transformations.py

import warnings
from collections import defaultdict, OrderedDict, namedtuple
from copy import deepcopy
from types import MappingProxyType
import pickle
import hashlib
import sympy as sp
from sympy.logic.boolalg import Boolean
from pystencils.simp.assignment_collection import AssignmentCollection
from pystencils.assignment import Assignment
from pystencils.field import AbstractField, FieldType, Field
from pystencils.data_types import TypedSymbol, PointerType, StructType, get_base_type, reinterpret_cast_func, \
    cast_func, pointer_arithmetic_func, get_type_of_expression, collate_types, create_type
from pystencils.kernelparameters import FieldPointerSymbol
from pystencils.slicing import normalize_slice
import pystencils.astnodes as ast


class NestedScopes:
    """Symbol visibility model using nested scopes

    - every accessed symbol that was not defined before, is added as a "free parameter"
    - free parameters are global, i.e. they are not in scopes
    - push/pop adds or removes a scope

    >>> s = NestedScopes()
    >>> s.access_symbol("a")
    >>> s.is_defined("a")
    False
    >>> s.free_parameters
    {'a'}
    >>> s.define_symbol("b")
    >>> s.is_defined("b")
    True
    >>> s.push()
    >>> s.is_defined_locally("b")
    False
    >>> s.define_symbol("c")
    >>> s.pop()
    >>> s.is_defined("c")
    False
    """

    def __init__(self):
        self.free_parameters = set()
        self._defined = [set()]

    def access_symbol(self, symbol):
        if not self.is_defined(symbol):
            self.free_parameters.add(symbol)

    def define_symbol(self, symbol):
        self._defined[-1].add(symbol)

    def is_defined(self, symbol):
        return any(symbol in scopes for scopes in self._defined)

    def is_defined_locally(self, symbol):
        return symbol in self._defined[-1]

    def push(self):
        self._defined.append(set())

    def pop(self):
        self._defined.pop()
        assert self.depth >= 1

    @property
    def depth(self):
        return len(self._defined)


def filtered_tree_iteration(node, node_type, stop_type=None):
    for arg in node.args:
        if isinstance(arg, node_type):
            yield arg
        elif stop_type and isinstance(node, stop_type):
            continue

        yield from filtered_tree_iteration(arg, node_type)


def generic_visit(term, visitor):
    if isinstance(term, AssignmentCollection):
        new_main_assignments = generic_visit(term.main_assignments, visitor)
        new_subexpressions = generic_visit(term.subexpressions, visitor)
        return term.copy(new_main_assignments, new_subexpressions)
    elif isinstance(term, list):
        return [generic_visit(e, visitor) for e in term]
    elif isinstance(term, Assignment):
        return Assignment(term.lhs, generic_visit(term.rhs, visitor))
    elif isinstance(term, sp.Matrix):
        return term.applyfunc(lambda e: generic_visit(e, visitor))
    else:
        return visitor(term)


def unify_shape_symbols(body, common_shape, fields):
    """Replaces symbols for array sizes to ensure they are represented by the same unique symbol.

    When creating a kernel with variable array sizes, all passed arrays must have the same size.
    This is ensured when the kernel is called. Inside the kernel this means that only on symbol has to be used instead
    of one for each field. For example shape_arr1[0]  and shape_arr2[0] must be equal, so they should also be
    represented by the same symbol.

    Args:
        body: ast node, for the kernel part where substitutions is made, is modified in-place
        common_shape: shape of the field that was chosen
        fields: all fields whose shapes should be replaced by common_shape
    """
    substitutions = {}
    for field in fields:
        assert len(field.spatial_shape) == len(common_shape)
        if not field.has_fixed_shape:
            for common_shape_component, shape_component in zip(common_shape, field.spatial_shape):
                if shape_component != common_shape_component:
                    substitutions[shape_component] = common_shape_component
    if substitutions:
        body.subs(substitutions)


def get_common_shape(field_set):
    """Takes a set of pystencils Fields and returns their common spatial shape if it exists. Otherwise
    ValueError is raised"""
    nr_of_fixed_shaped_fields = 0
    for f in field_set:
        if f.has_fixed_shape:
            nr_of_fixed_shaped_fields += 1

    if nr_of_fixed_shaped_fields > 0 and nr_of_fixed_shaped_fields != len(field_set):
        fixed_field_names = ",".join([f.name for f in field_set if f.has_fixed_shape])
        var_field_names = ",".join([f.name for f in field_set if not f.has_fixed_shape])
        msg = "Mixing fixed-shaped and variable-shape fields in a single kernel is not possible\n"
        msg += "Variable shaped: %s \nFixed shaped:    %s" % (var_field_names, fixed_field_names)
        raise ValueError(msg)

    shape_set = set([f.spatial_shape for f in field_set])
    if nr_of_fixed_shaped_fields == len(field_set):
        if len(shape_set) != 1:
            raise ValueError("Differently sized field accesses in loop body: " + str(shape_set))

    shape = list(sorted(shape_set, key=lambda e: str(e[0])))[0]
    return shape


def make_loop_over_domain(body, iteration_slice=None, ghost_layers=None, loop_order=None):
    """Uses :class:`pystencils.field.Field.Access` to create (multiple) loops around given AST.

