "
+                         f"{self._shape}, {fmt}.")
+    elif isinstance(fmt, Format):
+      if fmt.rank() != self.order:
+        raise ValueError("Inconsistent shape and format: "
+                         f"{self._shape}, {fmt}.")
+      else:
+        self._format = fmt
+    else:
+      raise ValueError(f"Invalid format argument: {fmt}.")
+
+  def __init__(self,
+               value_or_shape: Optional[Union[List[int], Tuple[int, ...], float,
+                                              int]] = None,
+               fmt: Optional[Union[ModeFormat, List[ModeFormat],
+                                   Format]] = None,
+               dtype: Optional[DType] = None,
+               name: Optional[str] = None):
+    """The tensor constructor interface defined by TACO API.
+
+    Args:
+      value_or_shape: This argument is optional and can be int, float,
+        List[int], or Tuple[int, ...]. If this argument is an int or float,
+        creates a scalar tensor and initializes it with the value. If this
+        argument is a list or tuple of int, uses it as the shape to create a
+        tensor.
+      fmt: This argument can be a ModeFormat, List[ModeFormat], or format. If
+        this argument is a ModeFormat, uses this ModeFormat for all the tensor
+        dimensions. If this argument is a list of ModeFormat, the len of the
+        list should equal to the rank of the tensor. If this argument is a
+        format, uses it for the format of the tensor.
+      dtype: An object of dtype, representing the data type of the tensor.
+      name: A string name of the tensor. If a name is not given, creates a
+        unique name for the tensor.
+
+    Raises:
+      ValueError: If there is any inconsistency among the input arguments.
+    """
+    # Take care of the argument default values.
+    fmt = fmt or ModeFormat.COMPRESSED
+    dtype = dtype or DType(Type.FLOAT64)
+    self._name = name or self._get_unique_name()
+
+    self._dtype = dtype
+    # We currently use _coords and _values to host the sparse tensor value with
+    # COO format, and _dense_storage to host the dense tensor value. We haven't
+    # implement the conversion between the two storages yet. This will be
+    # improved in a follow up CL.
+    self._coords = []
+    self._values = []
+    self._dense_storage = None
+    self._stats = _Stats()
+    if value_or_shape is None or isinstance(value_or_shape, int) or isinstance(
+        value_or_shape, float):
+      # Create a scalar tensor and ignore the fmt parameter.
+      self._shape = []
+      self._format = _make_format([], [])
+      if value_or_shape is not None:
+        self._dense_storage = np.array(value_or_shape, dtype=self._dtype.value)
+    elif (isinstance(value_or_shape, tuple) or isinstance(
+        value_or_shape, list)) and _all_instance_of(value_or_shape, int):
+      # Create a tensor with the specified shape and format.
+      self._shape = list(value_or_shape)
+      self._init_format(fmt)
+    else:
+      raise ValueError("Invalid first argument. "
+                       "Must be a tuple or list for a shape or a single value"
+                       f"if initializing a scalar tensor: {value_or_shape}.")
+
+  def __repr__(self) -> str:
+    value_str = (f"{repr(self._dense_storage)})" if self.is_dense() else
+                 f"{repr(self._coords)} {repr(self._values)})")
+    return (f"Tensor(_name={repr(self._name)} "
+            f"_dtype={repr(self._dtype)} : ") + value_str
+
+  def insert(self, coords: List[int], val: Union[float, int]) -> None:
+    """Inserts a value to the given coordinate.
+
+    Args:
+      coords: A list of integer coordinates. The length of the list must be the
+        same as the rank of the tensor.
+      val: A value being inserted. It is either an integral or a floating point
+        value. This value will be converted to the data type of the tensor.
+
+    Raises:
+      ValueError: When there is any problem in the parameters.
+    """
+    if not isinstance(coords, list):
+      raise ValueError(f"Non list coordinate detected: {coords}.")
+    if not _all_instance_of(coords, int):
+      raise ValueError(f"Non integer coordinate detected: {coords}.")
+    if (len(coords) != self.order or
+        any([c < 0 or c >= self._shape[i] for i, c in enumerate(coords)])):
+      raise ValueError("Invalid coordinate for rank: "
+                       f"{self.order}, {coords}.")
+
+    if not isinstance(val, int) and not isinstance(val, float):
+      raise ValueError(f"Value is neither int nor float: {val}.")
+
+    self._coords.append(tuple(coords))
+    self._values.append(self._dtype.value(val))
+
+  def is_dense(self) -> bool:
+    """Returns true if all the Tensor dimensions have a dense format."""
+    return self._format.is_dense()
+
+  def to_array(self) -> np.ndarray:
+    """Returns the numpy array for the Tensor.
+
+    This is currenly only implemented for dense Tensor.
+    """
+    if not self.is_dense():
+      raise ValueError("Conversion from non-dense Tensor "
+                       "to numpy array not supported yet.")
+    return self._dense_storage
+
+  @staticmethod
+  def from_array(array: np.ndarray) -> "Tensor":
+    """Returns a dense tensor with the value copied from the input array.
+
+    We currently only support the conversion of float64 numpy arrays to Tensor.
+
+    Args:
+      array: The numpy array that provides the data type, shape and value for
+        the tensor.
+
+    Returns:
+      A Tensor object.
+
+    Raises:
+      ValueError if the data type of the numpy array is not float64.
+    """
+    if array.dtype != np.float64:
+      raise ValueError(f"Expected float64 value type: {array.dtype}.")
+    tensor = Tensor(array.shape, ModeFormat.DENSE)
+    tensor._dense_storage = np.copy(array)
+    return tensor
+
+  @staticmethod
+  def from_coo(
+      coordinates: List[Tuple[int, ...]],
+      values: List[_AnyRuntimeType],
+      fmt: Format,
+      dtype: DType,
+  ) -> "Tensor":
+    """Converts coordinates and values to a sparse tensor representation.
+
+    Args:
+      coordinates: A list of coordinates with non-zero values.
+      values: The non-zero values.
+      fmt: The tensor storage format.
+      dtype: The tensor element data type.
+
+    Returns:
+      A tensor with the given non-zero values and storage format. The shape of
+      the tensor has the minimum size for each dimension to make the given
+      coordinates valid.
+    """
+    assert (isinstance(coordinates, List) and
+            _all_instance_of(coordinates, Tuple))
+    assert (isinstance(values, List) and _all_instance_of(values, dtype.value))
+    assert isinstance(fmt, Format)
+
+    rank = fmt.rank()
+    assert all(len(c) == rank and _all_instance_of(c, int) for c in coordinates)
+
+    # Find the maximum coordinate value for each dimension.
+    max_coordinate = list(map(max, zip(*coordinates)))
+    # The size of each dimension is one more that such a maximum coordinate
+    # value.
+    shape = [c + 1 for c in max_coordinate]
+    tensor = Tensor(shape, fmt)
+    tensor._coords = coordinates
+    tensor._values = values
+
+    return tensor
+
+  @property
+  def dtype(self) -> DType:
+    """Returns the data type for the Tensor."""
+    return self._dtype
+
+  @property
+  def format(self) -> Format:
+    """Returns the storage format for the Tensor."""
+    return self._format
+
+  @property
+  def name(self) -> str:
+    """Returns the name for the Tensor."""
+    return self._name
+
+  @property
+  def order(self) -> int:
+    """Returns the rank of the Tensor."""
+    return len(self._shape)
+
+  @property
+  def shape(self) -> List[int]:
+    """Returns the shape of the Tensor."""
+    return self._shape
+
+  def __getitem__(self, key) -> "Access":
+    """Verifies and processes a tensor access.
