Understanding Jaxprs
====================

Updated: May 3, 2020 (for commit f1a46fe).

Conceptually, one can think of JAX transformations as first tracing the Python
function to be transformed into a small and well-behaved intermediate form,
the jaxpr, that is then transformed accordingly, and ultimately compiled and executed.
One of the reasons JAX can pack so much power into such a small software package
is that it starts with a familiar and flexible programming interface (Python with NumPy)
and it uses the actual Python interpreter to do most of the heavy lifting to distill the
essence of the computation into a simple statically-typed expression language
with limited higher-order features: the jaxpr language.

Not all Python programs can be processed this way, but it turns out that many
scientific computing and machine learning programs do have this property.

Before we proceed, it is important to point out that not all JAX transformations
materialize a jaxpr as described above; some, e.g., differentiation,
will apply transformations incrementally during tracing.
Nevertheless, if one wants to understand how JAX works internally, or to
make use of the result of JAX tracing, it is useful to understand jaxpr.

A jaxpr instance represents a function with one of more typed parameters (input variables)
and one or more typed results. The results depend only on the input
variables; there are no free variables captured from enclosing scopes.
The inputs and outputs have types, which in JAX are represented as abstract
values. There are two related representations in the code for jaxprs. The main
one is :py:class:`jax.core.TypedJaxpr` and is what you obtain when you
use :py:func:`jax.make_jaxpr` to inspect jaxprs. It has the following
fields:

  * ``jaxpr``: is the actual computation content of the actual function (described below).
  * ``literals`` is a list of constants. For various reasons, during tracing JAX
    will collect the non-scalar constants that arise and will replace them with
    variables, e.g., constants that appear in the Python program, or the result of
    constant folding such constants. The variables that stand for these constants
    are mentioned separately in the enclosed ``jaxpr``.
    When applying a ``TypedJaxpr`` to some actual
    arguments, one must pass first the ``literals`` followed by the actual arguments.
  * ``in_avals`` and ``out_avals`` are the types of the input variables
    (excluding the ones that correspond to the ``literals``), and of the output values.
    These types are called in JAX abstract values, e.g., ``ShapedArray(float32[10,10])``.

The most interesting part of the TypedJaxpr is the actual execution content,
represented as a :py:class:`jax.core.Jaxpr` as printed using the following
grammar::

   jaxpr ::= { lambda Var* ; Var+.
               let Eqn*
               in  [Expr+] }

where:
  * The parameter of the jaxpr are shown as two lists of variables separated by
    ``;``. The first set of variables are the ones that have been introduced
    to stand for constants that have been hoisted out. These are called the
    `constvars`. The second list of variables are the real input variables.
  * ``Eqn*`` is a list of equations, defining intermediate variables referring to
    intermediate expressions. Each equation defines one or more variables as the
    result of applying a primitive on some atomic expressions. Each equation uses only
    input variables and intermediate variables defined by previous equations.
  * ``Expr+``: is a list of output atomic expressions for the jaxpr.

Equations are printed as follows::

  Eqn  ::= let Var+ = Primitive [ Param* ] Expr+

where:
  * ``Var+”`` are one or more intermediate variables to be defined as the
    output of a primitive invocation (some primitives can return multiple values)
  * ``Expr+`` are one or more atomic expressions, each either a variable or a
    literal constant. A special form of an atomic expression is the `unit`
    expression, printed as ``*`` and standing for a value that is not needed
    in the rest of the computation and has been elided.
  * ``Param*`` are zero or more named parameters to the primitive, printed in
    square brackets. Each parameter is shown as ``Name = Value``.


Most jaxpr primitives are first-order (they take just one or more Expr as arguments)::

  Primitive := add | sub | sin | mul | ...


The jaxpr primitives are documented in the :py:mod:`jax.lax` module.