    Args:
        body: Block object with inner loop contents
        iteration_slice: if not None, iteration is done only over this slice of the field
        ghost_layers: a sequence of pairs for each coordinate with lower and upper nr of ghost layers
             if None, the number of ghost layers is determined automatically and assumed to be equal for a
             all dimensions
        loop_order: loop ordering from outer to inner loop (optimal ordering is same as layout)

    Returns:
        tuple of loop-node, ghost_layer_info
    """
    # find correct ordering by inspecting participating FieldAccesses
    field_accesses = body.atoms(AbstractField.AbstractAccess)
    field_accesses = {e for e in field_accesses if not e.is_absolute_access}

    # exclude accesses to buffers from field_list, because buffers are treated separately
    field_list = [e.field for e in field_accesses if not FieldType.is_buffer(e.field)]
    fields = set(field_list)

    if loop_order is None:
        loop_order = get_optimal_loop_ordering(fields)

    shape = get_common_shape(fields)
    unify_shape_symbols(body, common_shape=shape, fields=fields)

    if iteration_slice is not None:
        iteration_slice = normalize_slice(iteration_slice, shape)

    if ghost_layers is None:
        required_ghost_layers = max([fa.required_ghost_layers for fa in field_accesses])
        ghost_layers = [(required_ghost_layers, required_ghost_layers)] * len(loop_order)
    if isinstance(ghost_layers, int):
        ghost_layers = [(ghost_layers, ghost_layers)] * len(loop_order)

    current_body = body
    for i, loop_coordinate in enumerate(reversed(loop_order)):
        if iteration_slice is None:
            begin = ghost_layers[loop_coordinate][0]
            end = shape[loop_coordinate] - ghost_layers[loop_coordinate][1]
            new_loop = ast.LoopOverCoordinate(current_body, loop_coordinate, begin, end, 1)
            current_body = ast.Block([new_loop])
        else:
            slice_component = iteration_slice[loop_coordinate]
            if type(slice_component) is slice:
                sc = slice_component
                new_loop = ast.LoopOverCoordinate(current_body, loop_coordinate, sc.start, sc.stop, sc.step)
                current_body = ast.Block([new_loop])
            else:
                assignment = ast.SympyAssignment(ast.LoopOverCoordinate.get_loop_counter_symbol(loop_coordinate),
                                                 sp.sympify(slice_component))
                current_body.insert_front(assignment)

    return current_body, ghost_layers


def create_intermediate_base_pointer(field_access, coordinates, previous_ptr):
    r"""
    Addressing elements in structured arrays is done with :math:`ptr\left[ \sum_i c_i \cdot s_i \right]`
    where :math:`c_i` is the coordinate value and :math:`s_i` the stride of a coordinate.
    The sum can be split up into multiple parts, such that parts of it can be pulled before loops.
    This function creates such an access for coordinates :math:`i \in \mbox{coordinates}`.
    Returns a new typed symbol, where the name encodes which coordinates have been resolved.

    Args:
        field_access: instance of :class:`pystencils.field.Field.Access` which provides strides and offsets
        coordinates: mapping of coordinate ids to its value, where stride*value is calculated
        previous_ptr: the pointer which is de-referenced

    Returns
        tuple with the new pointer symbol and the calculated offset

    Examples:
        >>> field = Field.create_generic('myfield', spatial_dimensions=2, index_dimensions=1)
        >>> x, y = sp.symbols("x y")
        >>> prev_pointer = TypedSymbol("ptr", "double")
        >>> create_intermediate_base_pointer(field[1,-2](5), {0: x}, prev_pointer)
        (ptr_01, _stride_myfield_0*x + _stride_myfield_0)
        >>> create_intermediate_base_pointer(field[1,-2](5), {0: x, 1 : y }, prev_pointer)
        (ptr_01_1m2, _stride_myfield_0*x + _stride_myfield_0 + _stride_myfield_1*y - 2*_stride_myfield_1)
    """
    field = field_access.field
    offset = 0
    name = ""
    list_to_hash = []
    for coordinate_id, coordinate_value in coordinates.items():
        offset += field.strides[coordinate_id] * coordinate_value

        if coordinate_id < field.spatial_dimensions:
            offset += field.strides[coordinate_id] * field_access.offsets[coordinate_id]
            if type(field_access.offsets[coordinate_id]) is int:
                name += "_%d%d" % (coordinate_id, field_access.offsets[coordinate_id])
            else:
                list_to_hash.append(field_access.offsets[coordinate_id])
        else:
            if type(coordinate_value) is int:
                name += "_%d%d" % (coordinate_id, coordinate_value)
            else:
                list_to_hash.append(coordinate_value)

    if len(list_to_hash) > 0:
        name += hashlib.md5(pickle.dumps(list_to_hash)).hexdigest()[:16]

    name = name.replace("-", 'm')
    new_ptr = TypedSymbol(previous_ptr.name + name, previous_ptr.dtype)
    return new_ptr, offset


def parse_base_pointer_info(base_pointer_specification, loop_order, spatial_dimensions, index_dimensions):
    """
    Creates base pointer specification for :func:`resolve_field_accesses` function.

