





'''simple docstring'''





from typing import Any





def   lowerCAmelCase_				( __a			,					__a			,					__a			,					__a			,					__a			,					)  ->	list:







  """simple docstring"""





  _validation(
      __a			,					__a			,					__a			,					__a			,					__a			,					)
  # Creates data structures and fill initial step
  lowerCamelCase__:   dict    						={}
  lowerCamelCase__:   dict    						={}
  for state in states_space:
    lowerCamelCase__:   Dict    						=observations_space[0]
    lowerCamelCase__:   Optional[Any]    						=(
        initial_probabilities[state] * emission_probabilities[state][observation]
    )
    lowerCamelCase__:   Tuple    						=None

  # Fills the data structure with the probabilities of
  # different transitions and pointers to previous states
  for o in range(1			,					len(__a	)	):
    lowerCamelCase__:   Optional[int]    						=observations_space[o]
    lowerCamelCase__:   List[Any]    						=observations_space[o - 1]
    for state in states_space:
      # Calculates the argmax for probability function
      lowerCamelCase__:   Union[str, Any]    						=""
      lowerCamelCase__:   str    						=-1
      for k_state in states_space:
        lowerCamelCase__:   int    						=(
            probabilities[(k_state, prior_observation)]
            * transition_probabilities[k_state][state]
            * emission_probabilities[state][observation]
        )
        if probability > max_probability:
          lowerCamelCase__:   Tuple    						=probability
          lowerCamelCase__:   Dict    						=k_state

            # Update probabilities and pointers dicts
      lowerCamelCase__:   Any    						=(
          probabilities[(arg_max, prior_observation)]
          * transition_probabilities[arg_max][state]
          * emission_probabilities[state][observation]
      )

      lowerCamelCase__:   str    						=arg_max

    # The final observation
  lowerCamelCase__:   Optional[int]    						=observations_space[len(__a	) - 1]

  # argmax for given final observation
  lowerCamelCase__:   str    						=""
  lowerCamelCase__:   Union[str, Any]    						=-1
  for k_state in states_space:
    lowerCamelCase__:   Tuple    						=probabilities[(k_state, final_observation)]
    if probability > max_probability:
      lowerCamelCase__:   Optional[int]    						=probability
      lowerCamelCase__:   List[Any]    						=k_state
  lowerCamelCase__:   Dict    						=arg_max

  # Process pointers backwards
  lowerCamelCase__:   int    						=last_state
  lowerCamelCase__:   List[Any]    						=[]
  for o in range(len(__a	) - 1			,					-1			,					-1	):
    result.append(__a	)
    lowerCamelCase__:   Union[str, Any]    						=pointers[previous, observations_space[o]]
  result.reverse()

  return result





def   lowerCAmelCase_				( __a			,					__a			,					__a			,					__a			,					__a			,					)  ->	None:







  """simple docstring"""





  _validate_not_empty(
      __a			,					__a			,					__a			,					__a			,					__a			,					)
  _validate_lists(__a			,					__a	)
  _validate_dicts(
      __a			,					__a			,					__a	)





def   lowerCAmelCase_				( __a			,					__a			,					__a			,					__a			,					__a			,					)  ->	None:







  """simple docstring"""





  if not all(
      [
          observations_space,
          states_space,
          initial_probabilities,
          transition_probabilities,
          emission_probabilities,
      ]	):
    raise ValueError("There's an empty parameter"	)





def   lowerCAmelCase_				( __a			,					__a	)  ->	None:







  """simple docstring"""





  _validate_list(__a			,					"observations_space"	)
  _validate_list(__a			,					"states_space"	)





def   lowerCAmelCase_				( __a			,					__a	)  ->	None:







  """simple docstring"""





  if not isinstance(_object			,					__a	):
    lowerCamelCase__:   Optional[int]    						=F"""{var_name} must be a list"""
    raise ValueError(__a	)
  else:
    for x in _object:
      if not isinstance(__a			,					__a	):
        lowerCamelCase__:   int    						=F"""{var_name} must be a list of strings"""
        raise ValueError(__a	)





def   lowerCAmelCase_				( __a			,					__a			,					__a			,					)  ->	None:







  """simple docstring"""





  _validate_dict(__a			,					"initial_probabilities"			,					__a	)
  _validate_nested_dict(__a			,					"transition_probabilities"	)
  _validate_nested_dict(__a			,					"emission_probabilities"	)





def   lowerCAmelCase_				( __a			,					__a	)  ->	None:







  """simple docstring"""





  _validate_dict(_object			,					__a			,					__a	)
  for x in _object.values():
    _validate_dict(__a			,					__a			,					__a			,					__a	)





def   lowerCAmelCase_				( __a			,					__a			,					__a			,					__a = False	)  ->	None:







  """simple docstring"""





  if not isinstance(_object			,					__a	):
    lowerCamelCase__:   List[str]    						=F"""{var_name} must be a dict"""
    raise ValueError(__a	)
  if not all(isinstance(__a			,					__a	) for x in _object	):
    lowerCamelCase__:   Any    						=F"""{var_name} all keys must be strings"""
    raise ValueError(__a	)
  if not all(isinstance(__a			,					__a	) for x in _object.values()	):
    lowerCamelCase__:   Dict    						="nested dictionary " if nested else ""
    lowerCamelCase__:   Optional[Any]    						=F"""{var_name} {nested_text}all values must be {value_type.__name__}"""
    raise ValueError(__a	)


if __name__ == "__main__":
     from doctest import testmod

     testmod()








def   lowerCAmelCase_				( __a			,					__a	)  ->	float:







  """simple docstring"""





  if density <= 0:
    raise ValueError("Impossible fluid density"	)
  if bulk_modulus <= 0:
    raise ValueError("Impossible bulk modulus"	)

  return (bulk_modulus / density) ** 0.5


if __name__ == "__main__":
     import doctest

     doctest.testmod()

