

from __future__ import annotations

import random
from collections.abc import Iterable






class 							A_				:


     def __init__(							self    :	str    ,						UpperCAmelCase    :	list[str]      )							->			None:
      __lowerCAmelCase:     dict[str, bool | None]     			=   {literal: None for literal in literals}


     def __str__(							self    :	Optional[int]      )							->			str:
      return "{" + " , ".join(self.literals      ) + "}"


     def __len__(							self    :	Dict      )							->			int:
      return len(self.literals      )


     def 						UpperCAmelCase						(							self    :	Optional[Any]    ,						UpperCAmelCase    :	dict[str, bool | None]      )							->			None:
      for literal in self.literals:
       __lowerCAmelCase:     List[str]     			=   literal[:2]
       if symbol in model:
        __lowerCAmelCase:     Optional[Any]     			=   model[symbol]
       else:
        continue
       if value is not None:
        # Complement assignment if literal is in complemented form
        if literal.endswith('\''      ):
         __lowerCAmelCase:     List[Any]     			=   not value
       __lowerCAmelCase:     Tuple     			=   value


     def 						UpperCAmelCase						(							self    :	List[Any]    ,						UpperCAmelCase    :	dict[str, bool | None]      )							->			bool | None:
      for literal in self.literals:
       __lowerCAmelCase:     int     			=   literal.rstrip('\''      ) if literal.endswith('\''      ) else literal + '\''
       if symbol in self.literals:
        return True

      self.assign(UpperCAmelCase      )
      for value in self.literals.values():
       if value in (True, None):
        return value
      return any(self.literals.values()      )







class 							A_				:


     def __init__(							self    :	List[Any]    ,						UpperCAmelCase    :	Iterable[Clause]      )							->			None:
      __lowerCAmelCase:     int     			=   list(UpperCAmelCase      )


     def __str__(							self    :	List[Any]      )							->			str:
      return "{" + " , ".join(str(UpperCAmelCase      ) for clause in self.clauses      ) + "}"







def _a				( )					->					Clause:

 """simple docstring"""






 __lowerCAmelCase:     Tuple     			=   []
 __lowerCAmelCase:     Union[str, Any]     			=   random.randint(1					,      5	)
 __lowerCAmelCase:     Optional[int]     			=   'A'
 __lowerCAmelCase:     Union[str, Any]     			=   0
 while i < no_of_literals:
  __lowerCAmelCase:     int     			=   random.randint(1					,      5	)
  __lowerCAmelCase:     str     			=   base_var + str(SCREAMING_SNAKE_CASE	)
  __lowerCAmelCase:     int     			=   random.randint(0					,      1	)
  if var_complement == 1:
   var_name += "'"
  if var_name in literals:
   i -= 1
  else:
   literals.append(SCREAMING_SNAKE_CASE	)
  i += 1
 return Clause(SCREAMING_SNAKE_CASE	)







def _a				( )					->					Formula:

 """simple docstring"""






 __lowerCAmelCase:     set[Clause]     			=   set()
 __lowerCAmelCase:     int     			=   random.randint(1					,      10	)
 while len(SCREAMING_SNAKE_CASE	) < no_of_clauses:
  clauses.add(generate_clause()	)
 return Formula(SCREAMING_SNAKE_CASE	)







def _a				( SCREAMING_SNAKE_CASE	:    Formula	)					->					tuple[list[Clause], list[str]]:

 """simple docstring"""






 __lowerCAmelCase:     Any     			=   formula.clauses
 __lowerCAmelCase:     int     			=   []
 for clause in formula.clauses:
  for literal in clause.literals:
   __lowerCAmelCase:     Union[str, Any]     			=   literal[:2]
   if symbol not in symbols_set:
    symbols_set.append(SCREAMING_SNAKE_CASE	)
 return clauses, symbols_set







def _a				( SCREAMING_SNAKE_CASE	:    list[Clause]					,      SCREAMING_SNAKE_CASE	:    list[str]					,      SCREAMING_SNAKE_CASE	:    dict[str, bool | None]	)					->					tuple[list[str], dict[str, bool | None]]:

 """simple docstring"""






