#
# Copyright (c) 2006-2011, Prometheus Research, LLC
# See `LICENSE` for license information, `AUTHORS` for the list of authors.
#


"""
:mod:`htsql.tr.code`
====================

This module declares space and code nodes.
"""


from ..util import (maybe, listof, tupleof, Clonable, Comparable, Printable)
from ..entity import TableEntity, ColumnEntity, Join
from ..domain import Domain, BooleanDomain
from .syntax import IdentifierSyntax
from .binding import Binding, QueryBinding, SegmentBinding
from .signature import Signature, Bag, Formula


class Expression(Comparable, Clonable, Printable):
    """
    Represents an expression node.

    This is an abstract class; its subclasses are divided into two categories:
    space nodes (see :class:`Space`) and code nodes (see :class:`Code`).
    There are also several expression node types that do not belong to either
    of these categories.

    An expression tree (a DAG) is an intermediate stage of the HTSQL
    translator.  An expression tree is translated from a binding tree by
    the *encoding* process.  It is then translated to a frame structure
    by the *compiling* and *assembling* processes.

    The following adapters are associated with the encoding process and
    generate new code and space nodes::

        Encode: (Binding, EncodingState) -> Code
        Relate: (Binding, EncodingState) -> Space

    See :class:`htsql.tr.encode.Encode` and :class:`htsql.tr.encode.Relate`
    for more detail.

    The compiling process works as follows.  Space nodes (and also unit nodes)
    are translated to frame nodes via several intermediate steps::

        Compile: (Space, CompilingState) -> Term
        Assemble: (Term, AssemblingState) -> Frame

    Code nodes are directly translated to phrase nodes::

        Evaluate: (Code, AssemblingState) -> Phrase

    See :class:`htsql.tr.compile.Compile, :class:`htsql.tr.assemble.Assemble`,
    :class:`htsql.tr.assemble.Evaluate` for more detail.

    Expression nodes support equality by value (as opposed to to equality
    by identity, which is the default for class instances).  That is, two
    expression nodes are equal if they are of the same type and all their
    (essential) attributes are equal.  Some attributes (e.g. `binding) may
    be considered not essential and do not participate in comparison.  To
    facilitate expression comparison, :class:`htsql.domain.Domain` objects
    also support equality by value.  By-value semantics is respected when
    expression nodes are used as dictionary keys.

    The constructor arguments:

    `binding` (:class:`htsql.tr.binding.Binding`)
        The binding node that gave rise to the expression; should be used
        only for presentation or error reporting.

    `equality_vector` (an immutable tuple or ``None``)
        Encapsulates all essential attributes of a node.  Two expression
        nodes are considered equal if they are of the same type and their
        equality vectors are equal.  If ``None``, the node is compared by
        identity.

        Note that the `binding` attribute is not essential and should not
        be a part of the equality vector.

    Other attributes:

    `syntax` (:class:`htsql.tr.syntax.Syntax`)
        The syntax node that gave rise to the expression; for debugging
        purposes only.

    `mark` (:class:`htsql.mark.Mark`)
        The location of the node in the original query; for error reporting.

    `hash` (an integer)
        The node hash; if two nodes are considered equal, their hashes
        must be equal too.
    """

    def __init__(self, binding, equality_vector=None):
        assert isinstance(binding, Binding)
        super(Expression, self).__init__(equality_vector)
        self.binding = binding
        self.syntax = binding.syntax
        self.mark = binding.syntax.mark

    def __str__(self):
        # Display the syntex node that gave rise to the expression.
        return str(self.syntax)


class QueryExpr(Expression):
    """
    Represents a whole HTSQL query.

    `segment` (:class:`SegmentExpr` or ``None``)
        The query segment.
    """

    def __init__(self, segment, binding):
        assert isinstance(segment, maybe(SegmentExpr))
        assert isinstance(binding, QueryBinding)
        super(QueryExpr, self).__init__(binding)
        self.segment = segment


class SegmentExpr(Expression):
    """
    Represents a segment of an HTSQL query.

    `space` (:class:`Space`)
        The space rendered by the segment.

    `elements` (a list of :class:`Code` objects)
        The elements rendered by the segment.
    """

    def __init__(self, space, elements, binding):
        assert isinstance(space, Space)
        assert isinstance(elements, listof(Code))
        assert isinstance(binding, SegmentBinding)
        super(SegmentExpr, self).__init__(binding)
        self.space = space
        self.elements = elements


class Family(object):

    is_scalar = False
    is_table = False
    is_kernel = False


class ScalarFamily(Family):

    is_scalar = True


class TableFamily(Family):

    is_table = True

    def __init__(self, table):
        assert isinstance(table, TableEntity)
        self.table = table


class KernelFamily(Family):

    is_kernel = True

    def __init__(self, seed, seed_baseline, kernel):
        assert isinstance(seed, Space)
        assert isinstance(seed_baseline, Space)
        assert seed_baseline.is_axis and seed.concludes(seed_baseline)
        assert isinstance(kernel, listof(Code))
        self.seed = seed
        self.seed_baseline = seed_baseline
        self.kernel = kernel


class Space(Expression):
    """
    Represents a space node.

    A space is an expression that represents an (ordered multi-) set of rows.
    Among others, we consider the following kinds of spaces:

    *The scalar space* `I`
        A space with only one row.

    *A table space* `T`
        Given a table `T`, the space consists of all rows of the table.

    *A direct table space* `A * T`
        Given a table `T` and another space `A`, the direct table space
        consists of pairs `(a, t)` where `a` runs over rows of `A` and
        `t` runs over rows of `T`.

        Note that a table space is a special case of a direct table space:
        `T` is equivalent to `I * T`.

        Table `T` in `A * T` is called *the prominent table* of the space.

    *A fiber table space* `A . T` or `A .j T`
        Given a space `A` with the prominent table `S`, another table `T`
        and a join condition `j` between tables `S` and `T`, the fiber
        table space consists of pairs `(a, t)` from `A * T` satisfying
        the join condition `j`.

        Table `T` is called the prominent table of `A . T`.