For example, here is the jaxpr produced for the function ``func1`` below

>>> from jax import make_jaxpr
>>> import jax.numpy as jnp
>>> def func1(first, second):
...    temp = first + jnp.sin(second) * 3.
...    return jnp.sum(temp)
...
>>> print(make_jaxpr(func1)(jnp.zeros(8), jnp.ones(8)))
{ lambda  ; a b.
  let c = sin b
      d = mul c 3.0
      e = add a d
      f = reduce_sum[ axes=(0,) ] e
  in (f,) }

Here there are no constvars, ``a`` and ``b`` are the input variables
and they correspond respectively to
``first`` and ``second`` function parameters. The scalar literal ``3.0`` is kept
inline.
The ``reduce_sum`` primitive has named parameters ``axes`` and ``input_shape``, in
addition to the operand ``e``.

Note that JAX traces through Python-level control-flow and higher-order functions
when it extracts the jaxpr. This means that just because a Python program contains
functions and control-flow, the resulting jaxpr does not have
to contain control-flow or higher-order features.
For example, when tracing the function ``func3`` JAX will inline the call to
``inner`` and the conditional ``if second.shape[0] > 4``, and will produce the same
jaxpr as before

>>> def func2(inner, first, second):
...   temp = first + inner(second) * 3.
...   return jnp.sum(temp)
...
>>> def inner(second):
...   if second.shape[0] > 4:
...     return jnp.sin(second)
...   else:
...     assert False
...
>>> def func3(first, second):
...   return func2(inner, first, second)
...
>>> print(make_jaxpr(func3)(jnp.zeros(8), jnp.ones(8)))
{ lambda  ; a b.
  let c = sin b
      d = mul c 3.0
      e = add a d
      f = reduce_sum[ axes=(0,) ] e
  in (f,) }


Handling PyTrees
----------------

In jaxpr there are no tuple types; instead primitives take multiple inputs
and produce multiple outputs. When processing a function that has structured
inputs or outputs, JAX will flatten those and in jaxpr they will appear as lists
of inputs and outputs. For more details, please see the documentation for
PyTrees (:doc:`notebooks/JAX_pytrees`).

For example, the following code produces an identical jaxpr to what we saw
before (with two input vars, one for each element of the input tuple)


>>> def func4(arg):  # Arg is a pair
...   temp = arg[0] + jnp.sin(arg[1]) * 3.
...   return jnp.sum(temp)
...
>>> print(make_jaxpr(func4)((jnp.zeros(8), jnp.ones(8))))
{ lambda  ; a b.
  let c = sin b
      d = mul c 3.0
      e = add a d
      f = reduce_sum[ axes=(0,) ] e
  in (f,) }



Constant Vars
--------------

ConstVars arise when the computation contains array constants, either
from the Python program, or from constant-folding. For example, the function
``func6`` below

>>> def func5(first, second):
...   temp = first + jnp.sin(second) * 3. - jnp.ones(8)
...   return temp
...
>>> def func6(first):
...   return func5(first, jnp.ones(8))
...

JAX produces the following jaxpr

>>> print(make_jaxpr(func6)(jnp.ones(8)))
{ lambda b d ; a.
  let c = add a b
      e = sub c d
  in (e,) }

When tracing ``func6``, the function ``func5`` is invoked with a constant value
(``np.ones(8)``) for the second argument. As a result, the sub-expression
``jnp.sin(second) * 3.`` is constant-folded.
There are two ConstVars, ``b`` (standing for ``jnp.sin(second) * 3.``) and ``d``
(standing for ``jnp.ones(8)``). Unfortunately, it is not easy to tell from the
jaxpr notation what constants the constant variables stand for.

Higher-order primitives
-----------------------

jaxpr includes several higher-order primitives. They are more complicated because
they include sub-jaxprs.

Conditionals
^^^^^^^^^^^^

JAX traces through normal Python conditionals. To capture a
conditional expression for dynamic execution, one must use the
:py:func:`jax.lax.switch` and :py:func:`jax.lax.cond` constructors,
which have the signatures::

  lax.switch(index: int, branches: Sequence[A -> B], operand: A) -> B

  lax.cond(pred: bool, true_body: A -> B, false_body: A -> B, operand: A) -> B

Both of these will bind a primitive called ``cond`` internally. The
``cond`` primitive in jaxprs reflects the more general signature of
:py:func:`lax.switch`: it takes an integer denoting the index of the branch
to execute (clamped into valid indexing range).