    Specification of how many and which intermediate pointers are created for a field access.
    For example [ (0), (2,3,)]  creates on base pointer for coordinates 2 and 3 and writes the offset for coordinate
    zero directly in the field access. These specifications are defined dependent on the loop ordering.
    This function translates more readable version into the specification above.

    Allowed specifications:
        - "spatialInner<int>" spatialInner0 is the innermost loop coordinate,
          spatialInner1 the loop enclosing the innermost
        - "spatialOuter<int>" spatialOuter0 is the outermost loop
        - "index<int>": index coordinate
        - "<int>": specifying directly the coordinate

    Args:
        base_pointer_specification: nested list with above specifications
        loop_order: list with ordering of loops from outer to inner
        spatial_dimensions: number of spatial dimensions
        index_dimensions: number of index dimensions

    Returns:
        list of tuples that can be passed to :func:`resolve_field_accesses`

    Examples:
        >>> parse_base_pointer_info([['spatialOuter0'], ['index0']], loop_order=[2,1,0],
        ...                         spatial_dimensions=3, index_dimensions=1)
        [[0], [3], [1, 2]]
    """
    result = []
    specified_coordinates = set()
    loop_order = list(reversed(loop_order))
    for spec_group in base_pointer_specification:
        new_group = []

        def add_new_element(elem):
            if elem >= spatial_dimensions + index_dimensions:
                raise ValueError("Coordinate %d does not exist" % (elem,))
            new_group.append(elem)
            if elem in specified_coordinates:
                raise ValueError("Coordinate %d specified two times" % (elem,))
            specified_coordinates.add(elem)
        for element in spec_group:
            if type(element) is int:
                add_new_element(element)
            elif element.startswith("spatial"):
                element = element[len("spatial"):]
                if element.startswith("Inner"):
                    index = int(element[len("Inner"):])
                    add_new_element(loop_order[index])
                elif element.startswith("Outer"):
                    index = int(element[len("Outer"):])
                    add_new_element(loop_order[-index])
                elif element == "all":
                    for i in range(spatial_dimensions):
                        add_new_element(i)
                else:
                    raise ValueError("Could not parse " + element)
            elif element.startswith("index"):
                index = int(element[len("index"):])
                add_new_element(spatial_dimensions + index)
            else:
                raise ValueError("Unknown specification %s" % (element,))

        result.append(new_group)

    all_coordinates = set(range(spatial_dimensions + index_dimensions))
    rest = all_coordinates - specified_coordinates
    if rest:
        result.append(list(rest))

    return result


def get_base_buffer_index(ast_node, loop_counters=None, loop_iterations=None):
    """Used for buffer fields to determine the linearized index of the buffer dependent on loop counter symbols.

    Args:
        ast_node: ast before any field accesses are resolved
        loop_counters: for CPU kernels: leave to default 'None' (can be determined from loop nodes)
                       for GPU kernels: list of 'loop counters' from inner to outer loop
        loop_iterations: number of iterations of each loop from inner to outer, for CPU kernels leave to default

    Returns:
        base buffer index - required by 'resolve_buffer_accesses' function
    """
    if loop_counters is None or loop_iterations is None:
        loops = [l for l in filtered_tree_iteration(ast_node, ast.LoopOverCoordinate, ast.SympyAssignment)]
        loops.reverse()
        parents_of_innermost_loop = list(parents_of_type(loops[0], ast.LoopOverCoordinate, include_current=True))
        assert len(loops) == len(parents_of_innermost_loop)
        assert all(l1 is l2 for l1, l2 in zip(loops, parents_of_innermost_loop))

        loop_iterations = [(l.stop - l.start) / l.step for l in loops]
        loop_counters = [l.loop_counter_symbol for l in loops]

    field_accesses = ast_node.atoms(AbstractField.AbstractAccess)
    buffer_accesses = {fa for fa in field_accesses if FieldType.is_buffer(fa.field)}
    loop_counters = [v * len(buffer_accesses) for v in loop_counters]

    base_buffer_index = loop_counters[0]
    stride = 1
    for idx, var in enumerate(loop_counters[1:]):
        cur_stride = loop_iterations[idx]
        stride *= int(cur_stride) if isinstance(cur_stride, float) else cur_stride
        base_buffer_index += var * stride
    return base_buffer_index


def resolve_buffer_accesses(ast_node, base_buffer_index, read_only_field_names=set()):

    def visit_sympy_expr(expr, enclosing_block, sympy_assignment):
        if isinstance(expr, AbstractField.AbstractAccess):
            field_access = expr

            # Do not apply transformation if field is not a buffer
            if not FieldType.is_buffer(field_access.field):
                return expr

            buffer = field_access.field
            field_ptr = FieldPointerSymbol(buffer.name, buffer.dtype, const=buffer.name in read_only_field_names)

            buffer_index = base_buffer_index
            if len(field_access.index) > 1:
                raise RuntimeError('Only indexing dimensions up to 1 are currently supported in buffers!')