 __lowerCAmelCase:     str     			=   []
 __lowerCAmelCase:     dict[str, bool | None]     			=   {}
 __lowerCAmelCase:     List[str]     			=   []

 for clause in clauses:
  if clause.evaluate(SCREAMING_SNAKE_CASE	):
   continue
  for literal in clause.literals:
   literals.append(SCREAMING_SNAKE_CASE	)

 for s in symbols:
  __lowerCAmelCase:     List[str]     			=   s + '\''
  if (s in literals and sym not in literals) or (
      s not in literals and sym in literals
  ):
   pure_symbols.append(SCREAMING_SNAKE_CASE	)
 for p in pure_symbols:
  __lowerCAmelCase:     Tuple     			=   None
 for s in pure_symbols:
  __lowerCAmelCase:     Optional[int]     			=   s + '\''
  if s in literals:
   __lowerCAmelCase:     List[str]     			=   True
  elif sym in literals:
   __lowerCAmelCase:     Optional[Any]     			=   False
 return pure_symbols, assignment







def _a				( SCREAMING_SNAKE_CASE	:    list[Clause]					,      SCREAMING_SNAKE_CASE	:    dict[str, bool | None]	)					->					tuple[list[str], dict[str, bool | None]]:

 """simple docstring"""






 __lowerCAmelCase:     str     			=   []
 for clause in clauses:
  if len(SCREAMING_SNAKE_CASE	) == 1:
   unit_symbols.append(list(clause.literals.keys()	)[0]	)
  else:
   __lowerCAmelCase			,							__lowerCAmelCase:     Any     			=   0, 0
   for literal, value in clause.literals.items():
    if value is False:
     f_count += 1
    elif value is None:
     __lowerCAmelCase:     Dict     			=   literal
     n_count += 1
   if f_count == len(SCREAMING_SNAKE_CASE	) - 1 and n_count == 1:
    unit_symbols.append(SCREAMING_SNAKE_CASE	)
 __lowerCAmelCase:     dict[str, bool | None]     			=   {}
 for i in unit_symbols:
  __lowerCAmelCase:     Optional[int]     			=   i[:2]
  __lowerCAmelCase:     Tuple     			=   len(SCREAMING_SNAKE_CASE	) == 2
 __lowerCAmelCase:     List[str]     			=   [i[:2] for i in unit_symbols]

 return unit_symbols, assignment







def _a				( SCREAMING_SNAKE_CASE	:    list[Clause]					,      SCREAMING_SNAKE_CASE	:    list[str]					,      SCREAMING_SNAKE_CASE	:    dict[str, bool | None]	)					->					tuple[bool | None, dict[str, bool | None] | None]:

 """simple docstring"""






 __lowerCAmelCase:     str     			=   True
 for clause in clauses:
  __lowerCAmelCase:     Optional[Any]     			=   clause.evaluate(SCREAMING_SNAKE_CASE	)
  if clause_check is False:
   return False, None
  elif clause_check is None:
   __lowerCAmelCase:     List[str]     			=   False
   continue

 if check_clause_all_true:
  return True, model

 try:
  __lowerCAmelCase			,							__lowerCAmelCase:     Any     			=   find_pure_symbols(SCREAMING_SNAKE_CASE					,      SCREAMING_SNAKE_CASE					,      SCREAMING_SNAKE_CASE	)
 except RecursionError:
  print('raises a RecursionError and is'	)
  return None, {}
 __lowerCAmelCase:     int     			=   None
 if len(SCREAMING_SNAKE_CASE	) > 0:
  __lowerCAmelCase			,							__lowerCAmelCase:     Optional[int]     			=   pure_symbols[0], assignment[pure_symbols[0]]

 if p:
  __lowerCAmelCase:     Optional[Any]     			=   model
  __lowerCAmelCase:     List[Any]     			=   value
  __lowerCAmelCase:     int     			=   list(SCREAMING_SNAKE_CASE	)
  if p in tmp_symbols:
   tmp_symbols.remove(SCREAMING_SNAKE_CASE	)
  return dpll_algorithm(SCREAMING_SNAKE_CASE					,      SCREAMING_SNAKE_CASE					,      SCREAMING_SNAKE_CASE	)