For example:

>>> from jax import lax
>>>
>>> def one_of_three(index, arg):
...   return lax.switch(index, [lambda x: x + 1.,
...                             lambda x: x - 2.,
...                             lambda x: x + 3.],
...                     arg)
...
>>> print(make_jaxpr(one_of_three)(1, 5.))
{ lambda  ; a b.
  let c = clamp 0 a 2
      d = cond[ branches=( { lambda  ; a.
                             let b = add a 1.0
                             in (b,) }
                           { lambda  ; a.
                             let b = sub a 2.0
                             in (b,) }
                           { lambda  ; a.
                             let b = add a 3.0
                             in (b,) } )
                linear=(False,) ] c b
  in (d,) }

The cond primitive has a number of parameters:

  * `branches` are jaxprs that correspond to the branch
    functionals. In this example, those functionals each take one
    input variable, corresponding to ``x``.
  * `linear` is a tuple of booleans that is used internally by the
    auto-differentiation machinery to encode which of the input
    parameters are used linearly in the conditional.

The above instance of the cond primitive takes two operands.  The first
one (``c``) is the branch index, then ``b`` is the operand (``arg``) to
be passed to whichever jaxpr in ``branches`` is selected by the branch
index.

Another example, using :py:func:`lax.cond`:

>>> from jax import lax
>>>
>>> def func7(arg):
...   return lax.cond(arg >= 0.,
...                   lambda xtrue: xtrue + 3.,
...                   lambda xfalse: xfalse - 3.,
...                   arg)
...
>>> print(make_jaxpr(func7)(5.))
{ lambda  ; a.
  let b = ge a 0.0
      c = convert_element_type[ new_dtype=int32
                                old_dtype=bool ] b
      d = cond[ branches=( { lambda  ; a.
                             let b = sub a 3.0
                             in (b,) }
                           { lambda  ; a.
                             let b = add a 3.0
                             in (b,) } )
                linear=(False,) ] c a
  in (d,) }


In this case, the boolean predicate is converted to an integer index
(0 or 1), and ``branches`` are jaxprs that correspond to the false and
true branch functionals, in that order. Again, each functional takes
one input variable, corresponding to ``xtrue`` and ``xfalse``
respectively.

The following example shows a more complicated situation when the input
to the branch functionals is a tuple, and the `false` branch functional
contains a constant ``jnp.ones(1)`` that is hoisted as a `constvar`

>>> def func8(arg1, arg2):  # arg2 is a pair
...   return lax.cond(arg1 >= 0.,
...                   lambda xtrue: xtrue[0],
...                   lambda xfalse: jnp.ones(1) + xfalse[1],
...                   arg2)
...
>>> print(make_jaxpr(func8)(5., (jnp.zeros(1), 2.)))
{ lambda f ; a b c.
  let d = ge a 0.0
      e = convert_element_type[ new_dtype=int32
                                old_dtype=bool ] d
      g = cond[ branches=( { lambda  ; c a b.
                             let d = add c b
                             in (d,) }
                           { lambda  ; e_ a b.
                             let 
                             in (a,) } )
                linear=(False, False, False) ] e f b c
  in (g,) }

The top-level jaxpr has one `constvar` ``f`` (corresponding to
``jnp.ones(1)`` from the body of the first (false) branch) and three
input variables ``a b c`` (corresponding to ``arg1`` and the two
elements of ``arg2``; note that ``arg2`` has been flattened).  The
``false_jaxpr`` has three input variables (``c`` corresponding to the
constant for ``jnp.ones(1)``, and ``a b`` for the two elements of
``arg2`` that are passed to ``false_jaxpr``).  The ``true_jaxpr`` has
three input variables. The first (``e_``) is an unused argument
matching the constant first argument ``c`` of ``false_jaxpr``
(required for the jaxpr signatures to match). The subsequent two
correspond to the two elements of ``arg2`` that is passed to
``true_jaxpr``.

