
from __future__ import annotations

import random
from collections.abc import Iterable






class     SCREAMING_SNAKE_CASE__      :

							"""simple docstring"""






							def __init__(					self       ,		SCREAMING_SNAKE_CASE_     )->				None:





													'''simple docstring'''






													__UpperCamelCase       					=     {literal: None for literal in literals}



							def __str__(					self     )->				str:





													'''simple docstring'''






													return "{" + " , ".join(self.literals     ) + "}"



							def __len__(					self     )->				int:





													'''simple docstring'''






													return len(self.literals     )



							def       A__						(					self       ,		SCREAMING_SNAKE_CASE_     )->				None:





													'''simple docstring'''






													for literal in self.literals:
																			__UpperCamelCase       					=     literal[:2]
																			if symbol in model:
																									__UpperCamelCase       					=     model[symbol]
																			else:
																									continue
																			if value is not None:
																									# Complement assignment if literal is in complemented form
																									if literal.endswith('''\''''     ):
																															__UpperCamelCase       					=     not value
																			__UpperCamelCase       					=     value



							def       A__						(					self       ,		SCREAMING_SNAKE_CASE_     )->				bool | None:





													'''simple docstring'''






													for literal in self.literals:
																			__UpperCamelCase       					=     literal.rstrip('''\''''     ) if literal.endswith('''\''''     ) else literal + '''\''''
																			if symbol in self.literals:
																									return True

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






class     SCREAMING_SNAKE_CASE__      :

							"""simple docstring"""






							def __init__(					self       ,		SCREAMING_SNAKE_CASE_     )->				None:





													'''simple docstring'''






													__UpperCamelCase       					=     list(SCREAMING_SNAKE_CASE_     )



							def __str__(					self     )->				str:





													'''simple docstring'''






													return "{" + " , ".join(str(SCREAMING_SNAKE_CASE_     ) for clause in self.clauses     ) + "}"





def 		A_	(							) ->      Clause:



						'''simple docstring'''

						__UpperCamelCase       					=     []
						__UpperCamelCase       					=     random.randint(1	, 5	)
						__UpperCamelCase       					=     '''A'''
						__UpperCamelCase       					=     0
						while i < no_of_literals:
												__UpperCamelCase       					=     random.randint(1	, 5	)
												__UpperCamelCase       					=     base_var + str(snake_case	)
												__UpperCamelCase       					=     random.randint(0	, 1	)
												if var_complement == 1:
																		var_name += "'"
												if var_name in literals:
																		i -= 1
												else:
																		literals.append(snake_case	)
												i += 1
						return Clause(snake_case	)




def 		A_	(							) ->      Formula:



						'''simple docstring'''

						__UpperCamelCase       					=     set()
						__UpperCamelCase       					=     random.randint(1	, 10	)
						while len(snake_case	) < no_of_clauses:
												clauses.add(generate_clause()	)
						return Formula(snake_case	)




def 		A_	(							snake_case							:				Formula	) ->      tuple[list[Clause], list[str]]:



						'''simple docstring'''

						__UpperCamelCase       					=     formula.clauses
						__UpperCamelCase       					=     []
						for clause in formula.clauses:
												for literal in clause.literals:
																		__UpperCamelCase       					=     literal[:2]
																		if symbol not in symbols_set:
																								symbols_set.append(snake_case	)
						return clauses, symbols_set




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



						'''simple docstring'''

						__UpperCamelCase       					=     []
						__UpperCamelCase       					=     {}
						__UpperCamelCase       					=     []

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

						for s in symbols:
												__UpperCamelCase       					=     s + '''\''''
												if (s in literals and sym not in literals) or (
												    s not in literals and sym in literals
												):
																		pure_symbols.append(snake_case	)
						for p in pure_symbols:
												__UpperCamelCase       					=     None
						for s in pure_symbols:
												__UpperCamelCase       					=     s + '''\''''
												if s in literals:
																		__UpperCamelCase       					=     True
												elif sym in literals:
																		__UpperCamelCase       					=     False
						return pure_symbols, assignment