 __lowerCAmelCase			,							__lowerCAmelCase:     Optional[Any]     			=   find_unit_clauses(SCREAMING_SNAKE_CASE					,      SCREAMING_SNAKE_CASE	)
 __lowerCAmelCase:     Dict     			=   None
 if len(SCREAMING_SNAKE_CASE	) > 0:
  __lowerCAmelCase			,							__lowerCAmelCase:     int     			=   unit_symbols[0], assignment[unit_symbols[0]]
 if p:
  __lowerCAmelCase:     Optional[int]     			=   model
  __lowerCAmelCase:     List[str]     			=   value
  __lowerCAmelCase:     Optional[Any]     			=   list(SCREAMING_SNAKE_CASE	)
  if p in tmp_symbols:
   tmp_symbols.remove(SCREAMING_SNAKE_CASE	)
  return dpll_algorithm(SCREAMING_SNAKE_CASE					,      SCREAMING_SNAKE_CASE					,      SCREAMING_SNAKE_CASE	)
 __lowerCAmelCase:     List[Any]     			=   symbols[0]
 __lowerCAmelCase:     Tuple     			=   symbols[1:]
 __lowerCAmelCase			,							__lowerCAmelCase:     Union[str, Any]     			=   model, model
 __lowerCAmelCase			,							__lowerCAmelCase:     Tuple     			=   True, False

 return dpll_algorithm(SCREAMING_SNAKE_CASE					,      SCREAMING_SNAKE_CASE					,      SCREAMING_SNAKE_CASE	) or dpll_algorithm(SCREAMING_SNAKE_CASE					,      SCREAMING_SNAKE_CASE					,      SCREAMING_SNAKE_CASE	)


if __name__ == "__main__":
  import doctest

  doctest.testmod()

  _a	 =     generate_formula()
  print(f"The formula {formula} is", end=''' ''')

  _a ,			_a	 =     generate_parameters(formula)
  _a ,			_a	 =     dpll_algorithm(clauses, symbols, {})

  if solution:
    print(f"satisfiable with the assignment {model}.")
  else:
    print('''not satisfiable.''')








import math






class 							A_				:


     def __init__(							self    :	Any    ,						UpperCAmelCase    :	int    ,						UpperCAmelCase    :	int    ,						UpperCAmelCase    :	int    ,						UpperCAmelCase    :	Any      )							->			Any:
      __lowerCAmelCase:     Optional[Any]     			=   x
      __lowerCAmelCase:     Any     			=   y
      __lowerCAmelCase:     Tuple     			=   step_size
      __lowerCAmelCase:     Dict     			=   function_to_optimize


     def 						UpperCAmelCase						(							self    :	Dict      )							->			int:
      return self.function(self.x    ,						self.y      )


     def 						UpperCAmelCase						(							self    :	str      )							->			int:
      __lowerCAmelCase:     Union[str, Any]     			=   self.step_size
      return [
          SearchProblem(UpperCAmelCase    ,						UpperCAmelCase    ,						UpperCAmelCase    ,						self.function      )
          for x, y in (
              (self.x - step_size, self.y - step_size),
              (self.x - step_size, self.y),
              (self.x - step_size, self.y + step_size),
              (self.x, self.y - step_size),
              (self.x, self.y + step_size),
              (self.x + step_size, self.y - step_size),
              (self.x + step_size, self.y),
              (self.x + step_size, self.y + step_size),
          )
      ]


     def __hash__(							self    :	Optional[Any]      )							->			int:
      return hash(str(self      )      )


     def __eq__(							self    :	Dict    ,						UpperCAmelCase    :	Tuple      )							->			Any:
      if isinstance(UpperCAmelCase    ,						UpperCAmelCase      ):
       return hash(str(self      )      ) == hash(str(UpperCAmelCase      )      )
      return False


     def __str__(							self    :	Optional[Any]      )							->			Dict:
      return F'''x: {self.x} y: {self.y}'''







def _a				( SCREAMING_SNAKE_CASE	:    List[str]					,      SCREAMING_SNAKE_CASE	:    bool = True					,      SCREAMING_SNAKE_CASE	:    float = math.inf					,      SCREAMING_SNAKE_CASE	:    float = -math.inf					,      SCREAMING_SNAKE_CASE	:    float = math.inf					,      SCREAMING_SNAKE_CASE	:    float = -math.inf					,      SCREAMING_SNAKE_CASE	:    bool = False					,      SCREAMING_SNAKE_CASE	:    int = 1_00_00					,      )					->					SearchProblem:

