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Layout Algebra

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Layout Algebra

CuTe provides an algebra of layouts — operations that combine layouts in structured ways. In the CuTe IR dialect, every operation in this algebra is exposed as an op:

  • Composition — functional composition of layouts.
  • Coalesce — simplify a layout without changing its semantics.
  • Complement — produce a "filler" layout that is disjoint from a given one.
  • Inverse (left and right) — invert a layout's coordinate mapping.
  • Slice and Dice — split a layout by an underscore pattern.
  • Products — replicate a layout according to another layout (6 variants).
  • Divides — split a layout according to another (4 variants).
  • Tile to shape — replicate a block layout to cover a target shape (cute.tile_to_shape).

Tiling and partitioning utilities are frequently used in kernels that build on these algebraic operations.

Throughout these documents we use the same notation as the source operations: A ∘ B for composition, A ⊗ B for product, and A ⊘ B for divide. See :doc:01_cute_types for the layout type itself.

This chapter is split into two sections:

.. toctree:: :maxdepth: 1 :hidden:

02_layout_algebra/core_ops 02_layout_algebra/products_and_divides

  • :doc:Core layout algebra ops <02_layout_algebra/core_ops> — coalesce, composition, complement, inverses, slice / dice.
  • :doc:Products and divides <02_layout_algebra/products_and_divides> — the replication and splitting families that drive CuTe's tiling and partitioning.