            if len(field_access.index) > 0:
                cell_index = field_access.index[0]
                buffer_index += cell_index

            result = ast.ResolvedFieldAccess(field_ptr, buffer_index, field_access.field, field_access.offsets,
                                             field_access.index)

            return visit_sympy_expr(result, enclosing_block, sympy_assignment)
        else:
            if isinstance(expr, ast.ResolvedFieldAccess):
                return expr

            new_args = [visit_sympy_expr(e, enclosing_block, sympy_assignment) for e in expr.args]
            kwargs = {'evaluate': False} if type(expr) in (sp.Add, sp.Mul, sp.Piecewise) else {}
            return expr.func(*new_args, **kwargs) if new_args else expr

    def visit_node(sub_ast):
        if isinstance(sub_ast, ast.SympyAssignment):
            enclosing_block = sub_ast.parent
            assert type(enclosing_block) is ast.Block
            sub_ast.lhs = visit_sympy_expr(sub_ast.lhs, enclosing_block, sub_ast)
            sub_ast.rhs = visit_sympy_expr(sub_ast.rhs, enclosing_block, sub_ast)
        else:
            for i, a in enumerate(sub_ast.args):
                visit_node(a)

    return visit_node(ast_node)


def resolve_field_accesses(ast_node, read_only_field_names=set(),
                           field_to_base_pointer_info=MappingProxyType({}),
                           field_to_fixed_coordinates=MappingProxyType({})):
    """
    Substitutes :class:`pystencils.field.Field.Access` nodes by array indexing

    Args:
        ast_node: the AST root
        read_only_field_names: set of field names which are considered read-only
        field_to_base_pointer_info: a list of tuples indicating which intermediate base pointers should be created
                                    for details see :func:`parse_base_pointer_info`
        field_to_fixed_coordinates: map of field name to a tuple of coordinate symbols. Instead of using the loop
                                    counters to index the field these symbols are used as coordinates

    Returns
        transformed AST
    """
    field_to_base_pointer_info = OrderedDict(sorted(field_to_base_pointer_info.items(), key=lambda pair: pair[0]))
    field_to_fixed_coordinates = OrderedDict(sorted(field_to_fixed_coordinates.items(), key=lambda pair: pair[0]))

    def visit_sympy_expr(expr, enclosing_block, sympy_assignment):
        if isinstance(expr, AbstractField.AbstractAccess):
            field_access = expr
            field = field_access.field

            if field_access.indirect_addressing_fields:
                new_offsets = tuple(visit_sympy_expr(off, enclosing_block, sympy_assignment)
                                    for off in field_access.offsets)
                new_indices = tuple(visit_sympy_expr(ind, enclosing_block, sympy_assignment)
                                    if isinstance(ind, sp.Basic) else ind
                                    for ind in field_access.index)
                field_access = Field.Access(field_access.field, new_offsets,
                                            new_indices, field_access.is_absolute_access)

            if field.name in field_to_base_pointer_info:
                base_pointer_info = field_to_base_pointer_info[field.name]
            else:
                base_pointer_info = [list(range(field.index_dimensions + field.spatial_dimensions))]

            field_ptr = FieldPointerSymbol(field.name, field.dtype, const=field.name in read_only_field_names)

            def create_coordinate_dict(group_param):
                coordinates = {}
                for e in group_param:
                    if e < field.spatial_dimensions:
                        if field.name in field_to_fixed_coordinates:
                            if not field_access.is_absolute_access:
                                coordinates[e] = field_to_fixed_coordinates[field.name][e]
                            else:
                                coordinates[e] = 0
                        else:
                            if not field_access.is_absolute_access:
                                coordinates[e] = ast.LoopOverCoordinate.get_loop_counter_symbol(e)
                            else:
                                coordinates[e] = 0
                        coordinates[e] *= field.dtype.item_size
                    else:
                        if isinstance(field.dtype, StructType):
                            assert field.index_dimensions == 1
                            accessed_field_name = field_access.index[0]
                            assert isinstance(accessed_field_name, str)
                            coordinates[e] = field.dtype.get_element_offset(accessed_field_name)
                        else:
                            coordinates[e] = field_access.index[e - field.spatial_dimensions]

                return coordinates

            last_pointer = field_ptr

            for group in reversed(base_pointer_info[1:]):
                coord_dict = create_coordinate_dict(group)
                new_ptr, offset = create_intermediate_base_pointer(field_access, coord_dict, last_pointer)
                if new_ptr not in enclosing_block.symbols_defined:
                    new_assignment = ast.SympyAssignment(new_ptr, last_pointer + offset, is_const=False)
                    enclosing_block.insert_before(new_assignment, sympy_assignment)
                last_pointer = new_ptr

            coord_dict = create_coordinate_dict(base_pointer_info[0])
            _, offset = create_intermediate_base_pointer(field_access, coord_dict, last_pointer)
            result = ast.ResolvedFieldAccess(last_pointer, offset, field_access.field,
                                             field_access.offsets, field_access.index)

            if isinstance(get_base_type(field_access.field.dtype), StructType):
                new_type = field_access.field.dtype.get_element_type(field_access.index[0])
                result = reinterpret_cast_func(result, new_type)

            return visit_sympy_expr(result, enclosing_block, sympy_assignment)
        else:
            if isinstance(expr, ast.ResolvedFieldAccess):
                return expr

            new_args = [visit_sympy_expr(e, enclosing_block, sympy_assignment) for e in expr.args]
            kwargs = {'evaluate': False} if type(expr) in (sp.Add, sp.Mul, sp.Piecewise) else {}
            return expr.func(*new_args, **kwargs) if new_args else expr