 """simple docstring"""






 __lowerCAmelCase:     Dict     			=   search_prob
 __lowerCAmelCase:     str     			=   []  # list to store the current score at each iteration
 __lowerCAmelCase:     Optional[int]     			=   0
 __lowerCAmelCase:     int     			=   False
 __lowerCAmelCase:     Any     			=   set()
 while not solution_found and iterations < max_iter:
  visited.add(SCREAMING_SNAKE_CASE	)
  iterations += 1
  __lowerCAmelCase:     int     			=   current_state.score()
  scores.append(SCREAMING_SNAKE_CASE	)
  __lowerCAmelCase:     Any     			=   current_state.get_neighbors()
  __lowerCAmelCase:     Tuple     			=   -math.inf
  __lowerCAmelCase:     Any     			=   math.inf
  __lowerCAmelCase:     Dict     			=   None  # to hold the next best neighbor
  for neighbor in neighbors:
   if neighbor in visited:
    continue  # do not want to visit the same state again
   if (
       neighbor.x > max_x
       or neighbor.x < min_x
       or neighbor.y > max_y
       or neighbor.y < min_y
   ):
    continue  # neighbor outside our bounds
   __lowerCAmelCase:     List[str]     			=   neighbor.score() - current_score
   if find_max:  # finding max
    # going to direction with greatest ascent
    if change > max_change and change > 0:
     __lowerCAmelCase:     int     			=   change
     __lowerCAmelCase:     int     			=   neighbor
   else:  # finding min
    # to direction with greatest descent
    if change < min_change and change < 0:
     __lowerCAmelCase:     str     			=   change
     __lowerCAmelCase:     str     			=   neighbor
  if next_state is not None:
   # we found at least one neighbor which improved the current state
   __lowerCAmelCase:     Optional[Any]     			=   next_state
  else:
   # since we have no neighbor that improves the solution we stop the search
   __lowerCAmelCase:     Dict     			=   True

 if visualization:
  from matplotlib import pyplot as plt

  plt.plot(range(SCREAMING_SNAKE_CASE	)					,      SCREAMING_SNAKE_CASE	)
  plt.xlabel('Iterations'	)
  plt.ylabel('Function values'	)
  plt.show()

 return current_state


if __name__ == "__main__":
  import doctest

  doctest.testmod()







  def _a				( SCREAMING_SNAKE_CASE	:    List[str]					,      SCREAMING_SNAKE_CASE	:    List[str]	)					->					Optional[Any]:

   """simple docstring"""






   return (x**2) + (y**2)

  # starting the problem with initial coordinates (3, 4)
  _a	 =     SearchProblem(x=3, y=4, step_size=1, function_to_optimize=test_fa)
  _a	 =     hill_climbing(prob, find_max=False)
  print(
      '''The minimum score for f(x, y) = x^2 + y^2 found via hill climbing: '''
      f"{local_min.score()}"
  )

  # starting the problem with initial coordinates (12, 47)
  _a	 =     SearchProblem(x=1_2, y=4_7, step_size=1, function_to_optimize=test_fa)
  _a	 =     hill_climbing(
      prob, find_max=False, max_x=1_0_0, min_x=5, max_y=5_0, min_y=-5, visualization=True
  )
  print(
      '''The minimum score for f(x, y) = x^2 + y^2 with the domain 100 > x > 5 '''
      f"and 50 > y > - 5 found via hill climbing: {local_min.score()}"
  )







  def _a				( SCREAMING_SNAKE_CASE	:    Tuple					,      SCREAMING_SNAKE_CASE	:    List[Any]	)					->					Dict:

   """simple docstring"""






   return (3 * x**2) - (6 * y)

  _a	 =     SearchProblem(x=3, y=4, step_size=1, function_to_optimize=test_fa)
  _a	 =     hill_climbing(prob, find_max=True)



  print(
      '''The maximum score for f(x, y) = x^2 + y^2 found via hill climbing: '''
      f"{local_min.score()}"
  )