+
+    In the tensor index notation, a tensor access T[i, j] is represented as
+    retrieving a value with key (i, j) from the tensor object T in Python. This
+    routine verifies the key for the tensor access and returns a tensor access
+    object.
+
+    Args:
+      key: The key used to access the tensor, which could be any Python object
+        from user inputs.
+
+    Returns:
+      The corresponding tensor access object.
+
+    Raises:
+      ValueError: If key is not an IndexVar or a tuple of IndexVar.
+    """
+    indices = _verify_and_normalize_indices(key)
+    return Access(self, indices)
+
+  def __setitem__(self, key, value) -> None:
+    """Verifies and processes a tensor assignment.
+
+    In the tensor index notation, a tensor assignment "T[i, j] = ..." is
+    represented as setting a value for a tensor object T via key (i, j) in
+    Python. This routine verifies the key, evaluates the value, and assigns the
+    value to the tensor.
+
+    We only support assignment of dense tensor currently.
+
+    Args:
+      key: The key used to access the tensor, which could be any Python object
+        from user inputs.
+      value: The value assigned to the tensor, which could be any Python object
+        from user inputs.
+
+    Raises:
+      ValueError: If tensor is not a dense tensor, or the key is not an IndexVar
+        or a tuple of IndexVar, or the length of the indices is not the same as
+        the rank of the tensor.
+    """
+    indices = _verify_and_normalize_indices(key)
+    if len(indices) != self.order:
+      raise ValueError("Mismatch between indices and tensor rank: "
+                       f"len({indices}) != {self.order}.")
+
+    result = value.evaluate(self, indices)
+    if self.is_dense():
+      assert isinstance(result, np.ndarray)
+      self._dense_storage = result
+    else:
+      assert _all_instance_of(result, np.ndarray) and len(result) == 2
+      assert (result[0].ndim, result[1].ndim) == (1, 2)
+      (self._values, self._coords) = result
+
+  def mlir_tensor_type(self) -> ir.RankedTensorType:
+    """Returns the MLIR type for the tensor."""
+    return _mlir_tensor_type(self._dtype, tuple(self._shape),
+                             self._format.mlir_tensor_attr())
+
+  def dense_dst_ctype_pointer(self) -> ctypes.pointer:
+    """Returns the ctypes pointer for the pointer to an MemRefDescriptor.
+
+    For a dense tensor output, the MLIR compiler allocates the storage for
+    the tensor. This routine returns the pointer to an MLIR MemRefDescriptor for
+    receiving the tensor.
+    """
+    assert self.is_dense()
+    mem_ref_desc = runtime.make_nd_memref_descriptor(
+        self.order, np.ctypeslib.as_ctypes_type(self.dtype.value))()
+    return ctypes.pointer(ctypes.pointer(mem_ref_desc))
+
+  def ctype_pointer(self) -> ctypes.pointer:
+    """Returns the ctypes pointer for the pointer to the input tensor."""
+    if self.is_dense():
+      if self._dense_storage is None:
+        self._dense_storage = np.zeros(self._shape, self._dtype.value)
+      return _ctype_pointer_from_array(self._dense_storage)
+
+    shape = np.array(self._shape, np.int64)
+    indices = np.array(self._coords, np.int64)
+    values = np.array(self._values, self._dtype.value)
+    ptr = utils.coo_tensor_to_sparse_tensor(_get_support_lib_name(), shape,
+                                            values, indices)
+    return ctypes.pointer(ctypes.cast(ptr, ctypes.c_void_p))
+
+  def get_coordinates_and_values(
+      self) -> Tuple[List[Tuple[int, ...]], List[_AnyRuntimeType]]:
+    """Returns the coordinates and values for the non-zero elements."""
+    if not self.is_dense():
+      return (self._coords, self._values)
+
+    # Coordinates for non-zero elements, grouped by dimensions.
+    coords_by_dims = self._dense_storage.nonzero()
+    # Coordinates for non-zero elements, grouped by elements.
+    coords = np.transpose(coords_by_dims)
+    values = self._dense_storage[coords_by_dims]
+    return (coords, values)
+
+  def _record_stats(self, structop: "_StructOpInfo"):
+    """Collects information for temporary tensors."""
+    # Exclude user specified destination tensors.
+    if structop.dst_name == self.name:
+      return
+
+    self._stats.add_element(structop)
+
+
+def _emit_operand(op_def: lang.LinalgOpDef, indices: Tuple[IndexVar, ...],
+                  name: str, kind: lang.OperandKind) -> lang.OperandDef:
+  """Emits an operand for a tensor access in the current linalg operation.
+
+  Args:
+    op_def: A LinalgOpDef representing the current linalg dialect operation.
+    indices: A tuple of IndexVar used to access the tensor.
+    name: A unique string name of the tensor.
+    kind: An OperandKind for the operand.
+
+  Returns:
+    An OperandDef representing the operand.
+  """
+  dim_sym = _mlir_symbols_from_index_vars(indices)
+  opnd = lang.OperandDef(kind, lang.T, dim_sym)
+  op_def.add_operand(name, opnd)
+  return opnd
+
+
[e-mail](frozen=True)
+class _DimInfo:
+  """Information for an operand dimension.
+
+  Attributes:
+    dim: An integer for the size of the dimension.
+    mode_format: A ModeFormat for the dimension sparsity.
+  """
+  dim: int
+  mode_format: ModeFormat
+
+
[e-mail]()
+class _ExprInfo:
+  """Expression information for validation and code generation.
+
+  Attributes:
+    src_indices: A tuple of IndexVar for the indices used by the tensors in the
+      expression tree.
+    dim_infos: A tuple of _DimInfo, representing the dimension information
+      corresponding to the src_indices.
+    reduce_indices: A set of IndexVar for the indices reduced by the expression.
+    acc_reduce_indices: An accumulated set of IndexVar for the indices reduced
+      by the expression and its children.
+    structop_info: Information to support the code generation for a structured
+      op in the linalg dialect, if the corresponding expression node is the root
+      of a subtree for a structured op.
+    mlir_value: The MLIR value generated for the structured op.
+  """
+  src_indices: Tuple[IndexVar, ...]
+  dim_infos: Tuple[_DimInfo, ...]
+  reduce_indices: Optional[Set[IndexVar]] = None
+  acc_reduce_indices: Optional[Set[IndexVar]] = None
+  structop_info: Optional[_StructOpInfo] = None
+  mlir_value: Optional[ir.Value] = None
+
+  def __post_init__(self) -> None:
+    """Verifies and fix up attribute values.
+
+    Verifies the consistency of the attributes and modifies the default values
+    to support convenient initializer syntax.
+    """
+    assert len(self.src_indices) == len(self.dim_infos)
+    self.reduce_indices = self.reduce_indices or set()
+    self.acc_reduce_indices = self.acc_reduce_indices or set()
+
+
+class IndexExpr(abc.ABC):
+  """The index notation base class.
+
+  We support the TACO API index_expression class with an alias of this class.
+  """
+
+  def _verify_operand_and_build_expr(self, rhs, op: _BinaryOp) -> "_BinaryExpr":
+    """Verifies the RHS operand and returns a binary expression.
+
+    Args:
+      rhs: The RHS of the binary operation, which could be any Python object
+        from user inputs.
+      op: A _BinaryOp object representing the binary operator.
+
+    Raises:
+      ValueError: If rhs is not an IndexExpr.