    *A filtered space* `A ? p`
        Given a space `A` and a predicate `p` defined on `A`, the
        filtered space consists of rows of `A` satisfying condition `p`.

    *An ordered space* `A [e,...]`
        Given a space `A` and a list of expressions `e,...`, the
        ordered space consists of rows of `A` reordered by the values
        of expressions.

    Note that all these examples (except for the scalar space) share the same
    form: they take an existing space, called *the base space* and apply some
    operation to produce a new space.  Thus *any space could be expressed
    as an application of a series of elementary operations to the scalar
    space*.

    Each subclass of :class:`Space` represents an operation that is applied
    to a base space.  We could classify the operations (and therefore
    :class:`Space` subclasses) into two groups: those which keep the row
    shape of the base space and those which expand it.  The latter are
    called *axis spaces*.  We also regard the scalar space as an axis space.

    Take an arbitrary space `A` and consider it as a sequence of
    operations applied to the scalar space.  If we then reapply only
    the axis operations from the sequence, we obtain a new space `A'`,
    which we call *the inflation* of `A`.  Note that `A` is a subset
    of the inflated space `A'`.

    Now we can establish how different spaces are related to each other.
    For that we will introduce a notion of *convergency* between two
    arbitrary spaces.  Informally, convergency describes how two spaces
    `A` and `B` can be naturally attached to each other.  When rows
    in `A` and `B` have the same shape, convergency is reduced to equality,
    that is, a row from `A` converges to an equal row from `B` if the latter
    exists.  When rows in `A` and `B` have different shapes, we need
    to determine their longest common prefix.  Then a row from `A`
    converges to all rows from `B` that share the same prefix values.

    Formally, for each pair of spaces `A` and `B`, we define a relation
    `<->` ("converges to") on rows from `A` and `B`, that is, a subset
    of the Cartesian product `A x B`, by the following rules:

    (1) For any space `A`, `<->` is the identity relation on `A`,
        that is, each row converges only to itself.

        For two spaces `A` and `B` where `A` is a subset of `B`,
        each row from `A` converges to an equal row from `B`.  In
        particular, this defines `<->` on any space `A` and its inflated
        space `A'`, as well as on any non-axis space `A` and its base
        space `B`.

    (2) Suppose `A` and `B` are spaces such that `A` is an axis space
        and `B` is the base of `A`.  It means that each element of `A`
        has the form `(b, t)` where `b` is some row from `B`. Then
        row `a` from `A` converges to row `b` from `B` if `a` has the
        form `a = (b, t)` for some `t`.

        By transitivity, we could extend `<->` on `A` and any of its
        *prefix spaces*, that is, the base of `A`, the base of the base
        of `A` and so on.  For instance, let `B` be the base space
        of `A` and let `C` be the base space of `B`.  Then `a` from `A`
        converges to `c` from `C` if there exists row `b` from `B`
        such that `a <-> b` and `b <-> c`.

        In particular, this defines `<->` on an arbitrary space `A`
        and the scalar space `I` since `I` is a prefix for any space.
        By the above definition, any row of `A` converges to the (only)
        row of `I`.

    (3) Finally, we are ready to define `<->` on an arbitrary pair
        of spaces `A` and `B`.  First, suppose that `A` and `B`
        share the same inflated space: `A' = B'`.  Then we could
        define `<->` on `A` and `B` transitively via `A'`: `a` from `A`
        converges to `b` from `B` if there exists `a'` from `A'` such
        that `a <-> a'` and `a' <-> b`.

        In the general case, find the longest prefixes `C` of `A`
        and `D` of `B` such that `C` and `D` have the same
        inflated space: `C' = D'`.  Rules `(1)` and `(2)` establish
        `<->` for the pairs `A` and `C`, `C` and `C' = D'`,
        `C' = D'` and `D`, and `D` and `B`.  We define `<->`
        on `A` and `B` transitively: `a` from `A` converges to
        `b` from `B` if there exist rows `c` from `C`,
        `c'` from `C' = D'`, `d` from `D` such that
        `a <-> c <-> c' <-> d <-> b`.

        Note that it is important that we take among the common inflated
        prefixes the longest one.  Any two spaces have a common inflated
        prefix: the scalar space.  If the scalar space is, indeed, the
        longest common inflated prefix of `A` and `B`, then each
        row of `A` converges to every row of `B`.

    Now we are ready to introduce several very important relations between
    spaces:

    `A` *spans* `B`
        A space `A` spans a space `B` if for every row `a` from `A`:

            `card { b` from `B | a <-> b } <= 1`.

        Informally, it means that the statement::

            SELECT * FROM A

        and the statement::

            SELECT * FROM A LEFT OUTER JOIN B ON (A <-> B)

        produce the same number of rows.

    `A` *dominates* `B`
        A space `A` dominates a space `B` if `A` spans `B` and
        for every row `b` from `B`:

            `card { a` from `A | a <-> b } >= 1`.

        Informally, it implies that the statement::

            SELECT * FROM B INNER JOIN A ON (A <-> B)

        and the statement::

            SELECT * FROM B LEFT OUTER JOIN A ON (A <-> B)

        produce the same number of rows.

    `A` *conforms* `B`
        A space `A` conforms a space `B` if `A` dominates `B`
        and `B` dominates `A`.  Alternatively, we could say
        `A` conforms `B` if the `<->` relation establishes
        a bijection between `A` and `B`.

        Informally, it means that the statement::

            SELECT * FROM A

        and the statement::

            SELECT * FROM B

        produce the same number of rows.

        Note that `A` conforming `B` is not the same as `A` being equal
        to `B`; even if `A` conforms `B`,  rows of `A` and `B` may
        have different shapes, therefore as sets, they are different.

    Now take an arbitrary space `A` and its base space `B`.  We say:

    `A` *contracts* `B`
        A space `A` contracts its base `B` if for any row from `B`
        there is no more than one converging row from `A`.

        Typically, it is non-axis spaces that contract their bases,
        although in some cases, an axis space could do it too.

    `A` *expands* `B`
        A space `A` expands its base `B` if for any row from `B`
        there is at least one converging row from `A`.