    def visit_node(sub_ast):
        if isinstance(sub_ast, ast.SympyAssignment):
            enclosing_block = sub_ast.parent
            assert type(enclosing_block) is ast.Block
            sub_ast.lhs = visit_sympy_expr(sub_ast.lhs, enclosing_block, sub_ast)
            sub_ast.rhs = visit_sympy_expr(sub_ast.rhs, enclosing_block, sub_ast)
        elif isinstance(sub_ast, ast.Conditional):
            enclosing_block = sub_ast.parent
            assert type(enclosing_block) is ast.Block
            sub_ast.condition_expr = visit_sympy_expr(sub_ast.condition_expr, enclosing_block, sub_ast)
            visit_node(sub_ast.true_block)
            if sub_ast.false_block:
                visit_node(sub_ast.false_block)
        else:
            for i, a in enumerate(sub_ast.args):
                visit_node(a)

    return visit_node(ast_node)


def move_constants_before_loop(ast_node):
    """Moves :class:`pystencils.ast.SympyAssignment` nodes out of loop body if they are iteration independent.

    Call this after creating the loop structure with :func:`make_loop_over_domain`
    """
    def find_block_to_move_to(node):
        """
        Traverses parents of node as long as the symbols are independent and returns a (parent) block
        the assignment can be safely moved to
        :param node: SympyAssignment inside a Block
        :return blockToInsertTo, childOfBlockToInsertBefore
        """
        assert isinstance(node.parent, ast.Block)

        last_block = node.parent
        last_block_child = node
        element = node.parent
        prev_element = node
        while element:
            if isinstance(element, ast.Block):
                last_block = element
                last_block_child = prev_element

            if isinstance(element, ast.Conditional):
                break
            else:
                critical_symbols = element.symbols_defined
            if node.undefined_symbols.intersection(critical_symbols):
                break
            prev_element = element
            element = element.parent
        return last_block, last_block_child

    def check_if_assignment_already_in_block(assignment, target_block, rhs_or_lhs=True):
        for arg in target_block.args:
            if type(arg) is not ast.SympyAssignment:
                continue
            if (rhs_or_lhs and arg.rhs == assignment.rhs) or (not rhs_or_lhs and arg.lhs == assignment.lhs):
                return arg
        return None

    def get_blocks(node, result_list):
        if isinstance(node, ast.Block):
            result_list.append(node)
        if isinstance(node, ast.Node):
            for a in node.args:
                get_blocks(a, result_list)

    all_blocks = []
    get_blocks(ast_node, all_blocks)
    for block in all_blocks:
        children = block.take_child_nodes()
        # Every time a symbol can be replaced in the current block because the assignment
        # was found in a parent block, but with a different lhs symbol (same rhs)
        # the outer symbol is inserted here as key.
        substitute_variables = {}
        for child in children:
            # Before traversing the next child, all symbols are substituted first.
            child.subs(substitute_variables)

            if not isinstance(child, ast.SympyAssignment):  # only move SympyAssignments
                block.append(child)
                continue

            target, child_to_insert_before = find_block_to_move_to(child)
            if target == block:     # movement not possible
                target.append(child)
            else:
                if isinstance(child, ast.SympyAssignment):
                    exists_already = check_if_assignment_already_in_block(child, target, False)
                else:
                    exists_already = False

                if not exists_already:
                    rhs_identical = check_if_assignment_already_in_block(child, target, True)
                    if rhs_identical:
                        # there is already an assignment out there with the same rhs
                        # -> replace all lhs symbols in this block with the lhs of the outer assignment
                        # -> remove the local assignment (do not re-append child to the former block)
                        substitute_variables[child.lhs] = rhs_identical.lhs
                    else:
                        target.insert_before(child, child_to_insert_before)
                elif exists_already and exists_already.rhs == child.rhs:
                    pass
                else:
                    # this variable already exists in outer block, but with different rhs
                    # -> symbol has to be renamed
                    assert isinstance(child.lhs, TypedSymbol)
                    new_symbol = TypedSymbol(sp.Dummy().name, child.lhs.dtype)
                    target.insert_before(ast.SympyAssignment(new_symbol, child.rhs), child_to_insert_before)
                    substitute_variables[child.lhs] = new_symbol


def split_inner_loop(ast_node: ast.Node, symbol_groups):
    """
    Splits inner loop into multiple loops to minimize the amount of simultaneous load/store streams

    Args:
        ast_node: AST root
        symbol_groups: sequence of symbol sequences: for each symbol sequence a new inner loop is created which
                       updates these symbols and their dependent symbols. Symbols which are in none of the symbolGroups
                       and which no symbol in a symbol group depends on, are not updated!
    """
    all_loops = ast_node.atoms(ast.LoopOverCoordinate)
    inner_loop = [l for l in all_loops if l.is_innermost_loop]
    assert len(inner_loop) == 1, "Error in AST: multiple innermost loops. Was split transformation already called?"
    inner_loop = inner_loop[0]
    assert type(inner_loop.body) is ast.Block
    outer_loop = [l for l in all_loops if l.is_outermost_loop]
    assert len(outer_loop) == 1, "Error in AST, multiple outermost loops."
    outer_loop = outer_loop[0]

