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Shape inference

Shape inference as discussed here is considered a specific instance of type inference for ShapedType. Type constraints are along (at least) three axis: 1) elemental type, 2) rank (including static or dynamic), 3) dimensions. While some operations have no compile time fixed shape (e.g., output shape is dictated by data) we could still have some knowledge of constraints/bounds in the system for that operation (e.g., the output of a tf.where is at most the size of the input data). That is, there are additional valuable constraints that could be captured even without full knowledge of the shape.

Type inference is currently modelled executionally for operation creation using the InferTypeOpInterface, while InferShapedTypeOpInterface is used to implement the shape and element type inference. The return type can often be deduced from the deduced return shape and elemental type (queryable from InferShapedTypeOpInterface) and so type inference for tensor types can be implemented with InferShapedTypeOpInterface.

Shape functions

The C++ interfaces are the base mechanism whereby shape inference is queried and executed, but not the intended way to specify shape constraints in general.

Initially the shape inference will be declaratively specified using:

Constraints on the operands of an operation directly. For example constraining the input type to be tensor/vector elements or that the elemental type be of a specific type (e.g., output of computing the size of a value is of elemental type i1) or class (e.g., float-like).
Constraints across operands and results of an operation.
- For example, specifying equality constraints on type/constituents of a type (shape and elemental type) between operands and results (e.g., the output type of an add is the same as those of the input operands).

NOTE: The C++ shape functions are an intermediate step until the shape dialect is more full-fledged, at which point the C++ functions should become the exceptional case.

Testing

Shape inference is currently tested alongside type inference by TestReturnTypeDriver in the test dialect. This driver performs two checks:

Verification that the return types specified matches the inferred types. This explicit check will be removed and made part of Op verification instead.
Test the creation of Ops without specifying the return type explicitly in function testCreateFunctions by creating new binary Ops (Op classes specified in TestReturnTypeDriver) using 1) all operands to testCreateFunctions as both operands, and 2) using combinations of input operands of the function.

Shape dialect

This section details the shape type inference dialect (shape). The initial focus will be on shape functions that describe shape functions could be used in runtime and compiler (for constructions of ops/refinement of shapes, reification of dynamic allocations for dialect including TF, TFLite, XLA & tensor compute dialect under discussion).

This will focus on the shape functions (e.g., determine the rank and dimensions of the output shape). As shown in the shaped container type, shape will be one of 3 components, the others being elemental type and attribute (which is currently left open with the intention of supporting extensions such as layouts or bounded shapes at a later point). This allows for decoupling of these:

Not all the information is needed for all analysis;
Not all shape functions need to provide all the information (e.g., one could define a base class function that only populates element type but composes with the others);
It allows reusing the constraints between, say, Tensor and Memref representation of an operation;

An argument could be made that these are metadata function instead of shape functions, with some considering shape and elemental types different and some considering them both as part of shape. But shape function is IMHO descriptive and metadata can span too large a range of potential uses/values.

Requirements

The requirements for the shape inference functions are determined by the requirements of shape inference, but we believe the requirements below still allow freedom to consider different shape inference approaches and so we do not impose a particular shape inference approach here.

Shape inference functions

Expressiveness shape functions need to support programs where tensors have shapes that are not known statically (for example, tensor<16x?xf32> or tensor<*xf32>*);
Shape error detection Many operations will have constraints on their operands. If the constraints are not satisfied or cannot be determined if satisfied statically, then a runtime check/assertion could be generated.
- This also aligns with the requirement that the shape function description should be usable by both the compiler and runtime.
- Shape error functions should be easy to understand, at least what constraint of the operation is violated. This also requires that shape function error messages should be configurable by the author of the shape function (e.g., the author would be able to give the semantic constraint invalidated rather the low-level check that failed).
- The static analysis may be used to eliminate run-time checks that are guaranteed to pass.
  - Ideally all would eventually (see section Inlining shape checking) be elided.
- Only reporting errors which are guaranteed to occur at runtime. If an error is only possible (rather than guaranteed) then we use a runtime assertion to fail and produce an error message with the invariant violated.
Shape functions usable by compiler and runtime.
- This does not mean the exact same C++ function, but rather the description should be consumable by either.
- Shape function description should not be constrained by either runtime or compiler's type system to handle types only used for analysis. That is, these two type systems differ and both should be supported, but the intersection of the two should not be required. As a particular example, if a compiler only wants to differentiate exact shapes vs dynamic shapes, then it need not consider a more generic shape latice even though the shape description supports it.
Declarative (e.g., analyzable at compile time, possible to generate different versions for different use cases)
- This may not strictly be a requirement, but a way to handle the former: a declarative specification could be reused by both while avoiding a need to map to or from a 3rd representation given these two systems have/and will have different types.
Shape inference functions are expressible at runtime
- User can define a shape function for a new operation dynamically at runtime, this allows for vendors to describe an operation and shape function dynamically.
  