        Note that it is possible that a space `A` both contracts and
        expands its base `B`, and also that `A` neither contracts
        nor expands `B`.  The former means that `A` conforms `B`.
        The latter holds, in particular, for the direct table space
        `A * T`.  `A * T` violates the contraction condition when
        `T` contains more than one row and violates the expansion
        condition when `T` has no rows.

    A few words about how rows of a space are ordered.  The default
    (also called *weak*) ordering rules are:

    - a table space `T = I * T` is sorted by the lexicographic order
      of the table primary key;

    - a non-axis space keeps the order of its base;

    - an axis space `A * T` or `A . T` respects the order its base `A`;
      rows with the same base element are sorted by the table order.

    An alternative sort order could be specified explicitly (also called
    *strong* ordering).  Whenever strong ordering is  specified, it
    overrides the weak ordering.  Thus, rows of an ordered space `A [e]`
    are sorted first by expression `e`, and then rows which are not
    differentiated by `e` are sorted using the weak ordering of `A`.
    However, if `A` already has a strong ordering, it must be respected.
    Therefore, the general rule for sorting `A [e]` is:

    - first, sort the space by the strong ordering of `A`;

    - then, by `e`;

    - finally, by the weak ordering of `A`.

    Class attributes:

    `is_axis` (Boolean)
        Indicates whether the space is an axis space, that is, the shape
        of the space rows differs from the shape of its base.

    `is_root` (Boolean)
        Indicates if the space is the root space.

    The constructor arguments:

    `base` (:class:`Space` or ``None``)
        The base space; ``None`` for the root space.

    `table` (:class:`htsql.entity.TableEntity` or ``None``)
        The prominent table of the space; ``None`` if the space has no
        prominent table.

    `is_contracting` (Boolean)
        Indicates if the space contracts its base space.

    `is_expanding` (Boolean)
        Indicates if the space expands its base space.

    Other attributes:

    `is_inflated` (Boolean)
        Indicates if the space is an inflation, that is, the space itself
        and all its prefixes are axis spaces.

    `root` (:class:`RootSpace`)
        The root scalar space.
    """

    is_axis = False
    is_root = False

    def __init__(self, base, family,
                 is_contracting, is_expanding,
                 binding, equality_vector=None):
        assert isinstance(base, maybe(Space))
        assert isinstance(family, Family)
        assert isinstance(is_contracting, bool)
        assert isinstance(is_expanding, bool)
        super(Space, self).__init__(binding, equality_vector)
        self.base = base
        self.family = family
        self.is_contracting = is_contracting
        self.is_expanding = is_expanding
        # Indicates that the space itself and all its prefixes are axes.
        self.is_inflated = (self.is_root or
                            (base.is_inflated and self.is_axis))
        # Extract the root scalar space from the base.
        self.root = (base.root if not self.is_root else self)

    def unfold(self):
        """
        Produces a list of prefix spaces.

        The method returns a list composed of the space itself,
        its base, the base of its base and so on.
        """
        prefixes = []
        prefix = self
        while prefix is not None:
            prefixes.append(prefix)
            # Note: `prefix.base` is None for the root space.
            prefix = prefix.base
        return prefixes

    def resembles(self, other):
        """
        Verifies if the spaces represent the same operation.

        Typically, it means that `self` and `other` have the same type
        and equal attributes, but may have different bases.
        """
        # We rely upon an assumption that the equality vector of a space node
        # is a tuple of all its essential attributes and the first element
        # of the tuple is the space base.  So we skip the base space and
        # compare the remaining attributes.
        return (isinstance(other, self.__class__) and
                self.equality_vector[1:] == other.equality_vector[1:])

    def inflate(self):
        """
        Produces the inflation of the space.

        If we represent a space as a series of operations sequentially
        applied to the scalar space, the inflation of the space is obtained
        by ignoring any non-axis operations and applying axis operations
        only.
        """
        # Shortcut: check if the space is already an inflation.
        if self.is_inflated:
            return self
        # This is going to become a new inflated space.
        space = None
        # Iterate over all prefixes starting from the scalar space.
        for prefix in reversed(self.unfold()):
            # Skip non-axis operations, reapply axis operations to
            # a new base.
            if prefix.is_axis:
                space = prefix.clone(base=space)
        # This is the inflated space now.
        return space

    def prune(self, other):
        """
        Prunes shared non-axis operations.

        Given spaces `A` and `B`, this function produces a new space
        `A'` such that `A` is a subset of `A'` and the convergence
        of `A` and `B` coincides with the convergence of `A'` and `B`.
        This is done by pruning any non-axis operations of `A` that
        also occurs in `B`.
        """
        # Sanity check on the argument.
        assert isinstance(other, Space)
        # Shortcut: we cannot further prune an inflated space.
        if self.is_inflated:
            return self
        # Unfold the spaces into individual operations.
        my_prefixes = self.unfold()
        their_prefixes = other.unfold()
        # This is going to become the pruned space.
        space = None
        # Iterate until the prefixes are exhausted or diverged.
        while my_prefixes and their_prefixes:
            # Get the next operation.
            my_prefix = my_prefixes[-1]
            their_prefix = their_prefixes[-1]
            # Compare the prefixes.
            if my_prefix.resembles(their_prefix):
                # So both prefixes represent the same operation.
                # If it is an axis operation, apply it; otherwise,
                # discard it.
                # FIXME: may break if the space contains a non-matching
                # `limit/offset` operation?
                if my_prefix.is_axis:
                    space = my_prefix.clone(base=space)
                my_prefixes.pop()
                their_prefixes.pop()
            elif not their_prefix.is_axis:
                # The prefixes represent different operations and `B`'s prefix
                # is not an axis.  Discard it, we will try the next prefix.
                # FIXME: we may miss an opportunity to compare `B`'s prefix
                # with other `A`'s prefixes.  It is not a big deal though,
                # we do not need to generate an optimal result here.
                their_prefixes.pop()
            elif not my_prefix.is_axis:
                # The prefixes represent different operations, `B`'s prefix
                # is an axis, and `A`'s prefix is not.  Here we apply the
                # `A`'s prefix.
                space = my_prefix.clone(base=space)
                my_prefixes.pop()
            else:
                # The prefixes are both axes and differ from each other.
                # At this point, the prefixes diverge and are not
                # comparable anymore.  Break from the loop.
                break
        # Reapply the unprocessed prefixes.
        while my_prefixes:
            my_prefix = my_prefixes.pop()
            space = my_prefix.clone(base=space)
        # We have a pruned space here.
        return space