    symbols_with_temporary_array = OrderedDict()
    assignment_map = OrderedDict((a.lhs, a) for a in inner_loop.body.args)

    assignment_groups = []
    for symbol_group in symbol_groups:
        # get all dependent symbols
        symbols_to_process = list(symbol_group)
        symbols_resolved = set()
        while symbols_to_process:
            s = symbols_to_process.pop()
            if s in symbols_resolved:
                continue

            if s in assignment_map:  # if there is no assignment inside the loop body it is independent already
                for new_symbol in assignment_map[s].rhs.atoms(sp.Symbol):
                    if not isinstance(new_symbol, AbstractField.AbstractAccess) and \
                            new_symbol not in symbols_with_temporary_array:
                        symbols_to_process.append(new_symbol)
            symbols_resolved.add(s)

        for symbol in symbol_group:
            if not isinstance(symbol, AbstractField.AbstractAccess):
                assert type(symbol) is TypedSymbol
                new_ts = TypedSymbol(symbol.name, PointerType(symbol.dtype))
                symbols_with_temporary_array[symbol] = sp.IndexedBase(new_ts,
                                                                      shape=(1,))[inner_loop.loop_counter_symbol]

        assignment_group = []
        for assignment in inner_loop.body.args:
            if assignment.lhs in symbols_resolved:
                new_rhs = assignment.rhs.subs(symbols_with_temporary_array.items())
                if not isinstance(assignment.lhs, AbstractField.AbstractAccess) and assignment.lhs in symbol_group:
                    assert type(assignment.lhs) is TypedSymbol
                    new_ts = TypedSymbol(assignment.lhs.name, PointerType(assignment.lhs.dtype))
                    new_lhs = sp.IndexedBase(new_ts, shape=(1,))[inner_loop.loop_counter_symbol]
                else:
                    new_lhs = assignment.lhs
                assignment_group.append(ast.SympyAssignment(new_lhs, new_rhs))
        assignment_groups.append(assignment_group)

    new_loops = [inner_loop.new_loop_with_different_body(ast.Block(group)) for group in assignment_groups]
    inner_loop.parent.replace(inner_loop, ast.Block(new_loops))

    for tmp_array in symbols_with_temporary_array:
        tmp_array_pointer = TypedSymbol(tmp_array.name, PointerType(tmp_array.dtype))
        alloc_node = ast.TemporaryMemoryAllocation(tmp_array_pointer, inner_loop.stop, inner_loop.start)
        free_node = ast.TemporaryMemoryFree(alloc_node)
        outer_loop.parent.insert_front(alloc_node)
        outer_loop.parent.append(free_node)


def cut_loop(loop_node, cutting_points):
    """Cuts loop at given cutting points.

    One loop is transformed into len(cuttingPoints)+1 new loops that range from
    old_begin to cutting_points[1], ..., cutting_points[-1] to old_end

    Modifies the ast in place

    Returns:
        list of new loop nodes
    """
    if loop_node.step != 1:
        raise NotImplementedError("Can only split loops that have a step of 1")
    new_loops = ast.Block([])
    new_start = loop_node.start
    cutting_points = list(cutting_points) + [loop_node.stop]
    for new_end in cutting_points:
        if new_end - new_start == 1:
            new_body = deepcopy(loop_node.body)
            new_body.subs({loop_node.loop_counter_symbol: new_start})
            new_loops.append(new_body)
        elif new_end - new_start == 0:
            pass
        else:
            new_loop = ast.LoopOverCoordinate(deepcopy(loop_node.body), loop_node.coordinate_to_loop_over,
                                              new_start, new_end, loop_node.step)
            new_loops.append(new_loop)
        new_start = new_end
    loop_node.parent.replace(loop_node, new_loops)
    return new_loops


def simplify_conditionals(node: ast.Node, loop_counter_simplification: bool = False) -> None:
    """Removes conditionals that are always true/false.

    Args:
        node: ast node, all descendants of this node are simplified
        loop_counter_simplification: if enabled, tries to detect if a conditional is always true/false
                                     depending on the surrounding loop. For example if the surrounding loop goes from
                                     x=0 to 10 and the condition is x < 0, it is removed.
                                     This analysis needs the integer set library (ISL) islpy, so it is not done by
                                     default.
    """
    for conditional in node.atoms(ast.Conditional):
        conditional.condition_expr = sp.simplify(conditional.condition_expr)
        if conditional.condition_expr == sp.true:
            conditional.parent.replace(conditional, [conditional.true_block])
        elif conditional.condition_expr == sp.false:
            conditional.parent.replace(conditional, [conditional.false_block] if conditional.false_block else [])
        elif loop_counter_simplification:
            try:
                # noinspection PyUnresolvedReferences
                from pystencils.integer_set_analysis import simplify_loop_counter_dependent_conditional
                simplify_loop_counter_dependent_conditional(conditional)
            except ImportError:
                warnings.warn("Integer simplifications in conditionals skipped, because ISLpy package not installed")


def cleanup_blocks(node: ast.Node) -> None:
    """Curly Brace Removal: Removes empty blocks, and replaces blocks with a single child by its child """
    if isinstance(node, ast.SympyAssignment):
        return
    elif isinstance(node, ast.Block):
        for a in list(node.args):
            cleanup_blocks(a)
        if len(node.args) <= 1 and isinstance(node.parent, ast.Block):
            node.parent.replace(node, node.args)
            return
    else:
        for a in node.args:
            cleanup_blocks(a)


class KernelConstraintsCheck:
    """Checks if the input to create_kernel is valid.