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



						'''simple docstring'''

						__UpperCamelCase       					=     []
						for clause in clauses:
												if len(snake_case	) == 1:
																		unit_symbols.append(list(clause.literals.keys()	)[0]	)
												else:
																		__UpperCamelCase					,							__UpperCamelCase       					=     0, 0
																		for literal, value in clause.literals.items():
																								if value is False:
																														f_count += 1
																								elif value is None:
																														__UpperCamelCase       					=     literal
																														n_count += 1
																		if f_count == len(snake_case	) - 1 and n_count == 1:
																								unit_symbols.append(snake_case	)
						__UpperCamelCase       					=     {}
						for i in unit_symbols:
												__UpperCamelCase       					=     i[:2]
												__UpperCamelCase       					=     len(snake_case	) == 2
						__UpperCamelCase       					=     [i[:2] for i in unit_symbols]

						return unit_symbols, assignment




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



						'''simple docstring'''

						__UpperCamelCase       					=     True
						for clause in clauses:
												__UpperCamelCase       					=     clause.evaluate(snake_case	)
												if clause_check is False:
																		return False, None
												elif clause_check is None:
																		__UpperCamelCase       					=     False
																		continue

						if check_clause_all_true:
												return True, model

						try:
												__UpperCamelCase					,							__UpperCamelCase       					=     find_pure_symbols(snake_case	, snake_case	, snake_case	)
						except RecursionError:
												print('''raises a RecursionError and is'''	)
												return None, {}
						__UpperCamelCase       					=     None
						if len(snake_case	) > 0:
												__UpperCamelCase					,							__UpperCamelCase       					=     pure_symbols[0], assignment[pure_symbols[0]]

						if p:
												__UpperCamelCase       					=     model
												__UpperCamelCase       					=     value
												__UpperCamelCase       					=     list(snake_case	)
												if p in tmp_symbols:
																		tmp_symbols.remove(snake_case	)
												return dpll_algorithm(snake_case	, snake_case	, snake_case	)

						__UpperCamelCase					,							__UpperCamelCase       					=     find_unit_clauses(snake_case	, snake_case	)
						__UpperCamelCase       					=     None
						if len(snake_case	) > 0:
												__UpperCamelCase					,							__UpperCamelCase       					=     unit_symbols[0], assignment[unit_symbols[0]]
						if p:
												__UpperCamelCase       					=     model
												__UpperCamelCase       					=     value
												__UpperCamelCase       					=     list(snake_case	)
												if p in tmp_symbols:
																		tmp_symbols.remove(snake_case	)
												return dpll_algorithm(snake_case	, snake_case	, snake_case	)
						__UpperCamelCase       					=     symbols[0]
						__UpperCamelCase       					=     symbols[1:]
						__UpperCamelCase					,							__UpperCamelCase       					=     model, model
						__UpperCamelCase					,							__UpperCamelCase       					=     True, False

						return dpll_algorithm(snake_case	, snake_case	, snake_case	) or dpll_algorithm(snake_case	, snake_case	, snake_case	)


if __name__ == "__main__":
				import doctest

				doctest.testmod()

				lowercase__							:							Tuple													=     generate_formula()
				print(F"The formula {formula} is", end=" ")

				lowercase__						,					lowercase__							:							Optional[int]													=     generate_parameters(formula)
				lowercase__						,					lowercase__							:							List[str]													=     dpll_algorithm(clauses, symbols, {})

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






from typing import Any

import numpy as np




def 		A_	(							snake_case							:				np.ndarray	) ->      bool:



						'''simple docstring'''

						assert np.shape(snake_case	)[0] == np.shape(snake_case	)[1]

						# If matrix not symmetric, exit right away.
						if np.allclose(snake_case	, matrix.T	) is False:
												return False