+    """
+    if not isinstance(rhs, IndexExpr):
+      raise ValueError(f"Expected IndexExpr: {rhs}")
+    return _BinaryExpr(op, self, rhs)
+
+  def __add__(self, rhs) -> "_BinaryExpr":
+    """Defines the operator +.
+
+    Args:
+      rhs: The value being added, which could be any Python object from user
+        inputs.
+
+    Returns:
+      A _BinaryExpr object representing the operation.
+
+    Raises:
+      ValueError: If rhs is not an IndexExpr.
+    """
+    return self._verify_operand_and_build_expr(rhs, operator.add)
+
+  def __mul__(self, rhs) -> "_BinaryExpr":
+    """Defines the operator *.
+
+    Args:
+      rhs: The value being multiplied, which could be any Python object from
+        user inputs.
+
+    Returns:
+      A _BinaryExpr object representing the operation.
+
+    Raises:
+      ValueError: If rhs is not an IndexExpr.
+    """
+    return self._verify_operand_and_build_expr(rhs, operator.mul)
+
+  def __sub__(self, rhs) -> "_BinaryExpr":
+    """Defines the operator -.
+
+    Args:
+      rhs: The value being subtracted, which could be any Python object from
+        user inputs.
+
+    Returns:
+      A _BinaryExpr object representing the operation.
+
+    Raises:
+      ValueError: If rhs is not an IndexExpr.
+    """
+    return self._verify_operand_and_build_expr(rhs, operator.sub)
+
+  @abc.abstractmethod
+  def _visit(self,
+             func: _ExprVisitor,
+             args,
+             *,
+             leaf_checker: _SubtreeLeafChecker = None) -> None:
+    """A post-order visitor.
+
+    Args:
+      func: A callable applied to each node in the expression tree.
+      args: The variable-length arguments passed to the callable. These
+        arguments are grouped as an iterable and will be unpacked before passing
+        to the callable. This is to enable the keyword argument only syntax
+        after this argument.
+      leaf_checker: A callable object to identify nodes that should be treated
+        as leaf nodes to support partial tree visiting.
+    """
+    pass
+
+  @abc.abstractmethod
+  def _emit_expression(
+      self,
+      expr_to_opnd: Dict["IndexExpr", lang.OperandDef],
+      expr_to_info: _ExprInfoDict,
+  ) -> lang.ScalarExpression:
+    """Emits MLIR for the expression tree.
+
+    Args:
+      expr_to_opnd: A dictionary for looking up structured op input operands for
+        the input nodes of the structured op.
+      expr_to_info: A dictionary for looking up code generation information for
+        expressions.
+
+    Returns:
+      A linalg dialect ScalarExpression for the expression.
+    """
+    pass
+
+  @abc.abstractmethod
+  def dtype(self) -> DType:
+    """Returns the data type for the result of the expression."""
+    pass
+
+  def _emit_structured_op(self, expr_to_info: _ExprInfoDict) -> None:
+    """Emits a structured op in the linalg dialect for the expression tree.
+
+    We define a DefineOpcallable in the domain specific language for the linalg
+    dialect and execute the callable to generate the structured op. Self is the
+    root of the expression tree for the structured op.
+
+    Args:
+      expr_to_info: A dictionary for looking up code generation information for
+        expressions.
+    """
+    op_info = expr_to_info[self].structop_info
+    op_name = op_info.dst_name
+    op_def = lang.LinalgOpDef(name=op_name)
+    op_callable = lang.DefinedOpCallable(op_name, op_def)
+
+    # Collect the input expression nodes for the structured op.
+    expr_inputs = []
+    self._visit(
+        _gather_structured_op_input,
+        (self, expr_to_info, expr_inputs),
+        leaf_checker=_is_structured_op_leaf,
+    )
+
+    # Create a linalg structured op operand for each input expression node and
+    # build a dictionary for looking up the information.
+    expr_to_input_opnd = {
+        e: _emit_structured_op_input(e, expr_to_info, op_def)
+        for e in expr_inputs
+    }
+
+    # Emit the expression tree, which produces the value assigned to the
+    # destination tensor.
+    value = self._emit_expression(expr_to_input_opnd, expr_to_info)
+    # Emit the structured op representation for the destination tensor.
+    dst_opnd = _emit_operand(op_def, op_info.dst_indices, op_info.dst_name,
+                             lang.OperandKind.OutputTensor)
+    dst_dim_syms = _mlir_dimensions_from_index_vars(op_info.dst_indices)
+    dst_use = lang.TensorUse(dst_opnd, dst_dim_syms)
+
+    expr_info = expr_to_info[self]
+    # If the structured op reduces some indices, explicitly represent the
+    # reduction. This is done by generating a ReduceFn for the dimensions being
+    # reduced in the linalg dialect and calling the function with the value
+    # being reduced. We only support add reduction currently.
+    if expr_info.reduce_indices:
+      reduce_dims = _mlir_dimensions_from_index_vars(expr_info.reduce_indices)
+      value = lang.ReduceFn.add[reduce_dims](value)
+
+    # Emit the assignment as a comprehension in the linalg dialect.
+    comp = lang.Comprehension((dst_use, value))
+    op_def.comprehensions.append(comp)
+
+    # The structured op in the linalg dialect requires an explicit
+    # initialization for the destination tensor. Emit MLIR to initialize the
+    # destination tensor.
+    init = op_info.emit_tensor_init()
+
+    # Collect MLIR values for the linalg input operands, with the assumption
+    # that dictionary preserves the insertion order.
+    args = [
+        expr_to_info[expr].mlir_value
+        for expr, opnd in expr_to_input_opnd.items()
+    ]
+    # Execute the DefineOpcallable object for the linalg dialect operation to
+    # emit MLIR for the linalg structured op.
+    expr_info.mlir_value = op_callable(*args, outs=[init])
+
+  def _identify_structured_ops(
+      self,
+      expr_to_info: _ExprInfoDict,
+      dst: Tensor,
+      dst_indices: Tuple[IndexVar, ...],
+  ) -> List["IndexExpr"]:
+    """Returns expression nodes for the roots of the identified structured ops.
+
+    A structured op in the linalg dialect only supports reduction performed on
+    the whole expression. If the expression tree contains reduction that are
+    performed on part of the expression tree, the expression tree needs to be
+    implemented with multiple structured ops. This routine identifies all the
+    expression nodes that contain reduction as the root of structured ops in the
+    linalg dialect.
+
+    Args:
+      expr_to_info: A dictionary for looking up code generation information for
+        expressions.
+      dst: A destination Tensor that accepts the value of the expression tree.
+      dst_indices: The indices used by the destination index expression.
+
+    Returns:
+      An ordered list of IndexExpr for the root expressions of the structured
+      ops, where child expressions go before parent expressions that use their
+      results.
+    """
+    reduce_indices = tuple(
+        set(expr_to_info[self].src_indices) - set(dst_indices))
+    for reduce_index in reduce_indices:
+      _mark_structured_op_root(self, reduce_index, expr_to_info)
+
+    self._visit(_accumulate_reduce_indices, (expr_to_info,))
+    structop_roots = []
+    self._visit(_gather_structured_op, (expr_to_info, structop_roots))
+
+    # Handle the root of the top level expression.
+    if not structop_roots or structop_roots[-1] != self:
+      # The top level expression is not a reduction. Add the top level
+      # expression as a structured op root.
+      structop_roots.append(self)
+
+    # Use user specified information for the destination tensor to build an
+    # _StructOpInfo for the top level expression.