    Test the following conditions:

    - SSA Form for pure symbols:
        -  Every pure symbol may occur only once as left-hand-side of an assignment
        -  Every pure symbol that is read, may not be written to later
    - Independence / Parallelization condition:
        - a field that is written may only be read at exact the same spatial position

    (Pure symbols are symbols that are not Field.Accesses)
    """
    FieldAndIndex = namedtuple('FieldAndIndex', ['field', 'index'])

    def __init__(self, type_for_symbol, check_independence_condition):
        self._type_for_symbol = type_for_symbol

        self.scopes = NestedScopes()
        self._field_writes = defaultdict(set)
        self.fields_read = set()
        self.check_independence_condition = check_independence_condition

    def process_assignment(self, assignment):
        # for checks it is crucial to process rhs before lhs to catch e.g. a = a + 1
        new_rhs = self.process_expression(assignment.rhs)
        new_lhs = self._process_lhs(assignment.lhs)
        return ast.SympyAssignment(new_lhs, new_rhs)

    def process_expression(self, rhs, type_constants=True):
        self._update_accesses_rhs(rhs)
        if isinstance(rhs, AbstractField.AbstractAccess):
            self.fields_read.add(rhs.field)
            self.fields_read.update(rhs.indirect_addressing_fields)
            return rhs
        elif isinstance(rhs, TypedSymbol):
            return rhs
        elif isinstance(rhs, sp.Symbol):
            return TypedSymbol(rhs.name, self._type_for_symbol[rhs.name])
        elif type_constants and isinstance(rhs, sp.Number):
            return cast_func(rhs, create_type(self._type_for_symbol['_constant']))
        elif isinstance(rhs, sp.Mul):
            new_args = [self.process_expression(arg, type_constants) if arg not in (-1, 1) else arg for arg in rhs.args]
            return rhs.func(*new_args) if new_args else rhs
        elif isinstance(rhs, sp.Indexed):
            return rhs
        else:
            if isinstance(rhs, sp.Pow):
                # don't process exponents -> they should remain integers
                return sp.Pow(self.process_expression(rhs.args[0], type_constants), rhs.args[1])
            else:
                new_args = [self.process_expression(arg, type_constants) for arg in rhs.args]
                return rhs.func(*new_args) if new_args else rhs

    @property
    def fields_written(self):
        return set(k.field for k, v in self._field_writes.items() if len(v))

    def _process_lhs(self, lhs):
        assert isinstance(lhs, sp.Symbol)
        self._update_accesses_lhs(lhs)
        if not isinstance(lhs, AbstractField.AbstractAccess) and not isinstance(lhs, TypedSymbol):
            return TypedSymbol(lhs.name, self._type_for_symbol[lhs.name])
        else:
            return lhs

    def _update_accesses_lhs(self, lhs):
        if isinstance(lhs, AbstractField.AbstractAccess):
            fai = self.FieldAndIndex(lhs.field, lhs.index)
            self._field_writes[fai].add(lhs.offsets)
            if len(self._field_writes[fai]) > 1:
                raise ValueError("Field {} is written at two different locations".format(lhs.field.name))
        elif isinstance(lhs, sp.Symbol):
            if self.scopes.is_defined_locally(lhs):
                raise ValueError("Assignments not in SSA form, multiple assignments to {}".format(lhs.name))
            if lhs in self.scopes.free_parameters:
                raise ValueError("Symbol {} is written, after it has been read".format(lhs.name))
            self.scopes.define_symbol(lhs)

    def _update_accesses_rhs(self, rhs):
        if isinstance(rhs, AbstractField.AbstractAccess) and self.check_independence_condition:
            writes = self._field_writes[self.FieldAndIndex(rhs.field, rhs.index)]
            for write_offset in writes:
                assert len(writes) == 1
                if write_offset != rhs.offsets:
                    raise ValueError("Violation of loop independence condition. Field "
                                     "{} is read at {} and written at {}".format(rhs.field, rhs.offsets, write_offset))
            self.fields_read.add(rhs.field)
        elif isinstance(rhs, sp.Symbol):
            self.scopes.access_symbol(rhs)


def add_types(eqs, type_for_symbol, check_independence_condition):
    """Traverses AST and replaces every :class:`sympy.Symbol` by a :class:`pystencils.typedsymbol.TypedSymbol`.