    def spans(self, other):
        """
        Verifies if the space spans another space.
        """
        # Sanity check on the argument.
        assert isinstance(other, Space)
        # Shortcut: any space spans itself.
        if self == other:
            return True
        # Extract axis prefixes from both spaces.
        my_axes = [prefix for prefix in self.unfold() if prefix.is_axis]
        their_axes = [prefix for prefix in other.unfold() if prefix.is_axis]
        # Iterate until the axes are exhausted or diverged.
        while my_axes and their_axes:
            # Check if the next pair of axes represent the same operation.
            if my_axes[-1].resembles(their_axes[-1]):
                # If so, go compare the next pair of axes.
                my_axes.pop()
                their_axes.pop()
            else:
                # Otherwise, the axes diverge.
                break
        # At this point, all processed pairs of axes converge by identity.
        # If the other space has no more axes left, it is spanned.  Otherwise,
        # it is spanned only if its remaining unprocessed axes represent
        # contracting operations.
        for their_axis in their_axes:
            if not their_axis.is_contracting:
                return False
        return True

    def conforms(self, other):
        """
        Verifies if the space conforms another space.
        """
        # Sanity check on the argument.
        assert isinstance(other, Space)
        # Shortcut: any space conforms itself.
        if self == other:
            return True
        # Unfold the spaces into individual operations.
        my_prefixes = self.unfold()
        their_prefixes = other.unfold()
        # Iterate until the prefixes are exhausted or diverged.
        while my_prefixes and their_prefixes:
            # Get the next pair of prefixes.
            my_prefix = my_prefixes[-1]
            their_prefix = their_prefixes[-1]
            # Compare the prefixes.
            if my_prefix.resembles(their_prefix):
                # If the prefixes represent the same operation, we could
                # proceed to the next pair of prefixes.
                my_prefixes.pop()
                their_prefixes.pop()
            elif (my_prefix.is_contracting and
                  my_prefix.is_expanding and
                  not my_prefix.is_axis):
                # Ok, the prefixes represent different operations, but
                # one of them is not an axis and does not change the
                # cardinality of its base.  We could skip this prefix
                # and proceed further.
                my_prefixes.pop()
            elif (their_prefix.is_contracting and
                  their_prefix.is_expanding and
                  not their_prefix.is_axis):
                # Same with the other prefix.
                their_prefixes.pop()
            else:
                # The prefixes start to diverge; break from the loop.
                break
        # If all prefixes are processed, the spaces conform each other.
        # Otherwise, they conform each other only if the remaining unprocessed
        # prefixes do not change the cardinality of their bases.
        for prefix in my_prefixes + their_prefixes:
            if not (prefix.is_contracting and prefix.is_expanding):
                return False
        return True

    def dominates(self, other):
        """
        Verifies if the space dominates another space.
        """
        # Sanity check on the argument.
        assert isinstance(other, Space)
        # Shortcut: any space dominates itself.
        if self == other:
            return True
        # Unfold the spaces into individual operations.
        my_prefixes = self.unfold()
        their_prefixes = other.unfold()
        # Iterate until the prefixes are exhausted or diverged.
        while my_prefixes and their_prefixes:
            # Get the next pair of prefixes.
            my_prefix = my_prefixes[-1]
            their_prefix = their_prefixes[-1]
            # Compare the prefixes.
            if my_prefix.resembles(their_prefix):
                # If the prefixes represent the same operation, we could
                # proceed to the next pair of prefixes.
                my_prefixes.pop()
                their_prefixes.pop()
            elif their_prefix.is_contracting and not their_prefix.is_axis:
                # We got prefixes representing different operations; however
                # the dominated prefix represents a non-axis operation that
                # does not increase the cardinality of its base.  Therefore
                # we could ignore this prefix and proceed further.
                their_prefixes.pop()
            else:
                # The prefixes start to diverge; break from the loop.
                break
        # If all prefixes are processed, the space dominates the other.
        # Otherwise, it is only possible if the remaining prefixes of
        # the space do not decrease the base cardinality while the
        # remaining prefixes of the other space do not increase the
        # base cardinality.
        for my_prefix in my_prefixes:
            if not my_prefix.is_expanding:
                return False
        for their_prefix in their_prefixes:
            if not their_prefix.is_contracting:
                return False
        return True

    def concludes(self, other):
        """
        Verifies if the other space is a prefix of the space.
        """
        # Sanity check on the argument.
        assert isinstance(other, Space)
        # Iterate over all prefixes of the space comparing them with
        # the given other space.
        space = self
        while space is not None:
            if space == other:
                return True
            space = space.base
        # None of the prefixes matched, the spaces must be unrelated.
        return False


class RootSpace(Space):
    """
    Represents a root scalar space.

    A scalar space `I` contains one row ``()``.  Any other space
    is generated by applying a sequence of elementary operations
    to `I`.

    `base` (always ``None``)
        The scalar space (and only the scalar space) has no base.
    """

    # Scalar space is an axis space.
    is_axis = True
    is_root = True

    def __init__(self, base, binding):
        # We keep `base` among constructor arguments despite it always being
        # equal to `None` to make
        #   space = space.clone(base=new_base)
        # work for all types of spaces.
        assert base is None
        # Note that we must satisfy the assumption that the first element
        # of the equality vector is the space base (used by `Space.resembles`).
        super(RootSpace, self).__init__(
                    base=None,
                    family=ScalarFamily(),
                    is_contracting=False,
                    is_expanding=False,
                    binding=binding,
                    equality_vector=(base,))

    def __str__(self):
        # Display a table expression in an algebraic form.
        return "I"


class ScalarSpace(Space):

    is_axis = True

    def __init__(self, base, binding):
        super(ScalarSpace, self).__init__(
                    base=base,
                    family=ScalarFamily(),
                    is_contracting=True,
                    is_expanding=True,
                    binding=binding,
                    equality_vector=(base,))

    def __str__(self):
        return "(%s * I)" % self.base


class TableSpace(Space):
    """
    Represents a table space.

