prompt: We are currently solving the following issue within our repository. Here is the issue text:
  --- BEGIN ISSUE ---
  Add a function for summation along axes
  `sum` function reduces all elements, while there are many cases that I want to take a summation over some axes. This is analogous to `numpy.sum` with the `axis` argument.
  
  The implementation should contain a general function for GPU to compute a reduction over specified axes.
  
  
  --- END ISSUE ---
  
  Below are some code segments, each from a relevant file. One or more of these files may contain bugs.
  --- BEGIN FILES ---
  Path: `chainer/functions/math/sum.py`
  Content:
  ```
  1 import numpy
  2 
  3 from chainer import cuda
  4 from chainer import function
  5 from chainer.utils import type_check
  6 
  7 
  8 class Sum(function.Function):
  9     """Sum of array elements over a given axis."""
  10 
  11     def __init__(self, axis=None):
  12         self.axis = axis
  13 
  14     def check_type_forward(self, in_types):
  15         type_check.expect(
  16             in_types.size() == 1,
  17             in_types[0].dtype == numpy.float32
  18         )
  19 
  20         if self.axis is not None:
  21             type_check.expect(
  22                 self.axis < in_types[0].ndim,
  23             )
  24 
  25     def forward(self, x):
  26         xp = cuda.get_array_module(*x)
  27         return xp.asarray(x[0].sum(axis=self.axis)),
  28 
  29     def backward(self, x, gy):
  30         xp = cuda.get_array_module(*x)
  31 
  32         gx = xp.empty_like(x[0])
  33         if self.axis is None:
  34             gx[:] = gy[0]
  35         else:
  36             gx[:] = xp.expand_dims(gy[0], axis=self.axis)
  37 
  38         return gx,
  39 
  40 
  41 def sum(x, axis=None):
  42     """Sum of array elements over a given axis.
  43 
  44     Args:
  45         x (~chainer.Variable): Elements to sum.
  46         axis (None or int): Axis which a sum is performed.
  47             The default (axis = None) is perform a sum over all the dimensions
  48             of the input array.
  49 
  50     Returns:
  51         ~chainer.Variable: Output variable.
  52 
  53     """
  54     return Sum(axis)(x)
  55
  ```
  --- END FILES ---
  
  Please first localize the bug based on the issue statement, and then generate a patch according to the `git diff` format fenced by three backticks.
  
  Here is an example:
  
  ```diff
  diff --git a/examples/server_async.py b/examples/server_async.py
  --- a/examples/server_async.py
  +++ b/examples/server_async.py
  @@ -313,4 +313,4 @@
   
   
   if __name__ == "__main__":
  -    asyncio.run(run_async_server("."), debug=True)
  +    asyncio.run(run_async_server(), debug=True)
  diff --git a/examples/server_sync.py b/examples/server_sync.py
  --- a/examples/server_sync.py
  +++ b/examples/server_sync.py
  @@ -313,5 +313,5 @@
   
   
   if __name__ == "__main__":
  -    server = run_sync_server(".")
  +    server = run_sync_server()
       server.shutdown()
  
  ```
  
golden_diff: diff --git a/chainer/functions/math/sum.py b/chainer/functions/math/sum.py
  --- a/chainer/functions/math/sum.py
  +++ b/chainer/functions/math/sum.py
  @@ -9,7 +9,17 @@
       """Sum of array elements over a given axis."""
   
       def __init__(self, axis=None):
  -        self.axis = axis
  +        if axis is None:
  +            self.axis = None
  +        elif isinstance(axis, int):
  +            self.axis = (axis,)
  +        elif isinstance(axis, tuple) and all(isinstance(a, int) for a in axis):
  +            if len(set(axis)) != len(axis):
  +                raise ValueError('duplicate value in axis: ({})'.format(
  +                    ', '.join(map(str, axis))))
  +            self.axis = axis
  +        else:
  +            raise TypeError('None, int or tuple of int are required')
   
       def check_type_forward(self, in_types):
           type_check.expect(
  @@ -18,9 +28,15 @@
           )
   
