Design Decisions in Pytato


  • reduction inames

  • finish trawling the design doc

  • expression nodes in index lambda
    • what pymbolic expression nodes are OK

    • reductions

    • function identifier scoping

    • piecewise def (use ISL?)


When this document refers to different ways of expressing a computation and transforming between them “without loss of information”, what is meant is that the transformation is valid

  • in both directions, and

  • for all possible inputs (including those with symbolic shapes).

Computation and Results

  • Results of computations either implement the Array interface or are a DictOfNamedArrays. The former are referred to as array expressions. The union type of both is referred to as an array result.

  • Array data is computed lazily, i.e., a representation of the desired computation is built, but computation/code generation is not carried out until instructed by the user. Evaluation/computation is never triggered implicitly.

  • Array.dtype is evaluated eagerly.

  • Array.shape is evaluated as eagerly as possible, however data-dependent name references in shapes are allowed. (This implies that the number of array axes must be statically known.)

    Consider the example of fancy indexing:

    A[A > 0]

    Here, the length of the resulting array depends on the data contained in A and cannot be statically determined at code generation time.

    In the case of data-dependent shapes, the shape is expressed in terms of scalar (i.e. having a Array.shape of ()) values with an integral Array.dtype (i.e. having dtype.kind == "i"). Such an expression marks the boundary between eager and lazy evaluation.

  • Array.shape is required to be an affine expression in terms of the instances of SizeParam in the computation graph. This permits shape inference to use Presburger arithmetic, meaning that shape inference is always decidable.

  • There is (deliberate) overlap in what various expression nodes can express, e.g.

    Expression capture (the “frontend”) should use the “highest-level” (most abstract) node type available that captures the user-intended operation. Lowering transformations (e.g. during code generation) may then convert these operations to a less abstract, more uniform representation.

    Operations that introduce nontrivial mappings on indices (e.g. reshape, strided slice, roll) are identified as potential candidates for being captured in their own high-level node vs. as an IndexLambda.

    Operations that can be expressed as IndexLambda without loss of information, should be expressed that way.

  • Every Array instance can be viewed a computation graph, where the Array instances form the nodes of the graph and there is an edge between a node and the array it uses. Since Array is an immutable type, the computation computation graphs would belong to the class of Directed Acyclic Graphs (DAGs). We choose the direction of the edges in the DAG to resemble the one typically seen in a data-flow graph, i.e. the successors of a node are its users and the predecessors of a node are the arrays that it uses.

    • Borrowing the notation from LLVM, we often refer to the direct successors of a node by users.


  • Input arrays, i.e. instances of InputArgumentBase, take Optional[str] as their names. If the name has not been provided, pytato assigns unique names to those arrays during lowering to a target IR.

  • No two non-identical array variables referenced in an expression may have the same name. pytato will detect such uses and raise an error. Here, “identical” is meant in the same-object a is b sense.

  • The (array) value associated with a name is immutable once evaluated. In-place slice assignment may be simulated by returning a new node realizing a “partial replacement”.

  • For arrays with data-dependent shapes, such as fancy indexing:

    A[A > 0]

    it may be necessary to automatically generate names, in this case to describe the shape of the index array used to realize the access A[A>0]. These will be drawn from the reserved namespace _pt_shp. Users may control the naming of these counts by assigning the tag pytato.tags.CountNamed, like so:

    A[(A > 0).tagged(CountNamed("mycount"))]
  • pytato.array.Placeholder expressions, like all array expressions, are considered read-only. When computation begins, the same actual memory may be supplied for multiple placeholder names, i.e. those arrays may alias.


    This does not preclude the arrays being declared with C’s *restrict qualifier in generated code, as they do not alias any data that is being modified.

Reserved Identifiers

  • Identifiers beginning with _pt_ are reserved for internal use by pytato. Any such internal use must be drawn from one of the following sub-regions, identified by their identifier prefixes:

    • _pt_shp: Used to automatically generate identifiers used in data-dependent shapes.

    • _pt_out: The default name of an unnamed output argument

    • _pt_in: The default name of an unnamed placeholder argument

    • _pt_data: Used to automatically generate identifiers for names of DataWrapper arguments that are not supplied by the user.

    • _pt_dist: Used to automatically generate identifiers for placeholders at distributed communication boundaries.

  • Identifiers used in index lambdas are also reserved. These include:

    • Identifiers matching the regular expression _[0-9]+. They are used as index (“iname”) placeholders.

    • Identifiers matching the regular expression _r[0-9]+. They are used as reduction indices.

    • Identifiers matching the regular expression _in[0-9]+. They are used as automatically generated names (if required) in pytato.array.IndexLambda.bindings.


In order to convey information about the computation from DAG construction time to processing/transformation/code generation time, each pytato.Array node may be tagged (via the pytato.Array.tags attribute) with an arbitrary number of informational “tags”. A tag is any subclass of pytools.tag.Tag. Guidelines for tag use:

  • Tags must not carry semantic information; i.e. a computation must have the same result even if all tags are stripped.

  • Tags may carry information related to efficient execution, i.e. it is permissible that evaluation of the expression is inefficient (even impractically so) without taking the information in the tags into account.

  • Tags should be descriptive, not prescriptive.

    For example:

    • Good: This array is the result of differentiation.

    • Bad: Unroll the loops in the code computing this result.

Metadata Propagation

Metadata (i.e. tags) is expected to be varied across application domains, not just in its information content, but also in the rules which might govern the way in which it propagates across the DAG. As an example, consider a Finite Element Method solver that uses pytato-arrays to store the DOFs.

  • A viable use of the tagging system in the solver could be to tag an array u to describe the first axis of an array denotes the mesh’s element indices. In this case, after performing an operation u_squared = u * u is semantically valid to tag u_squared also with the tag describing that it’s first denotes the mesh’s element indices.

  • Another viable use of the tagging system by the solver could be to encode the physical quantity that the array is storing. If u is an array tagged with metadata to describe that it represents the fluid’s velocity. Then, the operation u_squared = u * u should not propagate that u_squared is also an array storing the fluid’s velocity.

As a result, we choose to not propagate metadata (either upon construction, or via a dedicated, universally applied pass) and instead supply tools (such as Mapper) to permit users to implement their own tag/metadata propagation rules.

Memory layout

pytato arrays do not have a defined memory layout. Any operation in numpy that relies on memory layout information to do its job is undefined in pytato. At the most basic level, the attribute numpy.ndarray.strides is not available on subclasses of pytato.Array.

Dataclasses / attrs

dataclasses helps us reduce most of the boilerplate involved in instantiating a new type. However, dataclasses does not support keyword-only argument until Python-3.10. To overcome this, we prefer attrs which gives us all the required functionality of dataclasses and works with Python-3.8.

We have checks in place to avoid developer errors that could happen by using the defaults of these libraries. For eg. both dataclasses and attrs override the implementation of __eq__ for the class being implemented, which could potentially lead lead to an exponential complex operation.

Lessons learned

Namespace object

In pytato’s early days, there used to exist a Namespace type to define a namespace for all input names within an array expression. This was however removed in the later versions. As, in the process of associating names to array variables it would privately hold references to InputArgumentBase variables that could no longer be referenced by a user. This made it impossible for the garbage collector to deallocate large DataWrapper’s, unless the namespace itself went out-of-scope.


array expression

An object implementing the Array interface

array result

An array expression or an instance of DictOfNamedArrays.


Any string for which str.isidentifier() returns True. See also Reserved Identifiers.

placeholder name