+    expr_to_info[self].structop_info = _StructOpInfo(dst_indices,
+                                                     tuple(dst.shape),
+                                                     self.dtype(), dst.name,
+                                                     dst.format)
+
+    return structop_roots
+
+  def _validate_and_collect_expr_info(
+      self,
+      dst: Tensor,
+      dst_indices: Tuple[IndexVar, ...],
+  ) -> _ExprInfoDict:
+    """Propagates expression information for validation.
+
+    Propagates the indices used by child expression nodes to parent expression
+    nodes. Also collects and validates the sizes for the dimensions
+    corresponding to the indices.
+
+    Args:
+      dst: A destination Tensor that accepts the value of the expression tree.
+      dst_indices: The indices used by the destination index expression.
+
+    Raises:
+      ValueError if there is any inconsistency in indices or dimensional
+      values.
+
+    Returns:
+      A dictionary of (IndexExpr, _ExprInfo).
+    """
+    expr_to_info = {}
+    # Validate the expression tree and construct expression information.
+    self._visit(_validate_and_collect_expr_info, (expr_to_info,))
+
+    # Validate the destination dimension information.
+    info = expr_to_info[self]
+    index_to_dim_info = {i: d for i, d in zip(info.src_indices, info.dim_infos)}
+    for i, d, in zip(dst_indices, dst.shape):
+      if i not in index_to_dim_info:
+        raise ValueError("Destination IndexVar not used in the "
+                         f"source expression: {i}")
+      else:
+        if d != index_to_dim_info[i].dim:
+          raise ValueError(f"Inconsistent destination dimension for {i}: "
+                           f"{d} vs {index_to_dim_info[i].dim}")
+
+    return expr_to_info
+
+  def _emit_assignment(
+      self,
+      module: ir.Module,
+      dst: Tensor,
+      dst_indices: Tuple[IndexVar, ...],
+      expr_to_info: _ExprInfoDict,
+      input_accesses: List["Access"],
+  ) -> None:
+    """Emits an MLIR function for assigning the expression to a tensor."""
+    input_types = [a.tensor.mlir_tensor_type() for a in input_accesses]
+
+    # Build the kernel for the operations.
+    with ir.InsertionPoint(module.body):
+
+      @builtin.FuncOp.from_py_func(*input_types, name=_ENTRY_NAME)
+      def linalg_funcop(*args):
+        # Set up the mapping from the Access nodes to their MLIR values.
+        for e, mlir in zip(input_accesses, args):
+          expr_to_info[e].mlir_value = mlir
+
+        # Emit structured ops in the linalg dialect to implement the assignment.
+        for structop_root in self._identify_structured_ops(
+            expr_to_info, dst, dst_indices):
+          structop_root._emit_structured_op(expr_to_info)
+          dst._record_stats(expr_to_info[structop_root].structop_info)
+
+        # The function returns the MLIR value of the root expression.
+        return expr_to_info[self].mlir_value
+
+      linalg_funcop.func_op.attributes[
+          "llvm.emit_c_interface"] = ir.UnitAttr.get()
+
+  def evaluate(
+      self,
+      dst: Tensor,
+      dst_indices: Tuple[IndexVar, ...],
+  ) -> Union[np.ndarray, Tuple[np.ndarray, np.ndarray]]:
+    """Evaluates tensor assignment dst[dst_indices] = expression.
+
+    Args:
+      dst: The destination tensor.
+      dst_indices: The tuple of IndexVar used to access the destination tensor.
+
+    Returns:
+      The result of the dense tensor represented in numpy ndarray or the sparse
+      tensor represented by two numpy ndarray for its non-zero values and
+      indices.
+
+    Raises:
+      ValueError: If the expression is not proper or not supported.
+    """
+    expr_to_info = self._validate_and_collect_expr_info(dst, dst_indices)
+
+    # Compute a list of input accesses.
+    input_accesses = []
+    self._visit(_gather_input_accesses_index_vars, (input_accesses,))
+
+    support_lib = _get_support_lib_name()
+    # Build and compile the module to produce the execution engine.
+    with ir.Context(), ir.Location.unknown():
+      module = ir.Module.create()
+      self._emit_assignment(module, dst, dst_indices, expr_to_info,
+                            input_accesses)
+      compiled_module = _compile_mlir(module)
+
+      # We currently rely on an environment to pass in the full path of a
+      # supporting library for the execution engine.
+      engine = execution_engine.ExecutionEngine(
+          compiled_module, opt_level=_OPT_LEVEL, shared_libs=[support_lib])
+
+    # Gather the pointers for the input buffers.
+    input_pointers = [a.tensor.ctype_pointer() for a in input_accesses]
+    if dst.is_dense():
+      # The pointer to receive dense output is the first argument to the
+      # execution engine.
+      arg_pointers = [dst.dense_dst_ctype_pointer()] + input_pointers
+    else:
+      # The pointer to receive sparse output is the last argument to the
+      # execution engine. The pointer to receive a sparse tensor output is a
+      # pointer to pointer of char.
+      arg_pointers = input_pointers + [
+          ctypes.pointer(ctypes.pointer(ctypes.c_char(0)))
+      ]
+
+    # Invoke the execution engine to run the module and return the result.
+    engine.invoke(_ENTRY_NAME, *arg_pointers)
+
+    if dst.is_dense():
+      return runtime.ranked_memref_to_numpy(arg_pointers[0][0])
+
+    # Check and return the sparse tensor output.
+    rank, nse, shape, values, indices = utils.sparse_tensor_to_coo_tensor(
+        support_lib,
+        ctypes.cast(arg_pointers[-1][0], ctypes.c_void_p),
+        np.float64,
+    )
+    assert (np.equal(rank, dst.order)
+            and np.array_equal(shape, np.array(dst.shape)) and
+            np.equal(values.ndim, 1) and np.equal(values.shape[0], nse) and
+            np.equal(indices.ndim, 2) and np.equal(indices.shape[0], nse) and
+            np.equal(indices.shape[1], rank))
+    return (values, indices)
+
+
[e-mail](frozen=True)
+class Access(IndexExpr):
+  """The tensor access class.
+
+  We support the TACO API access class with an alias of this class.
+
+  Attributes:
+    tensor: A Tensor being accessed.
+    indices: A tuple of IndexVar, representing the indices used to access the
+      Tensor.
+  """
+  tensor: Tensor
+  indices: Tuple[IndexVar, ...]
+
+  def __post_init__(self) -> None:
+    """Verifies the tensor and indices for a tensor access.
+
+    Raises:
+       ValueError: If indices is not a list of IndexVar or the len of indices
+       doesn't equal to the rank of the tensor.
+    """
+    if (not isinstance(self.indices, tuple) or
+        not _all_instance_of(self.indices, IndexVar)):
+      raise ValueError(f"Indices contain non IndexVar: {str(self.indices)}.")
+    if self.tensor.order != len(self.indices):
+      raise ValueError("Invalid indices for rank: "
+                       f"str{self.tensor.order} != len({str(self.indices)}).")
+
+  def _emit_expression(
+      self,
+      expr_to_opnd: Dict[IndexExpr, lang.OperandDef],
+      expr_to_info: _ExprInfoDict,
+  ) -> lang.ScalarExpression:
+    """Emits a linalg dialect TensorUse expression for the tensor access."""
+    assert self in expr_to_opnd
+    dims = _mlir_dimensions_from_index_vars(self.indices)
+    return lang.TensorUse(expr_to_opnd[self], dims)
+
+  def _visit(self,
+             func: _ExprVisitor,
+             args,
+             *,
+             leaf_checker: _SubtreeLeafChecker = None) -> None:
+    if leaf_checker:
+      assert leaf_checker(self, *args)
+    func(self, *args)
+
+  def dtype(self) -> DType:
+    return self.tensor.dtype
+
+
+def _gather_input_accesses_index_vars(
+    expr: IndexExpr,
+    input_accesses: List[Access],
+) -> None:
+  """Collects Access nodes."""