           if self.axis is not None:
  -            type_check.expect(
  -                self.axis < in_types[0].ndim,
  -            )
  +            for axis in self.axis:
  +                if axis >= 0:
  +                    type_check.expect(
  +                        axis < in_types[0].ndim,
  +                    )
  +                else:
  +                    type_check.expect(
  +                        -axis - 1 < in_types[0].ndim,
  +                    )
   
       def forward(self, x):
           xp = cuda.get_array_module(*x)
  @@ -33,7 +49,15 @@
           if self.axis is None:
               gx[:] = gy[0]
           else:
  -            gx[:] = xp.expand_dims(gy[0], axis=self.axis)
  +            gy = gy[0]
  +            actual_axis = []
  +            for axis in self.axis:
  +                if axis < 0:
  +                    axis = len(gx.shape) + axis
  +                actual_axis.append(axis)
  +            for axis in sorted(actual_axis):
  +                gy = xp.expand_dims(gy, axis=axis)
  +            gx[:] = gy
   
           return gx,
   
  @@ -43,7 +67,7 @@
   
       Args:
           x (~chainer.Variable): Elements to sum.
  -        axis (None or int): Axis which a sum is performed.
  +        axis (None, int, or tuple of int): Axis which a sum is performed.
               The default (axis = None) is perform a sum over all the dimensions
               of the input array.
  