    A table space is a subset of a Cartesian product between the base
    space and a table.  This is an abstract class, see concrete subclasses
    :class:`DirectTableSpace` and :class:`FiberTableSpace`.

    `table` (:class:`htsql.entity.TableEntity`)
        The prominent table of the product.
    """

    # All subclasses of `TableSpace` are axis spaces.
    is_axis = True


class DirectTableSpace(TableSpace):
    """
    Represents a direct table space.

    A direct table space `A * T` consists of all pairs `(a, t)` where
    `a` is a row of the base space `A` and `t` is a row of the table `T`.

    `base` (:class:`Space`)
        The base space.

    `table` (:class:`htsql.entity.TableEntity`)
        The prominent table.
    """

    def __init__(self, base, table, binding):
        assert isinstance(base, Space) and base.family.is_scalar
        super(DirectTableSpace, self).__init__(
                    base=base,
                    family=TableFamily(table),
                    is_contracting=False,
                    is_expanding=False,
                    binding=binding,
                    equality_vector=(base, table))
        self.table = table

    def __str__(self):
        # Display:
        #   (<base> * schema.table)
        return "(%s * %s)" % (self.base, self.family.table)


class FiberTableSpace(TableSpace):
    """
    Represents a fiber table space.

    Let `A` be a space with the prominent table `S`, `j` be a join
    condition between tables `S` and `T`.  A fiber table space `A .j T`
    (or `A . T` when the join condition is implied) of the space `A`
    and the table `T` consists of all pairs `(a, t)`, where `a` is a row
    from `A` of the form `a = (..., s)` and `t` is a row from `T` such
    that `s` and `t` satisfy the join condition `j`.

    `base` (:class:`Space`)
        The base space.

    `join` (:class:`htsql.entity.Join`)
        The join condition.
    """

    def __init__(self, base, join, binding):
        assert isinstance(join, Join)
        # Check that the join origin is the prominent table of the base.
        assert isinstance(base, Space) and base.family.is_table
        assert base.family.table is join.origin
        super(FiberTableSpace, self).__init__(
                    base=base,
                    family=TableFamily(join.target),
                    is_contracting=join.is_contracting,
                    is_expanding=join.is_expanding,
                    binding=binding,
                    equality_vector=(base, join))
        self.join = join

    def __str__(self):
        # Display:
        #   (<base> . schema.table)
        return "(%s . %s)" % (self.base, self.family.table)


class QuotientSpace(Space):

    is_axis = True

    def __init__(self, base, seed, kernel, binding):
        assert isinstance(base, Space)
        assert isinstance(seed, Space)
        assert seed.spans(base)
        assert not base.spans(seed)
        seed_baseline = seed
        while not base.spans(seed_baseline.base):
            seed_baseline = seed_baseline.base
        assert isinstance(kernel, listof(Code))
        super(QuotientSpace, self).__init__(
                    base=base,
                    family=KernelFamily(seed, seed_baseline, kernel),
                    is_contracting=(not kernel),
                    is_expanding=(base.is_root and not kernel),
                    binding=binding,
                    equality_vector=(base, seed, tuple(kernel)))
        self.seed = seed
        self.seed_baseline = seed_baseline
        self.kernel = kernel


class ComplementSpace(Space):

    is_axis = True

    def __init__(self, base, binding, extra_codes=None):
        assert isinstance(base, Space)
        assert base.family.is_kernel
        assert isinstance(extra_codes, maybe(listof(Code)))
        super(ComplementSpace, self).__init__(
                    base=base,
                    family=base.family.seed.family,
                    is_contracting=False,
                    is_expanding=True,
                    binding=binding,
                    equality_vector=(base,))
        self.extra_codes = extra_codes


class MonikerSpace(Space):

    is_axis = True

    def __init__(self, base, seed, binding, extra_codes=None):
        assert isinstance(base, Space)
        assert isinstance(seed, Space)
        assert seed.spans(base)
        #assert not base.spans(seed)
        seed_baseline = seed
        while not seed_baseline.is_axis:
            seed_baseline = seed_baseline.base
        if not base.spans(seed_baseline):
            while not base.spans(seed_baseline.base):
                seed_baseline = seed_baseline.base
        super(MonikerSpace, self).__init__(
                    base=base,
                    family=seed.family,
                    is_contracting=base.spans(seed),
                    is_expanding=seed.dominates(base),
                    binding=binding,
                    equality_vector=(base, seed))
        self.seed = seed
        self.seed_baseline = seed_baseline
        self.extra_codes = extra_codes


class ForkedSpace(Space):

    is_axis = True

    def __init__(self, base, seed, kernel, binding, extra_codes=None):
        assert isinstance(base, Space)
        assert isinstance(seed, Space)
        assert isinstance(kernel, listof(Code))
        assert base.spans(seed) and seed.spans(base)
        # FIXME: this condition could be violated after the rewrite step:
        #assert base.family == seed.family
        assert all(base.spans(unit.space) for code in kernel
                                          for unit in code.units)
        assert isinstance(extra_codes, maybe(listof(Code)))
        seed_baseline = seed
        while not seed_baseline.is_axis:
            seed_baseline = seed_baseline.base
        super(ForkedSpace, self).__init__(
                    base=base,
                    family=base.family,
                    is_contracting=seed_baseline.is_contracting,
                    is_expanding=seed.dominates(base),
                    binding=binding,
                    equality_vector=(base, seed, tuple(kernel)))
        self.seed = seed
        self.seed_baseline = seed_baseline
        self.kernel = kernel
        self.extra_codes = extra_codes


class LinkedSpace(Space):