+  if isinstance(expr, Access) and expr not in input_accesses:
+    input_accesses.append(expr)
+
+
+def _op_to_callable(op: _BinaryOp) -> lang.ArithFnType:
+  """Returns the linalg dialect function object for the given operation."""
+  op_to_callable = {
+      operator.add: lang.ArithFn.add,
+      operator.sub: lang.ArithFn.sub,
+      operator.mul: lang.ArithFn.mul,
+  }
+  return op_to_callable[op]
+
+
[e-mail](frozen=True)
+class _BinaryExpr(IndexExpr):
+  """The representation for a binary operation.
+
+  Attributes:
+  op: A _BinaryOp representing the binary operation.
+  a: An IndexExpr representing the first operand of the operation.
+  b: An IndexExpr representing the second operand of the operation.
+  """
+  op: _BinaryOp
+  a: IndexExpr
+  b: IndexExpr
+
+  def __post_init__(self) -> None:
+    """Verifies that the operands being added are IndexExpr."""
+    assert isinstance(self.a, IndexExpr) and isinstance(self.b, IndexExpr)
+
+  def _emit_expression(
+      self,
+      expr_to_opnd: Dict[IndexExpr, lang.OperandDef],
+      expr_to_info: _ExprInfoDict,
+  ) -> lang.ScalarExpression:
+    """Emits the expression tree and returns the expression."""
+    # The current expression node is an internal node of the structured op.
+    if self not in expr_to_opnd:
+      a = self.a._emit_expression(expr_to_opnd, expr_to_info)
+      b = self.b._emit_expression(expr_to_opnd, expr_to_info)
+      return _op_to_callable(self.op)(a, b)
+
+    # The current expression is a leaf node of the structured op. That is, it is
+    # a temporary tensor generated by its child structured op.
+    op_info = expr_to_info[self].structop_info
+    assert op_info is not None
+    dims = _mlir_dimensions_from_index_vars(op_info.dst_indices)
+    return lang.TensorUse(expr_to_opnd[self], dims)
+
+  def _visit(self,
+             func: _ExprVisitor,
+             args,
+             *,
+             leaf_checker: _SubtreeLeafChecker = None) -> None:
+    """A post-order visitor."""
+    if leaf_checker is None or not leaf_checker(self, *args):
+      self.a._visit(func, args, leaf_checker=leaf_checker)
+      self.b._visit(func, args, leaf_checker=leaf_checker)
+    func(self, *args)
+
+  def dtype(self) -> DType:
+    """Returns the data type of the binary operation."""
+    return self.a.dtype()
+
+
+def _validate_and_collect_dim_info(
+    index_to_dim_info: Dict[IndexVar, _DimInfo],
+    indices: Tuple[IndexVar, ...],
+    dim_infos: Tuple[_DimInfo, ...],
+    expr: _BinaryExpr,
+) -> None:
+  """Validates and collects the dimension information for an index notation.
+
+  Validates (indices, dim_infos) against the information collected from other
+  source operands and is represented by index_to_dim_info. In particular, we
+  ensure that each IndexVar corresponds to only one dimension size. We also
+  aggregate the new information represented in (indices, dim_infos) to
+  index_to_dim_info.
+
+  Args:
+    index_to_dim: A dictionary of (IndexVar, _DimInfo) collected from the
+      previous operands.
+    indices: The IndexVars to be validated.
+    dim_infos: The dimension information for the IndexVars to be validated.
+    expr: The binary expression where (indices, dim_infos) is used.
+
+  Raises:
+    ValueError if there is any problem in the IndexVars or dimensional values.
+  """
+  assert len(indices) == len(dim_infos)
+  for i, d in zip(indices, dim_infos):
+    if i not in index_to_dim_info:
+      index_to_dim_info[i] = d
+    else:
+      if d.dim != index_to_dim_info[i].dim:
+        raise ValueError(f"Inconsistent source dimension for {i}: "
+                         f"{d.dim} vs {index_to_dim_info[i].dim}")
+      mode_format = _mode_format_estimator(expr.op)(
+          index_to_dim_info[i].mode_format, d.mode_format)
+      index_to_dim_info[i] = _DimInfo(d.dim, mode_format)
+
+
+def _validate_and_collect_expr_info(
+    expr: IndexExpr,
+    expr_to_info: _ExprInfoDict,
+) -> None:
+  """Validates dimension information and constructs _ExprInfo.
+
+  Validates that dimensional values for the same IndexVar are the same. Collects
+  a list of IndexVar used by the expression and their corresponding dimensional
+  values. Constructs an _ExprInfo object to record the information for the
+  IndexExpr.
+
+  This routine is passed to the post-order visitor as an _ExprVisitor object.
+
+  Args:
+    expr: The IndexExpr being validated.
+    expr_to_info: The dictionary of (IndexExpr, _ExprInfo) for recording the
+      expression information.
+
+  Raises:
+    ValueError if there is any problem in the IndexVars or dimensional values.
+  """
+  # Objects of class Access can be shared by different expressions. Avoid
+  # processing Access objects multiple times by skipping the processing if expr
+  # is already in the dictionary.
+  if expr in expr_to_info:
+    return
+
+  if isinstance(expr, Access):
+    src_indices = expr.indices
+    src_dims = tuple(expr.tensor.shape)
+    mode_formats = tuple(expr.tensor.format.format_pack.formats)
+    assert len(src_dims) == len(mode_formats)
+    dim_infos = tuple([_DimInfo(d, m) for d, m in zip(src_dims, mode_formats)])
+  else:
+    assert isinstance(expr, _BinaryExpr)
+    a_info = expr_to_info[expr.a]
+    index_to_dim_info = {
+        i: d for i, d in zip(a_info.src_indices, a_info.dim_infos)
+    }
+    b_info = expr_to_info[expr.b]
+    _validate_and_collect_dim_info(index_to_dim_info, b_info.src_indices,
+                                   b_info.dim_infos, expr)
+    # Here we rely on the fact that dictionaries keep the insertion order for
+    # keys and values.
+    src_indices = tuple(index_to_dim_info.keys())
+    dim_infos = tuple(index_to_dim_info.values())
+
+  expr_to_info[expr] = _ExprInfo(src_indices, dim_infos)
+
+
+def _mark_structured_op_root(
+    expr: IndexExpr,
+    reduce_index: IndexVar,
+    expr_to_info: _ExprInfoDict,
+) -> None:
+  """Identifies the root expression for a structured op in the linalg dialect.
+
+  An linalg structured op can only perform reduction on the whole expression.
+  For a TACO tensor algebra expression, the reduction on an IndexVar is done at
+  the smallest expression that contains all the uses of the IndexVar. If such an
+  expression is only part of the whole expression, we need to split this
+  sub-expression tree out from its parent and implement the sub-expression as a
+  structured op.
+
+  This routine identifies the root expression node for performing a reduction on
+  the given IndexVar. If the reduction of the given IndexVar should be performed
+  on expression X, then the IndexVar is added to expr_to_info[X].reduce_indices
+
+  Args:
+    expr: The root IndexExpr for the tensor algebra expression.
+    reduce_index: The IndexVar which we want to find out the proper expression
+      to perform a reduction.
+    expr_to_info: The dictionary to look up _ExprInfo for IndexExpr.