The actual operands to the cond primitive are: ``e f b c``, which
correspond in order to:

  * one operand for the predicate,
  * one constant (only used by ``false_jaxpr``, but passed to both),
    i.e., ``f``, which is a constvar for the top-level jaxpr
  * two operands passed to both jaxprs, i.e., ``b`` and ``c``, which are
    input vars, corresponding to ``arg2`` for the top-level jaxpr.

While
^^^^^

Just like for conditionals, Python loops are inlined during tracing.
If you want to capture a loop for dynamic execution, you must use one of several
special operations, :py:func:`jax.lax.while_loop` (a primitive)
and :py:func:`jax.lax.fori_loop`
(a helper that generates a while_loop primitive)::

    lax.while_loop(cond_fun: (C -> bool), body_fun: (C -> C), init: C) -> C
    lax.fori_loop(start: int, end: int, body: (int -> C -> C), init: C) -> C


In the above signature, “C” stands for the type of a the loop “carry” value.
For example, here is an example fori loop

>>> import numpy as np
>>>
>>> def func10(arg, n):
...   ones = jnp.ones(arg.shape)  # A constant
...   return lax.fori_loop(0, n,
...                        lambda i, carry: carry + ones * 3. + arg,
...                        arg + ones)
...
>>> print(make_jaxpr(func10)(np.ones(16), 5))
{ lambda c d ; a b.
  let e = add a d
      _ _ f = while[ body_jaxpr={ lambda  ; e g a b c.
                                  let d = add a 1
                                      f = add c e
                                      h = add f g
                                  in (d, b, h) }
                     body_nconsts=2
                     cond_jaxpr={ lambda  ; a b c.
                                  let d = lt a b
                                  in (d,) }
                     cond_nconsts=0 ] c a 0 b e
  in (f,) }

The top-level jaxpr has two constvars: ``c`` (corresponding to ``ones * 3.`` from the body
of the loop) and ``d`` (corresponding to the use of ``ones`` in the initial carry).
There are also two input variables (``a`` corresponding to ``arg`` and ``b`` corresponding
to ``n``).
The loop carry consists of three values, as seen in the body of ``cond_jaxpr``
(corresponding to the iteration index, iteration end, and the accumulated value carry).
Note that ``body_jaxpr`` takes 5 input variables. The first two are actually
constvars: ``e`` corresponding to ``ones * 3`` and ``g`` corresponding to the
captures use of ``arg`` in the loop body.
The parameter ``body_nconsts = 2`` specifies that there are 2 constants for the
``body_jaxpr``.
The other 3 input variables for ``body_jaxpr`` correspond to the flattened carry values.

The while primitive takes 5 arguments: ``c a 0 b e``, as follows:

  * 0 constants for ``cond_jaxpr`` (since ``cond_nconsts`` is 0)
  * 2 constants for ``body_jaxpr`` (``c``, and ``a``)
  * 3 parameters for the initial value of carry

Scan
^^^^

JAX supports a special form of loop over the elements of an array (with
statically known shape). The fact that there are a fixed number of iterations
makes this form of looping easily reverse-differentiable. Such loops are constructed
with the :py:func:`jax.lax.scan` operator::

  lax.scan(body_fun: (C -> A -> (C, B)), init_carry: C, in_arr: Array[A]) -> (C, Array[B])

Here ``C`` is the type of the scan carry, ``A`` is the element type of the input array(s),
and ``B`` is the element type of the output array(s).

For the example consider the function ``func11`` below

>>> def func11(arr, extra):
...   ones = jnp.ones(arr.shape)  #  A constant
...   def body(carry, aelems):
...     # carry: running dot-product of the two arrays
...     # aelems: a pair with corresponding elements from the two arrays
...     ae1, ae2 = aelems
...     return (carry + ae1 * ae2 + extra, carry)
...   return lax.scan(body, 0., (arr, ones))
...
>>> print(make_jaxpr(func11)(np.ones(16), 5.))
{ lambda c ; a b.
  let d e = scan[ jaxpr={ lambda  ; f a b c.
                          let d = mul b c
                              e = add a d
                              g = add e f
                          in (g, a) }
                  length=16
                  linear=(False, False, False, False)
                  num_carry=1
                  num_consts=1
                  reverse=False
                  unroll=1 ] b 0.0 a c
  in (d, e) }

The top-level jaxpr has one constvar ``c`` corresponding to the ``ones`` constant,
and two input variables corresponding to the arguments ``arr`` and ``extra``.
The body of the scan has 4 input variables, of which:

  * one (``f``) is a constant (since ``num_consts = 1``), and stands for the
    captured variable ``extra`` used in the loop body,
  * one (``a``) is the value of the carry (since ``num_carry = 1``)
  * The remaining 2 are the input values. ``b`` is the array element from the
    first array passed to lax.scan (``arr``) and ``c`` is the second array
    (``ones``).