    Additionally returns sets of all fields which are read/written

    Args:
        eqs: list of equations
        type_for_symbol: dict mapping symbol names to types. Types are strings of C types like 'int' or 'double'
        check_independence_condition: check that loop iterations are independent - this has to be skipped for indexed
                                      kernels

    Returns:
        ``fields_read, fields_written, typed_equations`` set of read fields, set of written fields,
         list of equations where symbols have been replaced by typed symbols
    """
    if isinstance(type_for_symbol, str) or not hasattr(type_for_symbol, '__getitem__'):
        type_for_symbol = typing_from_sympy_inspection(eqs, type_for_symbol)

    check = KernelConstraintsCheck(type_for_symbol, check_independence_condition)

    def visit(obj):
        if isinstance(obj, list) or isinstance(obj, tuple):
            return [visit(e) for e in obj]
        if isinstance(obj, sp.Eq) or isinstance(obj, ast.SympyAssignment) or isinstance(obj, Assignment):
            return check.process_assignment(obj)
        elif isinstance(obj, ast.Conditional):
            check.scopes.push()
            false_block = None if obj.false_block is None else visit(obj.false_block)
            result = ast.Conditional(check.process_expression(obj.condition_expr, type_constants=False),
                                     true_block=visit(obj.true_block), false_block=false_block)
            check.scopes.pop()
            return result
        elif isinstance(obj, ast.Block):
            check.scopes.push()
            result = ast.Block([visit(e) for e in obj.args])
            check.scopes.pop()
            return result
        elif isinstance(obj, ast.Node) and not isinstance(obj, ast.LoopOverCoordinate):
            return obj
        else:
            raise ValueError("Invalid object in kernel " + str(type(obj)))

    typed_equations = visit(eqs)

    return check.fields_read, check.fields_written, typed_equations


def insert_casts(node):
    """Checks the types and inserts casts and pointer arithmetic where necessary.

    Args:
        node: the head node of the ast

    Returns:
        modified AST
    """
    def cast(zipped_args_types, target_dtype):
        """
        Adds casts to the arguments if their type differs from the target type
        :param zipped_args_types: a zipped list of args and types
        :param target_dtype: The target data type
        :return: args with possible casts
        """
        casted_args = []
        for argument, data_type in zipped_args_types:
            if data_type.numpy_dtype != target_dtype.numpy_dtype:  # ignoring const
                casted_args.append(cast_func(argument, target_dtype))
            else:
                casted_args.append(argument)
        return casted_args

    def pointer_arithmetic(expr_args):
        """
        Creates a valid pointer arithmetic function
        :param expr_args: Arguments of the add expression
        :return: pointer_arithmetic_func
        """
        pointer = None
        new_args = []
        for arg, data_type in expr_args:
            if data_type.func is PointerType:
                assert pointer is None
                pointer = arg
        for arg, data_type in expr_args:
            if arg != pointer:
                assert data_type.is_int() or data_type.is_uint()
                new_args.append(arg)
        new_args = sp.Add(*new_args) if len(new_args) > 0 else new_args
        return pointer_arithmetic_func(pointer, new_args)

    if isinstance(node, sp.AtomicExpr) or isinstance(node, cast_func):
        return node
    args = []
    for arg in node.args:
        args.append(insert_casts(arg))
    # TODO indexed, LoopOverCoordinate
    if node.func in (sp.Add, sp.Mul, sp.Or, sp.And, sp.Pow, sp.Eq, sp.Ne, sp.Lt, sp.Le, sp.Gt, sp.Ge):
        # TODO optimize pow, don't cast integer on double
        types = [get_type_of_expression(arg) for arg in args]
        assert len(types) > 0
        target = collate_types(types)
        zipped = list(zip(args, types))
        if target.func is PointerType:
            assert node.func is sp.Add
            return pointer_arithmetic(zipped)
        else:
            return node.func(*cast(zipped, target))
    elif node.func is ast.SympyAssignment:
        lhs = args[0]
        rhs = args[1]
        target = get_type_of_expression(lhs)
        if target.func is PointerType:
            return node.func(*args)  # TODO fix, not complete
        else:
            return node.func(lhs, *cast([(rhs, get_type_of_expression(rhs))], target))
    elif node.func is ast.ResolvedFieldAccess:
        return node
    elif node.func is ast.Block:
        for old_arg, new_arg in zip(node.args, args):
            node.replace(old_arg, new_arg)
        return node
    elif node.func is ast.LoopOverCoordinate:
        for old_arg, new_arg in zip(node.args, args):
            node.replace(old_arg, new_arg)
        return node
    elif node.func is sp.Piecewise:
        expressions = [expr for (expr, _) in args]
        types = [get_type_of_expression(expr) for expr in expressions]
        target = collate_types(types)
        zipped = list(zip(expressions, types))
        casted_expressions = cast(zipped, target)
        args = [arg.func(*[expr, arg.cond]) for (arg, expr) in zip(args, casted_expressions)]

    return node.func(*args)


def remove_conditionals_in_staggered_kernel(function_node: ast.KernelFunction) -> None:
    """Removes conditionals of a kernel that iterates over staggered positions by splitting the loops at last element"""

    all_inner_loops = [l for l in function_node.atoms(ast.LoopOverCoordinate) if l.is_innermost_loop]
    assert len(all_inner_loops) == 1, "Transformation works only on kernels with exactly one inner loop"
    inner_loop = all_inner_loops.pop()

    for loop in parents_of_type(inner_loop, ast.LoopOverCoordinate, include_current=True):
        cut_loop(loop, [loop.stop - 1])

    simplify_conditionals(function_node.body, loop_counter_simplification=True)
    cleanup_blocks(function_node.body)

    move_constants_before_loop(function_node.body)
    cleanup_blocks(function_node.body)


# --