+  """
+  assert (isinstance(expr, _BinaryExpr))
+  a_info = expr_to_info[expr.a]
+  b_info = expr_to_info[expr.b]
+  expr_info = expr_to_info[expr]
+
+  if reduce_index in a_info.src_indices and reduce_index in b_info.src_indices:
+    expr_info.reduce_indices.add(reduce_index)
+    return
+
+  if reduce_index in a_info.src_indices:
+    _mark_structured_op_root(expr.a, reduce_index, expr_to_info)
+  elif reduce_index in b_info.src_indices:
+    _mark_structured_op_root(expr.b, reduce_index, expr_to_info)
+  else:
+    assert False, "Unreachable path"
+
+
+def _accumulate_reduce_indices(
+    expr: IndexExpr,
+    expr_to_info: _ExprInfoDict,
+) -> None:
+  """Propagates reduction indices from child expressions to parent expressions.
+
+  This routine is passed to the post-order visitor as an _ExprVisitor object.
+
+  Args:
+    expr: The IndexExpr being visited.
+    expr_to_info: The dictionary of (IndexExpr, _ExprInfo) for recording the
+      expression information.
+  """
+  assert expr in expr_to_info
+  expr_info = expr_to_info[expr]
+
+  if isinstance(expr, _BinaryExpr):
+    a_info = expr_to_info[expr.a]
+    b_info = expr_to_info[expr.b]
+    expr_info.acc_reduce_indices = (
+        a_info.acc_reduce_indices | b_info.acc_reduce_indices
+        | expr_info.reduce_indices)
+  else:
+    assert isinstance(expr, Access)
+
+
+def _gather_structured_op(
+    expr: IndexExpr,
+    expr_to_info: _ExprInfoDict,
+    structop_roots: List[IndexExpr],
+) -> None:
+  """Adds structured op root expression information to structop_roots.
+
+  This routine is passed to the post-order visitor as an _ExprVisitor object.
+
+  Args:
+    expr: The IndexExpr being visited.
+    expr_to_info: The dictionary to look up _ExprInfo for IndexExpr.
+    structop_roots: The resulting list of IndexExpr that are the roots for
+      linalg structured ops.
+  """
+  if not expr_to_info[expr].reduce_indices:
+    return
+
+  # If the expression is the root for reducing some indices, collect the indices
+  # and dimensions for the reduction result.
+  dst_indices = []
+  dst_dims = []
+  mode_fmts = []
+  for i, d in zip(expr_to_info[expr].src_indices, expr_to_info[expr].dim_infos):
+    if i not in expr_to_info[expr].acc_reduce_indices:
+      dst_indices.append(i)
+      dst_dims.append(d.dim)
+      mode_fmts.append(d.mode_format)
+
+  # Add the information to the dictionary.
+  op_info = _StructOpInfo(
+      tuple(dst_indices),
+      tuple(dst_dims),
+      expr.dtype(),
+      f"temp{len(structop_roots)}",
+      _make_format(mode_fmts),
+  )
+  expr_to_info[expr].structop_info = op_info
+
+  # Add the expression to the list of structured op roots.
+  structop_roots.append(expr)
+
+
+def _is_structured_op_leaf(
+    expr: IndexExpr,
+    root: IndexExpr,
+    expr_to_info: _ExprInfoDict,
+    *unused_args,
+) -> bool:
+  """Returns true iff the expression is a leaf node for a structured op.
+
+  The root of a structured op is a leaf of its parent structured op that uses
+  its result. An expression node is a leaf node for the current structured op if
+  it is an Access node or the root for a structured op that is not the current
+  structured op.
+
+  This routine is passed to the post-order visitor as a _SubtreeLeafChecker
+  object. Because the post-order visitor pass the same parameters to both
+  _SubtreeLeafChecker and _ExprVisitor, this routine may received unused
+  parameters.
+
+  Args:
+    expr: The IndexExpr being visited.
+    root: The root of the current structured op.
+    expr_to_info: The dictionary to look up _ExprInfo for IndexExpr.
+
+  Returns:
+    True if the current IndexExpr is a leaf for the current structured op.
+  """
+  return (expr != root and
+          expr_to_info[expr].structop_info is not None) or isinstance(
+              expr, Access)
+
+
+def _gather_structured_op_input(
+    expr: IndexExpr,
+    root: IndexExpr,
+    expr_to_info: _ExprInfoDict,
+    structop_inputs: List[IndexExpr],
+) -> None:
+  """Adds the IndexExpr to structop_inputs if it is an input.
+
+  If the current IndexExpr is an input for the current structured op, adds it to
+  structop_inputs. The current IndexExpr is an input if it is an Access node or
+  if it is the root for a structured op that is not the current structured op.
+
+  This routine is passed to the post-order visitor as an _ExprVisitor object.
+
+  Args:
+    expr: The IndexExpr being visited.
+    root: The root of the current structured op.
+    expr_to_info: The dictionary to look up _ExprInfo for IndexExpr.
+    structop_inputs: The resulting list of IndexExpr that provide input to the
+      current structured op.
+  """
+  if (expr != root and expr not in structop_inputs) and (
+      isinstance(expr, Access) or
+      (expr in expr_to_info and expr_to_info[expr].structop_info)):
+    structop_inputs.append(expr)
+
+
+def _emit_structured_op_input(
+    expr: IndexExpr,
+    expr_to_info: _ExprInfoDict,
+    op_def: lang.LinalgOpDef,
+) -> lang.OperandDef:
+  """Emits OperandDef in the linalg dialect for the input IndexExpr.
+
+  Args:
+    expr: The input IndexExpr for the current structured op.
+    expr_to_info: The dictionary to look up _ExprInfo for IndexExpr.
+    op_def: The linalg operation for the current structured op.
+
+  Returns:
+    An OperandDef in the linalg dialect for the input IndexExpr.
+  """
+  op_info = expr_to_info[expr].structop_info
+  if op_info:
+    # The input is a temporary tensor produced by another structured op.
+    indices = op_info.dst_indices
+    name = op_info.dst_name
+  else:
+    # The input is a user provided tensor.
+    assert isinstance(expr, Access)
+    indices = expr.indices
+    name = expr.tensor.name
+
+  dim_sym = _mlir_symbols_from_index_vars(indices)
+  opnd = lang.OperandDef(lang.OperandKind.InputTensor, lang.T, dim_sym)
+  op_def.add_operand(name, opnd)
+  return opnd

mlir/test/Integration/Dialect/SparseTensor/taco/tools/mlir_pytaco_api.py

+#  Part of the LLVM Project, under the Apache License v2.0 with LLVM Exceptions.
+#  See https://llvm.org/LICENSE.txt for license information.
+#  SPDX-License-Identifier: Apache-2.0 WITH LLVM-exception
+
+"""Supports the PyTACO API with the MLIR-PyTACO implementation.
+
+See http://tensor-compiler.org/ for TACO tensor compiler.
+
+This module exports the MLIR-PyTACO implementation through the language defined
+by PyTACO. In particular, it defines the function and type aliases and constants
+needed for the PyTACO API to support the execution of PyTACO programs using the
+MLIR-PyTACO implementation.
+"""
+
+from . import mlir_pytaco
+from . import mlir_pytaco_io
+
+# Functions defined by PyTACO API.
+get_index_vars = mlir_pytaco.get_index_vars
+from_array = mlir_pytaco.Tensor.from_array
+read = mlir_pytaco_io.read
+write = mlir_pytaco_io.write
+
+# Classes defined by PyTACO API.