    is_axis = True

    def __init__(self, base, seed, kernel, counter_kernel,
                 binding, extra_codes=None):
        assert isinstance(base, Space)
        assert isinstance(seed, Space)
        assert seed.spans(base)
        assert not base.spans(seed)
        assert isinstance(kernel, listof(Code))
        assert isinstance(counter_kernel, listof(Code))
        assert len(kernel) == len(counter_kernel)
        assert all(seed.spans(unit.space) for code in kernel
                                          for unit in code.units)
        assert all(base.spans(unit.space) for code in counter_kernel
                                          for unit in code.units)
        seed_baseline = seed
        if not base.spans(seed_baseline):
            while not base.spans(seed_baseline.base):
                seed_baseline = seed_baseline.base
        super(LinkedSpace, self).__init__(
                    base=base,
                    family=seed.family,
                    is_contracting=False,
                    is_expanding=False,
                    binding=binding,
                    equality_vector=(base, seed, tuple(kernel),
                                     tuple(counter_kernel)))
        self.seed = seed
        self.seed_baseline = seed_baseline
        self.kernel = kernel
        self.counter_kernel = counter_kernel
        self.extra_codes = extra_codes


class FilteredSpace(Space):
    """
    Represents a filtered space.

    A filtered space `A ? f`, where `A` is the base space and `f` is
    a predicate expression on `A`, consists of rows of `A` satisfying
    the condition `f`.

    `base` (:class:`Space`)
        The base space.

    `filter` (:class:`Code`)
        The predicate expression.
    """

    def __init__(self, base, filter, binding):
        assert isinstance(filter, Code)
        assert isinstance(filter.domain, BooleanDomain)
        super(FilteredSpace, self).__init__(
                    base=base,
                    family=base.family,
                    is_contracting=True,
                    is_expanding=False,
                    binding=binding,
                    equality_vector=(base, filter))
        self.filter = filter

    def __str__(self):
        # Display:
        #   (<base> ? <filter>)
        return "(%s ? %s)" % (self.base, self.filter)


class OrderedSpace(Space):
    """
    Represents an ordered space.

    An ordered space `A [e,...;p:q]` is a space with explicitly specified
    strong ordering.  It also may extract a slice of the base space.

    `base` (:class:`Space`)
        The base space.

    `order` (a list of pairs `(code, direction)`)
        Expressions to sort the space by.

        Here `code` is a :class:`Code` instance, `direction` is either
        ``+1`` (indicates ascending order) or ``-1`` (indicates descending
        order).

    `limit` (a non-negative integer or ``None``)
        If set, the space extracts the first `limit` rows from the base
        space (with respect to the space ordering).  The remaining rows
        are discarded.

    `offset` (a non-negative integer or ``None``)
        If set, indicates that when extracting rows from the base space,
        the first `offset` rows should be skipped.
    """

    # FIXME: Non-commutativity of the ordered space may affect `prune`
    # and other functions.  Add class attribute `is_commutative`?
    # Or override `resembles` to return `True` only for equal nodes?

    def __init__(self, base, order, limit, offset, binding):
        assert isinstance(order, listof(tupleof(Code, int)))
        assert isinstance(limit, maybe(int))
        assert isinstance(offset, maybe(int))
        assert limit is None or limit >= 0
        assert offset is None or offset >= 0
        super(OrderedSpace, self).__init__(
                    base=base,
                    family=base.family,
                    is_contracting=True,
                    is_expanding=(limit is None and offset is None),
                    binding=binding,
                    equality_vector=(base, tuple(order)))
        self.order = order
        self.limit = limit
        self.offset = offset

    def __str__(self):
        # Display:
        #   <base> [<code>,...;<offset>:<limit>+<offset>]
        indicators = []
        if self.order:
            indicator = ",".join(str(code) for code, dir in self.order)
            indicators.append(indicator)
        if self.limit is not None and self.offset is not None:
            indicator = "%s:%s+%s" % (self.offset, self.offset, self.limit)
            indicators.append(indicator)
        elif self.limit is not None:
            indicator = ":%s" % self.limit
            indicators.append(indicator)
        elif self.offset is not None:
            indicator = "%s:" % self.offset
            indicators.append(indicator)
        indicators = ";".join(indicators)
        return "%s [%s]" % (self.base, indicators)


class Code(Expression):
    """
    Represents a code expression.

    A code expression is a function on spaces.  Specifically, it is a
    functional (possibly of several variables) that maps a space
    (or a Cartesian product of several spaces) to some scalar domain.
    :class:`Code` is an abstract base class for all code expressions;
    see its subclasses for concrete types of expressions.

    Among all code expressions, we distinguish *unit expressions*:
    elementary functions on spaces.  There are several kinds of units:
    among them are columns and aggregate functions (see :class:`Unit`
    for more detail).  A non-unit code could be expressed as
    a composition of a scalar function and one or several units:

        `f = F(u(a),v(b),...)`,

    where

    - `f` is a code expression;
    - `F` is a scalar function;
    - `a`, `b`, ... are elements of spaces `A`, `B`, ...;
    - `u`, `v`, ... are unit expressions on `A`, `B`, ....

    Note: special forms like `COUNT` or `EXISTS` are also expressed
    as code nodes.  Since they are not regular functions, special care
    must be taken to properly wrap them with appropriate
    :class:`ScalarUnit` and/or :class:`AggregateUnit` instances.

    `domain` (:class:`htsql.domain.Domain`)
        The co-domain of the code expression.

    `units` (a list of :class:`Unit`)
        The unit expressions of which the code is composed.
    """

    def __init__(self, domain, units, binding, equality_vector=None):
        assert isinstance(domain, Domain)
        assert isinstance(units, listof(Unit))
        super(Code, self).__init__(binding, equality_vector)
        self.domain = domain
        self.units = units


class LiteralCode(Code):
    """
    Represents a literal value.

    `value` (valid type depends on the domain)
        The value.

