



from typing import Any
def 		UpperCamelCase	(   snake_case__       :    list       ,     snake_case__       :    list       ,     snake_case__       :    dict       ,     snake_case__       :    dict       ,     snake_case__       :    dict       ,     )	->     list:
   _validation(
       snake_case__       ,     snake_case__       ,     snake_case__       ,     snake_case__       ,     snake_case__       ,     )
   # Creates data structures and fill initial step
   UpperCamelCase    :  dict							=	{}
   UpperCamelCase    :  dict							=	{}
   for state in states_space:
      UpperCamelCase    :  str							=	observations_space[0]
      UpperCamelCase    :  int							=	(
          initial_probabilities[state] * emission_probabilities[state][observation]
      )
      UpperCamelCase    :  Optional[Any]							=	None

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

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

         UpperCamelCase    :  List[Any]							=	arg_max

    # The final observation
   UpperCamelCase    :  Dict							=	observations_space[len(snake_case__	) - 1]

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

   # Process pointers backwards
   UpperCamelCase    :  str							=	last_state
   UpperCamelCase    :  str							=	[]
   for o in range(len(snake_case__	) - 1       ,     -1       ,     -1	):
      result.append(snake_case__	)
      UpperCamelCase    :  Dict							=	pointers[previous, observations_space[o]]
   result.reverse()

   return result
def 		UpperCamelCase	(   snake_case__       :    Any       ,     snake_case__       :    Any       ,     snake_case__       :    Any       ,     snake_case__       :    Any       ,     snake_case__       :    Any       ,     )	->     None:
   _validate_not_empty(
       snake_case__       ,     snake_case__       ,     snake_case__       ,     snake_case__       ,     snake_case__       ,     )
   _validate_lists(snake_case__       ,     snake_case__	)
   _validate_dicts(
       snake_case__       ,     snake_case__       ,     snake_case__	)
def 		UpperCamelCase	(   snake_case__       :    Any       ,     snake_case__       :    Any       ,     snake_case__       :    Any       ,     snake_case__       :    Any       ,     snake_case__       :    Any       ,     )	->     None:
   if not all(
       [
           observations_space,
           states_space,
           initial_probabilities,
           transition_probabilities,
           emission_probabilities,
       ]	):
      raise ValueError('There\'s an empty parameter'	)
def 		UpperCamelCase	(   snake_case__       :    Any       ,     snake_case__       :    Any	)	->     None:
   _validate_list(snake_case__       ,     'observations_space'	)
   _validate_list(snake_case__       ,     'states_space'	)
def 		UpperCamelCase	(   snake_case__       :    Any       ,     snake_case__       :    str	)	->     None:
   if not isinstance(_object       ,     snake_case__	):
      UpperCamelCase    :  str							=	F"""{var_name} must be a list"""
      raise ValueError(snake_case__	)
   else:
      for x in _object:
         if not isinstance(snake_case__       ,     snake_case__	):
            UpperCamelCase    :  List[Any]							=	F"""{var_name} must be a list of strings"""
            raise ValueError(snake_case__	)
def 		UpperCamelCase	(   snake_case__       :    Any       ,     snake_case__       :    Any       ,     snake_case__       :    Any       ,     )	->     None:
   _validate_dict(snake_case__       ,     'initial_probabilities'       ,     snake_case__	)
   _validate_nested_dict(snake_case__       ,     'transition_probabilities'	)
   _validate_nested_dict(snake_case__       ,     'emission_probabilities'	)
def 		UpperCamelCase	(   snake_case__       :    Any       ,     snake_case__       :    str	)	->     None:
   _validate_dict(_object       ,     snake_case__       ,     snake_case__	)
   for x in _object.values():
      _validate_dict(snake_case__       ,     snake_case__       ,     snake_case__       ,     snake_case__	)
def 		UpperCamelCase	(   snake_case__       :    Any       ,     snake_case__       :    str       ,     snake_case__       :    type       ,     snake_case__       :    bool = False	)	->     None:
   if not isinstance(_object       ,     snake_case__	):
      UpperCamelCase    :  Dict							=	F"""{var_name} must be a dict"""
      raise ValueError(snake_case__	)
   if not all(isinstance(snake_case__       ,     snake_case__	) for x in _object	):
      UpperCamelCase    :  Tuple							=	F"""{var_name} all keys must be strings"""
      raise ValueError(snake_case__	)
   if not all(isinstance(snake_case__       ,     snake_case__	) for x in _object.values()	):
      UpperCamelCase    :  Tuple							=	'nested dictionary ' if nested else ''
      UpperCamelCase    :  str							=	F"""{var_name} {nested_text}all values must be {value_type.__name__}"""
      raise ValueError(snake_case__	)


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

    testmod()








import functools
from typing import Any
def 		UpperCamelCase	(   snake_case__       :    str       ,     snake_case__       :    list[str]	)	->     bool:

   # Validation
   if not isinstance(snake_case__       ,     snake_case__	) or len(snake_case__	) == 0:
      raise ValueError('the string should be not empty string'	)

   if not isinstance(snake_case__       ,     snake_case__	) or not all(
       isinstance(snake_case__       ,     snake_case__	) and len(snake_case__	) > 0 for item in words	):
      raise ValueError('the words should be a list of non-empty strings'	)

   # Build trie
   UpperCamelCase    :  dict[str, Any]							=	{}
   UpperCamelCase    :  List[str]							=	'WORD_KEEPER'

   for word in words:
      UpperCamelCase    :  List[str]							=	trie
      for c in word:
         if c not in trie_node:
            UpperCamelCase    :  int							=	{}

         UpperCamelCase    :  str							=	trie_node[c]

      UpperCamelCase    :  Tuple							=	True

   UpperCamelCase    :  List[Any]							=	len(snake_case__	)

   # Dynamic programming method
   @functools.cache
   def is_breakable(snake_case__       :    int	) -> bool:
      if index == len_string:
         return True

      UpperCamelCase    :  Dict							=	trie
      for i in range(snake_case__       ,     snake_case__	):
         UpperCamelCase    :  List[Any]							=	trie_node.get(string[i]       ,     snake_case__	)

         if trie_node is None:
            return False

         if trie_node.get(snake_case__       ,     snake_case__	) and is_breakable(i + 1	):
            return True

      return False

   return is_breakable(0	)


if __name__ == "__main__":
    import doctest

    doctest.testmod()