verification_info: {"golden_diff": "diff --git a/chainer/functions/math/sum.py b/chainer/functions/math/sum.py\n--- a/chainer/functions/math/sum.py\n+++ b/chainer/functions/math/sum.py\n@@ -9,7 +9,17 @@\n     \"\"\"Sum of array elements over a given axis.\"\"\"\n \n     def __init__(self, axis=None):\n-        self.axis = axis\n+        if axis is None:\n+            self.axis = None\n+        elif isinstance(axis, int):\n+            self.axis = (axis,)\n+        elif isinstance(axis, tuple) and all(isinstance(a, int) for a in axis):\n+            if len(set(axis)) != len(axis):\n+                raise ValueError('duplicate value in axis: ({})'.format(\n+                    ', '.join(map(str, axis))))\n+            self.axis = axis\n+        else:\n+            raise TypeError('None, int or tuple of int are required')\n \n     def check_type_forward(self, in_types):\n         type_check.expect(\n@@ -18,9 +28,15 @@\n         )\n \n         if self.axis is not None:\n-            type_check.expect(\n-                self.axis < in_types[0].ndim,\n-            )\n+            for axis in self.axis:\n+                if axis >= 0:\n+                    type_check.expect(\n+                        axis < in_types[0].ndim,\n+                    )\n+                else:\n+                    type_check.expect(\n+                        -axis - 1 < in_types[0].ndim,\n+                    )\n \n     def forward(self, x):\n         xp = cuda.get_array_module(*x)\n@@ -33,7 +49,15 @@\n         if self.axis is None:\n             gx[:] = gy[0]\n         else:\n-            gx[:] = xp.expand_dims(gy[0], axis=self.axis)\n+            gy = gy[0]\n+            actual_axis = []\n+            for axis in self.axis:\n+                if axis < 0:\n+                    axis = len(gx.shape) + axis\n+                actual_axis.append(axis)\n+            for axis in sorted(actual_axis):\n+                gy = xp.expand_dims(gy, axis=axis)\n+            gx[:] = gy\n \n         return gx,\n \n@@ -43,7 +67,7 @@\n \n     Args:\n         x (~chainer.Variable): Elements to sum.\n-        axis (None or int): Axis which a sum is performed.\n+        axis (None, int, or tuple of int): Axis which a sum is performed.\n             The default (axis = None) is perform a sum over all the dimensions\n             of the input array.\n", "issue": "Add a function for summation along axes\n`sum` function reduces all elements, while there are many cases that I want to take a summation over some axes. This is analogous to `numpy.sum` with the `axis` argument.\n\nThe implementation should contain a general function for GPU to compute a reduction over specified axes.\n\n", "before_files": [{"content": "import numpy\n\nfrom chainer import cuda\nfrom chainer import function\nfrom chainer.utils import type_check\n\n\nclass Sum(function.Function):\n    \"\"\"Sum of array elements over a given axis.\"\"\"\n\n    def __init__(self, axis=None):\n        self.axis = axis\n\n    def check_type_forward(self, in_types):\n        type_check.expect(\n            in_types.size() == 1,\n            in_types[0].dtype == numpy.float32\n        )\n\n        if self.axis is not None:\n            type_check.expect(\n                self.axis < in_types[0].ndim,\n            )\n\n    def forward(self, x):\n        xp = cuda.get_array_module(*x)\n        return xp.asarray(x[0].sum(axis=self.axis)),\n\n    def backward(self, x, gy):\n        xp = cuda.get_array_module(*x)\n\n        gx = xp.empty_like(x[0])\n        if self.axis is None:\n            gx[:] = gy[0]\n        else:\n            gx[:] = xp.expand_dims(gy[0], axis=self.axis)\n\n        return gx,\n\n\ndef sum(x, axis=None):\n    \"\"\"Sum of array elements over a given axis.\n\n    Args:\n        x (~chainer.Variable): Elements to sum.\n        axis (None or int): Axis which a sum is performed.\n            The default (axis = None) is perform a sum over all the dimensions\n            of the input array.\n\n    Returns:\n        ~chainer.Variable: Output variable.\n\n    \"\"\"\n    return Sum(axis)(x)\n", "path": "chainer/functions/math/sum.py"}], "after_files": [{"content": "import numpy\n\nfrom chainer import cuda\nfrom chainer import function\nfrom chainer.utils import type_check\n\n\nclass Sum(function.Function):\n    \"\"\"Sum of array elements over a given axis.\"\"\"\n\n    def __init__(self, axis=None):\n        if axis is None:\n            self.axis = None\n        elif isinstance(axis, int):\n            self.axis = (axis,)\n        elif isinstance(axis, tuple) and all(isinstance(a, int) for a in axis):\n            if len(set(axis)) != len(axis):\n                raise ValueError('duplicate value in axis: ({})'.format(\n                    ', '.join(map(str, axis))))\n            self.axis = axis\n        else:\n            raise TypeError('None, int or tuple of int are required')\n\n    def check_type_forward(self, in_types):\n        type_check.expect(\n            in_types.size() == 1,\n            in_types[0].dtype == numpy.float32\n        )\n\n        if self.axis is not None:\n            for axis in self.axis:\n                if axis >= 0:\n                    type_check.expect(\n                        axis < in_types[0].ndim,\n                    )\n                else:\n                    type_check.expect(\n                        -axis - 1 < in_types[0].ndim,\n                    )\n\n    def forward(self, x):\n        xp = cuda.get_array_module(*x)\n        return xp.asarray(x[0].sum(axis=self.axis)),\n\n    def backward(self, x, gy):\n        xp = cuda.get_array_module(*x)\n\n        gx = xp.empty_like(x[0])\n        if self.axis is None:\n            gx[:] = gy[0]\n        else:\n            gy = gy[0]\n            actual_axis = []\n            for axis in self.axis:\n                if axis < 0:\n                    axis = len(gx.shape) + axis\n                actual_axis.append(axis)\n            for axis in sorted(actual_axis):\n                gy = xp.expand_dims(gy, axis=axis)\n            gx[:] = gy\n\n        return gx,\n\n\ndef sum(x, axis=None):\n    \"\"\"Sum of array elements over a given axis.\n\n    Args:\n        x (~chainer.Variable): Elements to sum.\n        axis (None, int, or tuple of int): Axis which a sum is performed.\n            The default (axis = None) is perform a sum over all the dimensions\n            of the input array.\n\n    Returns:\n        ~chainer.Variable: Output variable.\n\n    \"\"\"\n    return Sum(axis)(x)\n", "path": "chainer/functions/math/sum.py"}]}