+dtype = mlir_pytaco.DType
+mode_format = mlir_pytaco.ModeFormat
+mode_ordering = mlir_pytaco.ModeOrdering
+mode_format_pack = mlir_pytaco.ModeFormatPack
+format = mlir_pytaco.Format
+index_var = mlir_pytaco.IndexVar
+tensor = mlir_pytaco.Tensor
+index_expression = mlir_pytaco.IndexExpr
+access = mlir_pytaco.Access
+
+# Data type constants defined by PyTACO API.
+int16 = mlir_pytaco.DType(mlir_pytaco.Type.INT16)
+int32 = mlir_pytaco.DType(mlir_pytaco.Type.INT32)
+int64 = mlir_pytaco.DType(mlir_pytaco.Type.INT64)
+float32 = mlir_pytaco.DType(mlir_pytaco.Type.FLOAT32)
+float64 = mlir_pytaco.DType(mlir_pytaco.Type.FLOAT64)
+
+# Storage format constants defined by the PyTACO API. In PyTACO, each storage
+# format constant has two aliasing names.
+compressed = mlir_pytaco.ModeFormat.COMPRESSED
+Compressed = mlir_pytaco.ModeFormat.COMPRESSED
+dense = mlir_pytaco.ModeFormat.DENSE
+Dense = mlir_pytaco.ModeFormat.DENSE

mlir/test/Integration/Dialect/SparseTensor/taco/tools/mlir_pytaco_io.py

+#  Part of the LLVM Project, under the Apache License v2.0 with LLVM Exceptions.
+#  See https://llvm.org/LICENSE.txt for license information.
+#  SPDX-License-Identifier: Apache-2.0 WITH LLVM-exception
+
+"""Experimental MLIR-PyTACO with sparse tensor support.
+
+See http://tensor-compiler.org/ for TACO tensor compiler.
+
+This module implements the PyTACO API for writing a tensor to a file or reading
+a tensor from a file.
+
+See the following links for Matrix Market Exchange (.mtx) format and FROSTT
+(.tns) format:
+  https://math.nist.gov/MatrixMarket/formats.html
+  http://frostt.io/tensors/file-formats.html
+"""
+
+from typing import List, TextIO
+
+from . import mlir_pytaco
+
+# Define the type aliases so that we can write the implementation here as if
+# it were part of mlir_pytaco.py.
+Tensor = mlir_pytaco.Tensor
+Format = mlir_pytaco.Format
+DType = mlir_pytaco.DType
+Type = mlir_pytaco.Type
+
+# Constants used in the implementation.
+_MTX_FILENAME_SUFFIX = ".mtx"
+_TNS_FILENAME_SUFFIX = ".tns"
+
+_MTX_HEAD = "%%MatrixMarket"
+_MTX_MATRIX = "matrix"
+_MTX_COORDINATE = "coordinate"
+_MTX_REAL = "real"
+_MTX_SYMMETRY = "symmetric"
+_MTX_GENERAL = "general"
+_SYMMETRY_FIELD_ID = 4
+
+# The TACO supported header for .mtx has the following five fields:
+# . %%MatrixMarket
+# . matrix | tensor
+# . coordinate | array
+# . real
+# . symmetric | general
+#
+# This is what we support currently.
+_SUPPORTED_HEADER_FIELDS = ((_MTX_HEAD,), (_MTX_MATRIX,), (_MTX_COORDINATE,),
+                            (_MTX_REAL,), (_MTX_GENERAL, _MTX_SYMMETRY))
+
+_A_SPACE = " "
+_MTX_COMMENT = "%"
+_TNS_COMMENT = "#"
+
+
+def _coordinate_from_strings(strings: List[str]) -> List[int]:
+  """"Return the coordinate represented by the input strings."""
+  # Coordinates are 1-based in the text file and 0-based in memory.
+  return [int(s) - 1 for s in strings]
+
+
+def _read_coordinate_format(file: TextIO, tensor: Tensor,
+                            is_symmetric: bool) -> None:
+  """Reads tensor values in coordinate format."""
+  rank = tensor.order
+  # Process the data for the tensor.
+  for line in file:
+    if not line:
+      continue
+
+    fields = line.split(_A_SPACE)
+    if rank != len(fields) - 1:
+      raise ValueError("The format and data have mismatched ranks: "
+                       f"{rank} vs {len(fields)-1}.")
+    coordinate = _coordinate_from_strings(fields[:-1])
+    value = float(fields[-1])
+    tensor.insert(coordinate, value)
+    if is_symmetric and coordinate[0] != coordinate[-1]:
+      coordinate.reverse()
+      tensor.insert(coordinate, value)
+
+
+def _read_mtx(file: TextIO, fmt: Format) -> Tensor:
+  """Inputs tensor from a text file with .mtx format."""
+  # The first line should have this five fields:
+  #   head tensor-kind format data-type symmetry
+  fields = file.readline().rstrip("\n").split(_A_SPACE)
+  tuple_to_str = lambda x: "|".join(x)
+  if len(fields) != len(_SUPPORTED_HEADER_FIELDS):
+    raise ValueError(
+        "Expected first line with theses fields "
+        f"{' '.join(map(tuple_to_str, _SUPPORTED_HEADER_FIELDS))}: "
+        f"{' '.join(fields)}")
+
+  for i, values in enumerate(_SUPPORTED_HEADER_FIELDS):
+    if fields[i] not in values:
+      raise ValueError(f"The {i}th field can only be one of these values "
+                       f"{tuple_to_str(values)}: {fields[i]}")
+
+  is_symmetric = (fields[_SYMMETRY_FIELD_ID] == _MTX_SYMMETRY)
+  # Skip leading empty lines or comment lines.
+  line = file.readline()
+  while not line or line[0] == _MTX_COMMENT:
+    line = file.readline()
+
+  # Process the first data line with dimensions and number of non-zero values.
+  fields = line.split(_A_SPACE)
+  rank = fmt.rank()
+  if rank != len(fields) - 1:
+    raise ValueError("The format and data have mismatched ranks: "
+                     f"{rank} vs {len(fields)-1}.")
+  shape = fields[:-1]
+  shape = [int(s) for s in shape]
+  num_non_zero = float(fields[-1])
+
+  # Read the tensor values in coordinate format.
+  tensor = Tensor(shape, fmt)
+  _read_coordinate_format(file, tensor, is_symmetric)
+  return tensor
+
+
+def _read_tns(file: TextIO, fmt: Format) -> Tensor:
+  """Inputs tensor from a text file with .tns format."""
+  rank = fmt.rank()
+  coordinates = []
+  values = []
+  dtype = DType(Type.FLOAT64)
+
+  for line in file:
+    # Skip empty lines and comment lines.
+    if not line or line[0] == _TNS_COMMENT:
+      continue
+
+    # Process each line with a coordinate and the value at the coordinate.
+    fields = line.split(_A_SPACE)
+    if rank != len(fields) - 1:
+      raise ValueError("The format and data have mismatched ranks: "
+                       f"{rank} vs {len(fields)-1}.")
+    coordinates.append(tuple(_coordinate_from_strings(fields[:-1])))
+    values.append(dtype.value(fields[-1]))
+
+  return Tensor.from_coo(coordinates, values, fmt, dtype)
+
+
+def _write_tns(file: TextIO, tensor: Tensor) -> None:
+  """Outputs a tensor to a file using .tns format."""
+  coords, non_zeros = tensor.get_coordinates_and_values()
+  assert len(coords) == len(non_zeros)
+  # Output a coordinate and the corresponding value in a line.
+  for c, v in zip(coords, non_zeros):
+    # The coordinates are 1-based in the text file and 0-based in memory.