    `domain` (:class:`htsql.domain.Domain`)
        The value type.
    """

    def __init__(self, value, domain, binding):
        super(LiteralCode, self).__init__(
                    domain=domain,
                    units=[],
                    binding=binding,
                    equality_vector=(value, domain))
        self.value = value

    def __str__(self):
        # The actual value is often more helpful than the expression
        # that generated it.
        return repr(self.value)


class CastCode(Code):
    """
    Represents a type conversion operator.

    `base` (:class:`Code`)
        The expression to convert.

    `domain` (:class:`htsql.domain.Domain`)
        The target domain.
    """

    def __init__(self, base, domain, binding):
        super(CastCode, self).__init__(
                    domain=domain,
                    units=base.units,
                    binding=binding,
                    equality_vector=(base, domain))
        self.base = base


class FormulaCode(Formula, Code):
    """
    Represents a formula code.

    A formula code represents a function or an operator call as a code node.

    `signature` (:class:`htsql.tr.signature.Signature`)
        The signature of the formula.

    `domain` (:class:`Domain`)
        The co-domain of the formula.

    `arguments` (a dictionary)
        The arguments of the formula.

        Note that all the arguments become attributes of the node object.
    """

    def __init__(self, signature, domain, binding, **arguments):
        assert isinstance(signature, Signature)
        # Check that the arguments match the formula signature.
        arguments = Bag(**arguments)
        assert arguments.admits(Code, signature)
        # Extract unit nodes from the arguments.
        units = []
        for cell in arguments.cells():
            units.extend(cell.units)
        equality_vector = (signature, domain, arguments.freeze())
        # The first two arguments are processed by the `Formula`
        # constructor, the rest of them go to the `Binding` constructor.
        super(FormulaCode, self).__init__(
                    signature, arguments,
                    domain=domain,
                    units=units,
                    binding=binding,
                    equality_vector=equality_vector)


class Unit(Code):
    """
    Represents a unit expression.

    A unit is an elementary function on a space.  There are several kinds
    of units; see subclasses :class:`ColumnUnit`, :class:`ScalarUnit`,
    :class:`AggregateUnit`, and :class:`CorrelatedUnit` for more detail.

    Units are divided into two categories: *primitive* and *compound*.

    A primitive unit is an intrinsic function of its space; no additional
    calculations are required to generate a primitive unit.  Currently,
    the only example of a primitive unit is :class:`ColumnUnit`.

    A compound unit requires calculating some non-intrinsic function
    on the target space.  There are three types of compound units:
    :class:`ScalarUnit`, :class:`AggregateUnit` and :class:`CorrelatedUnit`.
    They correspond respectively to a scalar function and two kinds of
    an aggregate function.

    Note that it is easy to *lift* a unit code from one space to another.
    Specifically, suppose a unit `u` is defined on a space `A` and `B`
    is another space such that `B` spans `A`.  Then for each row `b`
    from `B` there is no more than one row `a` from `A` such that `a <-> b`.
    Therefore we could define `u` on `B` as follows:

    - `u(b) = u(a)` if there exists `a` from `A` such that `a <-> b`;
    - `u(b) =` ``NULL`` if there is no rows in `A` convergent to `b`.

    When a space `B` spans the space `A` of a unit `u`, we say that
    `u` is *singular* on `B`.  By the previous argument, `u` could be
    lifted to `B`.  Thus any unit is well-defined not only on the
    space where it is originally defined, but also on any space where
    it is singular.

    Attributes:

    `space` (:class:`Space`)
        The space on which the unit is defined.

    `domain` (:class:`htsql.domain.Domain`)
        The unit co-domain.

    Class attributes:

    `is_primitive` (Boolean)
        If set, indicates that the unit is primitive.

    `is_compound` (Boolean)
        If set, indicates that the unit is compound.
    """

    is_primitive = False
    is_compound = False

    def __init__(self, space, domain, binding, equality_vector=None):
        assert isinstance(space, Space)
        super(Unit, self).__init__(
                    domain=domain,
                    units=[self],
                    binding=binding,
                    equality_vector=equality_vector)
        self.space = space

    def singular(self, space):
        """
        Verifies if the unit is singular (well-defined) on the given space.
        """
        return space.spans(self.space)


class PrimitiveUnit(Unit):
    """
    Represents a primitive unit.

    A primitive unit is an intrinsic function on a space.

    This is an abstract class; for the (only) concrete subclass, see
    :class:`ColumnUnit`.
    """

    is_primitive = True


class CompoundUnit(Unit):
    """
    Represents a compound unit.

    A compound unit is some non-intrinsic function on a space.

    This is an abstract class; for concrete subclasses, see
    :class:`ScalarUnit`, :class:`AggregateUnit`, :class:`CorrelatedUnit`.

    `code` (:class:`Code`)
        The expression to evaluate on the unit space.
    """

    is_compound = True

    def __init__(self, code, space, domain, binding, equality_vector=None):
        assert isinstance(code, Code)
        super(CompoundUnit, self).__init__(
                    space=space,
                    domain=domain,
                    binding=binding,
                    equality_vector=equality_vector)
        self.code = code


class ColumnUnit(PrimitiveUnit):
    """
    Represents a column unit.

    A column unit is a function on a space that returns a column of the
    prominent table of the space.

    `column` (:class:`htsql.entity.ColumnEntity`)
        The column produced by the unit.

    `space` (:class:`Space`)
        The unit space.  Note that the prominent table of the space
        must coincide with the table of the column.
    """

    def __init__(self, column, space, binding):
        assert isinstance(column, ColumnEntity)
        assert (space.family.is_table and
                (space.family.table.schema_name, space.family.table.name)
                    == (column.schema_name, column.table_name))
        super(ColumnUnit, self).__init__(
                    space=space,
                    domain=column.domain,
                    binding=binding,
                    equality_vector=(column, space))
        self.column = column


class ScalarUnit(CompoundUnit):
    """
    Represents a scalar unit.

    A scalar unit is an expression evaluated in the specified space.

    Recall that any expression has the following form:

        `F(u(a),v(b),...)`,

    where

    - `F` is a scalar function;
    - `a`, `b`, ... are elements of spaces `A`, `B`, ...;
    - `u`, `v`, ... are unit expressions on `A`, `B`, ....