  This requirement is on the wishlist.
Doesn't require graph-wide shape information (e.g., only require local information)
- Shape functions should be cheap to invoke on each kernel launch.
- Shape function can be dictated by arguments (operands, attributes and regions) only (e.g., same operands as the corresponding operation could be constructed & invoked with).
- Shape information that needs higher-level/graph information should use richer types (e.g., TensorList<F32>);
- The function should be invocable before/while constructing an op (e.g., can't rely on the op being constructed).
Shape functions should be pure functions.
Should support functions whose type is only known dynamically (e.g., read_from_file op)
- Without needing to invoke the op (e.g., reading a file once for determining the shape & then post to be able to actually consume the output of the file).
The shape function operation dialect should be interoperable with non-shape function dialect operations.
- There may be a common set of operations that satisfy most uses (e.g., merge, equal_type, arithmetic expressions, slice, concat, pattern matching on attributes such as padding etc.) that will be discovered and could cover a large percentage of the use cases. Among these there will be some which carry extra semantic info that could be used for symbolic constraints (e.g., checking equality of two dimensions resulting in setting an equality constraint) and higher-order interpretation for constraint solving.
  
  It is therefore beneficial (but not required) to reuse operations, especially as for statically known shapes, arbitrary arithmetic computations could still be performed. This means that the computations performed statically may or may not be supported by an arbitrary solver, but would still be allowed.
The shape function should be expandable such that symbolic equality and upper bound constraints (say) could be represented and may be propagated by shape inference.
- E.g., the shape functions may contain more information that is only useful when used from shape inference;
Shape functions are allowed to fail and report an error. The error reporting should report the location of the operation that failed with, where possible, a user actionable error message.
- These failures could become inlined and become runtime failures with runtime values and error messages.
- Reporting errors should be optional. E.g., The same function may be used as to query validity without reporting an error.

Non-goals

The shape dialect is an IR representations and not a programming language;
- While the functions should be readable, it doesn't carry the conveniences of a programming language. Deciding how people write these things, e.g. a mini dsl, a C++ API that generates them, extracting them programmatically from SetShapeFn calls, etc., is still TBD.
Describe the shape inference approach that will use the shape functions;
- The goal is that the shape functions and the constraints one could obtain from them are general enough that they would be useful for various analysis. But whether we follow very simple (e.g., only fully static information is used for shape output, unranked for everything else) to very advance (e.g., expression trees of symbolic constants) can be evaluated independently of this proposal and with concrete benefit analysis.
Describe the approach whereby error messages will be generated;
- While the shape functions will be able to emit errors optionally, it will be possible to dictate when they emit an error. This enables deciding whether or which error to emit: there have been proposals in the literature that the iteration order for shape inference affect the quality of the error message produced, and the shape functions do not mandate that.
Flow sensitive shape functions;
- To enable scalable/cheap shape inference, the shape functions do not intend to provide flow sensitive information. This facility could potentially be built as part of shome higher order analysis that reuse the shape functions/constraints due to the shape functions.
All static functions are usable for dynamic/unknown shapes;
- More involved computations can be performed with statically known shapes than what can be sensibly analyzed with unknown/symbolic variables.

Discussion

Inline shape inference checks

Shape functions should be lowerable to runtime checks for validity. E.g. verify as much as possible statically, but enable generating instructions to compute the shape dynamically and or falling back to runtime checks for attributes not verifiable at compile time. These checks inserted should ideally only check that which could not have been verified statically.

These inlined calls could interfere with optimization patterns/passes (e.g., shape inference should not insert constructs that interfere with optimization patterns) and so could be delayed until later (with another round of optimizations, constant folding, CSE, etc., that should remove redundant runtime operations).

Possibly Asked Questions

What about ODS specifications of operations?

In ODS we have been recording the constraints for the operands & attributes of an operation. Where these are sufficient to constrain the output shape (e.g., SameOperandAndResultType or broadcastable) we should generate the shape function from those. Where not, an explicit shape function should be specified (spelling TBD but currently considering using the MLIR textual form as serialization approach).

Why not extract the shape function from reference implementation?

This could be done in future! The extracted shape function would use the shape inference dialect, so we are starting there. Especially for operations described in a structured way, one could autogenerate the shape function.

How/in what language will the shape functions be authored?

TBD. open to many approaches and suggestions, starting on the IR produced by whatever language is the priority of this proposal.

What shape inference approach is being suggested here?

None. There are multiple different shape inference approaches that we could layer on top of these. From the most basic (always return unranked), to more useful (return fixed shape for constant inputs/arguments) to the more advanced (create logical conjuctions of algebraic statements between symbolic named values).

Open points

Should shape functions that produce dynamic outputs given all statically shaped inputs be marked specially? E.g., read from file.

TODO: Add examples here.

WIP/Future considerations

Shape functions are determined by attributes and could be arbitrarily complicated with a wide-range of specification possibilities. Equality relationships are common (e.g., the elemental type of the output matches the primitive type of the inputs, both inputs have exactly the same type [primitive type and shape]) and so these should be easy to specify. Algebraic relationships would also be common (e.g., a concat of [n,m] and [n,m] matrix along axis 0 is [n+n, m] matrix), while some ops only have defined shapes under certain cases (e.g., matrix multiplication of [a,b] and [c,d] is only defined if b == c).

Instead of specifying an additional mechanism to specify a shape transfer function, the reference implementation of the operation will be used to derive the shape function. The reference implementation is general and can support the arbitrary computations needed to specify output shapes.

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