+    plus_one_to_str = lambda x: str(x + 1)
+    file.write(f"{' '.join(map(plus_one_to_str,c))} {v}\n")
+
+
+def read(filename: str, fmt: Format) -> Tensor:
+  """Inputs a tensor from a given file.
+
+  The name suffix of the file specifies the format of the input tensor. We
+  currently only support .mtx format for support sparse tensors.
+
+  Args:
+    filename: A string input filename.
+    fmt: The storage format of the tensor.
+
+  Raises:
+    ValueError: If filename doesn't end with .mtx or .tns, or fmt is not an
+    instance of Format or fmt is not a sparse tensor.
+  """
+  if (not isinstance(filename, str) or
+      (not filename.endswith(_MTX_FILENAME_SUFFIX) and
+       not filename.endswith(_TNS_FILENAME_SUFFIX))):
+    raise ValueError("Expected string filename ends with "
+                     f"{_MTX_FILENAME_SUFFIX} or {_TNS_FILENAME_SUFFIX}: "
+                     f"{filename}.")
+  if not isinstance(fmt, Format) or fmt.is_dense():
+    raise ValueError(f"Expected a sparse Format object: {fmt}.")
+
+  with open(filename, "r") as file:
+    return (_read_mtx(file, fmt) if filename.endswith(_MTX_FILENAME_SUFFIX) else
+            _read_tns(file, fmt))
+
+
+def write(filename: str, tensor: Tensor) -> None:
+  """Outputs a tensor to a given file.
+
+  The name suffix of the file specifies the format of the output. We currently
+  only support .tns format.
+
+  Args:
+    filename: A string output filename.
+    tensor: The tensor to output.
+
+  Raises:
+    ValueError: If filename doesn't end with .tns or tensor is not a Tensor.
+  """
+  if (not isinstance(filename, str) or
+      not filename.endswith(_TNS_FILENAME_SUFFIX)):
+    raise ValueError("Expected string filename ends with"
+                     f" {_TNS_FILENAME_SUFFIX}: {filename}.")
+  if not isinstance(tensor, Tensor):
+    raise ValueError(f"Expected a Tensor object: {tensor}.")
+
+  with open(filename, "w") as file:
+    return _write_tns(file, tensor)

mlir/test/Integration/Dialect/SparseTensor/taco/tools/mlir_pytaco_utils.py

+#  Part of the LLVM Project, under the Apache License v2.0 with LLVM Exceptions.
+#  See https://llvm.org/LICENSE.txt for license information.
+#  SPDX-License-Identifier: Apache-2.0 WITH LLVM-exception
+
+# This file contains the utilities to process sparse tensor outputs.
+
+from typing import Tuple
+import ctypes
+import functools
+import numpy as np
+
+
[e-mail]_cache()
+def _get_c_shared_lib(lib_name: str) -> ctypes.CDLL:
+  """Loads and returns the requested C shared library.
+
+  Args:
+    lib_name: A string representing the C shared library.
+
+  Returns:
+    The C shared library.
+
+  Raises:
+    OSError: If there is any problem in loading the shared library.
+    ValueError: If the shared library doesn't contain the needed routines.
+  """
+  # This raises OSError exception if there is any problem in loading the shared
+  # library.
+  c_lib = ctypes.CDLL(lib_name)
+
+  try:
+    c_lib.convertToMLIRSparseTensor.restype = ctypes.c_void_p
+  except Exception as e:
+    raise ValueError("Missing function convertToMLIRSparseTensor from "
+                     f"the supporting C shared library: {e} ") from e
+
+  try:
+    c_lib.convertFromMLIRSparseTensor.restype = ctypes.c_void_p
+  except Exception as e:
+    raise ValueError("Missing function convertFromMLIRSparseTensor from "
+                     f"the C shared library: {e} ") from e
+
+  return c_lib
+
+
+def sparse_tensor_to_coo_tensor(
+    lib_name: str,
+    sparse_tensor: ctypes.c_void_p,
+    dtype: np.dtype,
+) -> Tuple[int, int, np.ndarray, np.ndarray, np.ndarray]:
+  """Converts an MLIR sparse tensor to a COO-flavored format tensor.
+
+  Args:
+     lib_name: A string for the supporting C shared library.
+     sparse_tensor: A ctypes.c_void_p to the MLIR sparse tensor descriptor.
+     dtype: The numpy data type for the tensor elements.
+
+  Returns:
+    A tuple that contains the following values for the COO-flavored format
+    tensor:
+    rank: An integer for the rank of the tensor.
+    nse: An interger for the number of non-zero values in the tensor.
+    shape: A 1D numpy array of integers, for the shape of the tensor.
+    values: A 1D numpy array, for the non-zero values in the tensor.
+    indices: A 2D numpy array of integers, representing the indices for the
+      non-zero values in the tensor.
+
+  Raises:
+    OSError: If there is any problem in loading the shared library.
+    ValueError: If the shared library doesn't contain the needed routines.
+  """
+  c_lib = _get_c_shared_lib(lib_name)
+
+  rank = ctypes.c_ulonglong(0)
+  nse = ctypes.c_ulonglong(0)
+  shape = ctypes.POINTER(ctypes.c_ulonglong)()
+  values = ctypes.POINTER(np.ctypeslib.as_ctypes_type(dtype))()
+  indices = ctypes.POINTER(ctypes.c_ulonglong)()
+  c_lib.convertFromMLIRSparseTensor(sparse_tensor, ctypes.byref(rank),
+                                    ctypes.byref(nse), ctypes.byref(shape),
+                                    ctypes.byref(values), ctypes.byref(indices))
+
+  # Convert the returned values to the corresponding numpy types.
+  shape = np.ctypeslib.as_array(shape, shape=[rank.value])
+  values = np.ctypeslib.as_array(values, shape=[nse.value])
+  indices = np.ctypeslib.as_array(indices, shape=[nse.value, rank.value])
+  return rank, nse, shape, values, indices
+
+
+def coo_tensor_to_sparse_tensor(lib_name: str, np_shape: np.ndarray,
+                                np_values: np.ndarray,
+                                np_indices: np.ndarray) -> int:
+  """Converts a COO-flavored format sparse tensor to an MLIR sparse tensor.
+
+  Args:
+     lib_name: A string for the supporting C shared library.
+     np_shape: A 1D numpy array of integers, for the shape of the tensor.
+     np_values: A 1D numpy array, for the non-zero values in the tensor.
+     np_indices: A 2D numpy array of integers, representing the indices for the
+       non-zero values in the tensor.
+
+  Returns:
+     An integer for the non-null ctypes.c_void_p to the MLIR sparse tensor
+     descriptor.
+
+  Raises:
+    OSError: If there is any problem in loading the shared library.
+    ValueError: If the shared library doesn't contain the needed routines.
+  """
+
+  rank = ctypes.c_ulonglong(len(np_shape))
+  nse = ctypes.c_ulonglong(len(np_values))
+  shape = np_shape.ctypes.data_as(ctypes.POINTER(ctypes.c_ulonglong))
+  values = np_values.ctypes.data_as(
+      ctypes.POINTER(np.ctypeslib.as_ctypes_type(np_values.dtype)))
+  indices = np_indices.ctypes.data_as(ctypes.POINTER(ctypes.c_ulonglong))
+
+  c_lib = _get_c_shared_lib(lib_name)
+  ptr = c_lib.convertToMLIRSparseTensor(rank, nse, shape, values, indices)
+  assert ptr is not None, "Problem with calling convertToMLIRSparseTensor"
+  return ptr