code: def _least_upper_bound(*nodes):
      """Compute the least upper bound of a set of nodes.
  
      Args:
          nodes: sequence of entries from dtypes + weak_types
  
      Returns:
          The type representing the least upper bound of the input nodes on the
          promotion lattice.
      """
      # This function computes the least upper bound of a set of nodes N within a
      # partially ordered set defined by the lattice generated above.
      # Given a partially ordered set S, let the set of upper bounds of n ∈ S be
      #   UB(n) ≡ {m ∈ S | n ≤ m}
      # Further, for a set of nodes N ⊆ S, let the set of common upper bounds be
      # given by
      #   CUB(N) ≡ {a ∈ S | ∀ b ∈ N: a ∈ UB(b)}
      # Then the least upper bound of N is defined as
      #   LUB(N) ≡ {c ∈ CUB(N) | ∀ d ∈ CUB(N), c ≤ d}
      # The definition of an upper bound implies that
      #   c ≤ d if and only if d ∈ UB(c),
      # so the LUB can be expressed:
      #   LUB(N) = {c ∈ CUB(N) | ∀ d ∈ CUB(N): d ∈ UB(c)}
      # or, equivalently:
      #   LUB(N) = {c ∈ CUB(N) | CUB(N) ⊆ UB(c)}
      # By definition, LUB(N) has a cardinality of 1 for a partially ordered set.
      # Note a potential algorithmic shortcut: from the definition of CUB(N),
      # we have
      #   ∀ c ∈ N: CUB(N) ⊆ UB(c)
      # So if N ∩ CUB(N) is nonempty, if follows that LUB(N) = N ∩ CUB(N).
      N = set(nodes)
      UB = LATTICE_UPPER_BOUNDS
      try:
          bounds = [UB[n] for n in N]
      except KeyError:
          dtype = next(n for n in N if n not in UB)
          raise ValueError(
              f"{dtype=} is not a valid dtype for Keras type promotion."
          )
      CUB = set.intersection(*bounds)
      LUB = (CUB & N) or {c for c in CUB if CUB.issubset(UB[c])}
      if len(LUB) == 1:
          return LUB.pop()
      elif len(LUB) == 0:
          msg = (
              f"Input dtypes {tuple(str(n) for n in nodes)} have no available "
              "implicit dtype promotion path. Try explicitly casting inputs to "
              "the desired output type."
          )
          raise ValueError(msg)
      else:
          # If we get here, it means the lattice is ill-formed.
          raise ValueError(
              f"Internal Type Promotion error: {nodes} do not have a unique "
              f"least upper bound on the specified lattice; options are {LUB}. "
              "This is an unexpected error in Keras's internal logic; "
              "please report it to the maintainers."
          )
  
file_path: keras/keras/backend/common/dtypes.py
imports: import functools
  from keras import backend
  from keras.api_export import keras_export
  from keras.backend.common.variables import ALLOWED_DTYPES
  from keras.backend.common.variables import standardize_dtype
label_desc: Use image node_id 3 for calling a global function with example usage: _least_upper_bound() and returns: LUB