The ``linear`` parameter describes for each of the input variables whether they
are guaranteed to be used linearly in the body. Once the scan goes through
linearization, more arguments will be linear.

The scan primitive takes 4 arguments: ``b 0.0 a c``, of which:

  * one is the free variable for the body
  * one is the initial value of the carry
  * The next 2 are the arrays over which the scan operates.

XLA_call
^^^^^^^^

The call primitive arises from JIT compilation, and it encapsulates
a sub-jaxpr along with parameters the specify the backend and the device the
computation should run. For example

>>> from jax import jit
>>>
>>> def func12(arg):
...   @jit
...   def inner(x):
...     return x + arg * jnp.ones(1)  # Include a constant in the inner function
...   return arg + inner(arg - 2.)
...
>>> print(make_jaxpr(func12)(1.))
{ lambda b ; a.
  let c = sub a 2.0
      d = xla_call[ backend=None
                    call_jaxpr={ lambda  ; c b a.
                                 let d = mul b c
                                     e = add a d
                                 in (e,) }
                    device=None
                    donated_invars=(False, False, False)
                    name=inner ] b a c
      e = add a d
  in (e,) }

The top-level constvar ``b`` refers to the ``jnp.ones(1)`` constant, and
the top-level input variable `a` refers to the ``arg`` parameter of ``func12``.
The ``xla_call`` primitive stands for a call to the jitted ``inner`` function.
The primitive has the function body in the ``call_jaxpr`` parameter, a jaxpr
with 3 input parameters:

  * ``c`` is a constvar and stands for the ``ones`` constant,
  * ``b`` corresponds to the free variable ``arg`` captured in the ``inner`` function,
  * ``a`` corresponds to the ``inner`` parameter ``x``.

The primitive takes three arguments ``b a c``.

XLA_pmap
^^^^^^^^

If you use the :py:func:`jax.pmap` transformation, the function to be
mapped is captured using the ``xla_pmap`` primitive. Consider this
example

>>> from jax import pmap
>>>
>>> def func13(arr, extra):
...   def inner(x):
...     # use a free variable "extra" and a constant jnp.ones(1)
...     return (x + extra + jnp.ones(1)) / lax.psum(x, axis_name='rows')
...   return pmap(inner, axis_name='rows')(arr)
...
>>> print(make_jaxpr(func13)(jnp.ones((1, 3)), 5.))
{ lambda c ; a b.
  let d = xla_pmap[ axis_name=rows
                    axis_size=1
                    backend=None
                    call_jaxpr={ lambda  ; d b a.
                                 let c = add a b
                                     e = add c d
                                     f = psum[ axis_index_groups=None
                                               axis_name=rows ] a
                                     g = div e f
                                 in (g,) }
                    devices=None
                    donated_invars=(False, False, False)
                    global_axis_size=None
                    mapped_invars=(True, False, True)
                    name=inner ] c b a
  in (d,) }

The top-level constvar ``c`` refers to the ``jnp.ones(1)`` constant.
The ``xla_pmap`` primitive specifies the name of the axis (parameter ``rows``)
and the body of the function to be mapped as the ``call_jaxpr`` parameter. The
value of this parameter is a Jaxpr with 3 input variables:

  * ``d`` stands for the constant ``jnp.ones(1)``,
  * ``b`` stands for the free variable ``extra``,
  * ``a`` stands for the parameter ``x`` of ``inner``.


The parameter ``mapped_invars`` specify which of the input variables should be
mapped and which should be broadcast. In our example, the value of ``extra``
is broadcast, the other input values are mapped.