    We require that the units of the expression are singular on the given
    space.  If so, the expression units `u`, `v`, ... could be lifted to
    the given slace (see :class:`Unit`).  The scalar unit is defined as

        `F(u(x),v(x),...)`,

    where `x` is a row of the space where the scalar unit is defined.

    `code` (:class:`Code`)
        The expression to evaluate.

    `space` (:class:`Space`)
        The space on which the unit is defined.
    """

    def __init__(self, code, space, binding):
        super(ScalarUnit, self).__init__(
                    code=code,
                    space=space,
                    domain=code.domain,
                    binding=binding,
                    equality_vector=(code, space))


class AggregateUnitBase(CompoundUnit):
    """
    Represents an aggregate unit.

    Aggregate units express functions on sets.  Specifically, let `A` and `B`
    be spaces such that `B` spans `A`, but  `A` does not span `B`, and
    let `g` be a function that takes subsets of `B` as an argument.  Then
    we could define an aggregate unit `u` on `A` as follows:

        `u(a) = g({b | a <-> b})`

    Here, for each row `a` from `A`, we take the subset of convergent
    rows from `B` and apply `g` to it; the result is the value of `u(a)`.

    The space `A` is the unit space, the space `B` is called *the plural
    space* of an aggregate unit, and `g` is called *the composite expression*
    of an aggregate unit.

    `code` (:class:`Code`)
        The composite expression of the aggregate unit.

    `plural_space` (:class:`Space`)
        The plural space of the aggregate unit, that is, the space
        which subsets form the argument of the composite expression.

    `space` (:class:`Space`)
        The space on which the unit is defined.
    """

    def __init__(self, code, plural_space, space, binding):
        assert isinstance(code, Code)
        assert isinstance(plural_space, Space)
        # FIXME: consider lifting the requirement that the plural
        # space spans the unit space.  Is it really necessary?
        assert plural_space.spans(space)
        assert not space.spans(plural_space)
        super(AggregateUnitBase, self).__init__(
                    code=code,
                    space=space,
                    domain=code.domain,
                    binding=binding,
                    equality_vector=(code, plural_space, space))
        self.plural_space = plural_space


class AggregateUnit(AggregateUnitBase):
    """
    Represents a regular aggregate unit.

    A regular aggregate unit is expressed in SQL using an aggregate
    expression with ``GROUP BY`` clause.
    """


class CorrelatedUnit(AggregateUnitBase):
    """
    Represents a correlated aggregate unit.

    A correlated aggregate unit is expressed in SQL using a correlated
    subquery.
    """


class KernelUnit(CompoundUnit):

    def __init__(self, code, space, binding):
        assert space.family.is_kernel
        super(KernelUnit, self).__init__(
                    code=code,
                    space=space,
                    domain=code.domain,
                    binding=binding,
                    equality_vector=(code, space))


class ComplementUnit(CompoundUnit):

    def __init__(self, code, space, binding):
        assert isinstance(space, ComplementSpace)
        super(ComplementUnit, self).__init__(
                    code=code,
                    space=space,
                    domain=code.domain,
                    binding=binding,
                    equality_vector=(code, space))


class MonikerUnit(CompoundUnit):

    def __init__(self, code, space, binding):
        assert isinstance(space, MonikerSpace)
        super(MonikerUnit, self).__init__(
                code=code,
                space=space,
                domain=code.domain,
                binding=binding,
                equality_vector=(code, space))


class ForkedUnit(CompoundUnit):

    def __init__(self, code, space, binding):
        assert isinstance(space, ForkedSpace)
        super(ForkedUnit, self).__init__(
                code=code,
                space=space,
                domain=code.domain,
                binding=binding,
                equality_vector=(code, space))


class LinkedUnit(CompoundUnit):

    def __init__(self, code, space, binding):
        assert isinstance(space, LinkedSpace)
        super(LinkedUnit, self).__init__(
                code=code,
                space=space,
                domain=code.domain,
                binding=binding,
                equality_vector=(code, space))


class BatchExpr(Expression):
    """
    Represents a collection of expression nodes.

    This is an auxiliary expression node used internally by the compiler.

    `collection` (a list of :class:`Expression`)
        A collection of expression nodes.
    """

    def __init__(self, collection, binding):
        assert isinstance(collection, listof(Expression))
        super(BatchExpr, self).__init__(binding)
        self.collection = collection

    def __str__(self):
        # Display the collection:
        #   (<expression>, <expression>, ...)
        return "(%s)" % ", ".join(str(expression)
                                  for expression in self.collection)


class ScalarBatchExpr(BatchExpr):
    """
    Represents a collection of sclar units sharing the same base space.

    This is an auxiliary expression node used internally by the compiler.

    `space` (:class:`Space`)
        The base space of the scalar units.

    `collection` (a list of :class:`ScalarUnit`)
        A collection of scalar units.  All units must have the same base
        space.
    """

    def __init__(self, space, collection, binding):
        assert isinstance(space, Space)
        assert isinstance(collection, listof(ScalarUnit))
        assert all(space == unit.space for unit in collection)
        super(ScalarBatchExpr, self).__init__(collection, binding)
        self.space = space


class AggregateBatchExpr(BatchExpr):
    """
    Represents a collection of aggregate units sharing the same base and
    plural spaces.

    This is an auxiliary expression node used internally by the compiler.

    `plural_space` (:class:`Space`)
        The plural space of the aggregates.

    `space` (:class:`Space`)
        The base space of the aggregates.

    `collection` (a list of :class:`AggregateUnit`)
        A collection of aggregate units.  All units must have the same
        base and plural spaces.
    """

    def __init__(self, plural_space, space, collection, binding):
        assert isinstance(plural_space, Space)
        assert isinstance(space, Space)
        assert isinstance(collection, listof(AggregateUnit))
        assert all(plural_space == unit.plural_space and space == unit.space
                   for unit in collection)
        super(AggregateBatchExpr, self).__init__(collection, binding)
        self.plural_space = plural_space
        self.space = space