						# Get eigenvalues and eignevectors for a symmetric matrix.
						__UpperCamelCase					,							__UpperCamelCase       					=     np.linalg.eigh(snake_case	)

						# Check sign of all eigenvalues.
						# np.all returns a value of type np.bool_
						return bool(np.all(eigen_values > 0	)	)




def 		A_	(							snake_case							:				int	) ->      Any:



						'''simple docstring'''

						__UpperCamelCase       					=     np.random.randn(snake_case	, snake_case	)
						__UpperCamelCase       					=     np.dot(snake_case	, random_matrix.T	)
						assert _is_matrix_spd(snake_case	)
						return spd_matrix




def 		A_	(							snake_case							:				np.ndarray	, snake_case							:				np.ndarray	, snake_case							:				int = 1000	, snake_case							:				float = 1e-8	, ) ->      Any:



						'''simple docstring'''

						assert np.shape(snake_case	)[0] == np.shape(snake_case	)[1]
						assert np.shape(snake_case	)[0] == np.shape(snake_case	)[0]
						assert _is_matrix_spd(snake_case	)

						# Initialize solution guess, residual, search direction.
						__UpperCamelCase       					=     np.zeros((np.shape(snake_case	)[0], 1)	)
						__UpperCamelCase       					=     np.copy(snake_case	)
						__UpperCamelCase       					=     np.copy(snake_case	)

						# Set initial errors in solution guess and residual.
						__UpperCamelCase       					=     1e9
						__UpperCamelCase       					=     1e9
						__UpperCamelCase       					=     1e9

						# Set iteration counter to threshold number of iterations.
						__UpperCamelCase       					=     0

						while error > tol:
												# Save this value so we only calculate the matrix-vector product once.
												__UpperCamelCase       					=     np.dot(snake_case	, snake_case	)

												# The main algorithm.

												# Update search direction magnitude.
												__UpperCamelCase       					=     np.dot(ra.T	, snake_case	) / np.dot(pa.T	, snake_case	)
												# Update solution guess.
												__UpperCamelCase       					=     xa + alpha * pa
												# Calculate new residual.
												__UpperCamelCase       					=     ra - alpha * w
												# Calculate new Krylov subspace scale.
												__UpperCamelCase       					=     np.dot(r.T	, snake_case	) / np.dot(ra.T	, snake_case	)
												# Calculate new A conjuage search direction.
												__UpperCamelCase       					=     r + beta * pa

												# Calculate errors.
												__UpperCamelCase       					=     np.linalg.norm(r - ra	)
												__UpperCamelCase       					=     np.linalg.norm(x - xa	)
												__UpperCamelCase       					=     np.maximum(snake_case	, snake_case	)

												# Update variables.
												__UpperCamelCase       					=     np.copy(snake_case	)
												__UpperCamelCase       					=     np.copy(snake_case	)
												__UpperCamelCase       					=     np.copy(snake_case	)

												# Update number of iterations.
												iterations += 1
												if iterations > max_iterations:
																		break

						return x




def 		A_	(							) ->      None:



						'''simple docstring'''

						__UpperCamelCase       					=     3
						__UpperCamelCase       					=     _create_spd_matrix(snake_case	)
						__UpperCamelCase       					=     np.random.randn(snake_case	, 1	)
						__UpperCamelCase       					=     np.dot(snake_case	, snake_case	)

						# Numpy solution.
						__UpperCamelCase       					=     np.linalg.solve(snake_case	, snake_case	)

						# Our implementation.
						__UpperCamelCase       					=     conjugate_gradient(snake_case	, snake_case	)

						# Ensure both solutions are close to x_true (and therefore one another).
						assert np.linalg.norm(x_numpy - x_true	) <= 1e-6
						assert np.linalg.norm(x_conjugate_gradient - x_true	) <= 1e-6


if __name__ == "__main__":
				import doctest

				doctest.testmod()
				test_conjugate_gradient()




