






from __future__ import annotations

from collections.abc import Sequence
from typing import Literal



def 					_lowerCamelCase(			lowerCAmelCase__				:		str    ,    lowerCAmelCase__				:		str      ):





				'''simple docstring'''







				SCREAMING_SNAKE_CASE_   :	int													=   list(lowerCAmelCase__      )
				SCREAMING_SNAKE_CASE_   :	Tuple													=   list(lowerCAmelCase__      )
				SCREAMING_SNAKE_CASE_   :	Optional[Any]													=   0
				for i in range(len(lowerCAmelCase__      )      ):
								if lista[i] != lista[i]:
												count += 1
												SCREAMING_SNAKE_CASE_   :	Dict													=   '_'
				if count > 1:
								return False
				else:
								return "".join(lowerCAmelCase__      )



def 					_lowerCamelCase(			lowerCAmelCase__				:		list[str]      ):





				'''simple docstring'''







				SCREAMING_SNAKE_CASE_   :	Any													=   []
				while True:
								SCREAMING_SNAKE_CASE_   :	List[Any]													=   ['$'] * len(lowerCAmelCase__      )
								SCREAMING_SNAKE_CASE_   :	int													=   []
								for i in range(len(lowerCAmelCase__      )      ):
												for j in range(i + 1    ,    len(lowerCAmelCase__      )      ):
																SCREAMING_SNAKE_CASE_   :	Optional[Any]													=   compare_string(binary[i]    ,    binary[j]      )
																if k is False:
																				SCREAMING_SNAKE_CASE_   :	Union[str, Any]													=   '*'
																				SCREAMING_SNAKE_CASE_   :	Any													=   '*'
																				temp.append('X'      )
								for i in range(len(lowerCAmelCase__      )      ):
												if checka[i] == "$":
																pi.append(binary[i]      )
								if len(lowerCAmelCase__      ) == 0:
												return pi
								SCREAMING_SNAKE_CASE_   :	List[str]													=   list(set(lowerCAmelCase__      )      )



def 					_lowerCamelCase(			lowerCAmelCase__				:		int    ,    lowerCAmelCase__				:		Sequence[float]      ):





				'''simple docstring'''







				SCREAMING_SNAKE_CASE_   :	Any													=   []
				for minterm in minterms:
								SCREAMING_SNAKE_CASE_   :	Any													=   ''
								for _ in range(lowerCAmelCase__      ):
												SCREAMING_SNAKE_CASE_   :	List[str]													=   str(minterm % 2      ) + string
												minterm //= 2
								temp.append(lowerCAmelCase__      )
				return temp



def 					_lowerCamelCase(			lowerCAmelCase__				:		str    ,    lowerCAmelCase__				:		str    ,    lowerCAmelCase__				:		int      ):





				'''simple docstring'''







				SCREAMING_SNAKE_CASE_   :	List[Any]													=   list(lowerCAmelCase__      )
				SCREAMING_SNAKE_CASE_   :	Any													=   list(lowerCAmelCase__      )
				SCREAMING_SNAKE_CASE_   :	List[str]													=   0
				for i in range(len(lowerCAmelCase__      )      ):
								if lista[i] != lista[i]:
												count_n += 1
				return count_n == count



def 					_lowerCamelCase(			lowerCAmelCase__				:		list[list[int]]    ,    lowerCAmelCase__				:		list[str]      ):





				'''simple docstring'''







				SCREAMING_SNAKE_CASE_   :	List[str]													=   []
				SCREAMING_SNAKE_CASE_   :	str													=   [0] * len(lowerCAmelCase__      )
				for i in range(len(chart[0]      )      ):
								SCREAMING_SNAKE_CASE_   :	Optional[int]													=   0
								SCREAMING_SNAKE_CASE_   :	Optional[int]													=   -1
								for j in range(len(lowerCAmelCase__      )      ):
												if chart[j][i] == 1:
																count += 1
																SCREAMING_SNAKE_CASE_   :	Tuple													=   j
								if count == 1:
												SCREAMING_SNAKE_CASE_   :	Optional[int]													=   1
				for i in range(len(lowerCAmelCase__      )      ):
								if select[i] == 1:
												for j in range(len(chart[0]      )      ):
																if chart[i][j] == 1:
																				for k in range(len(lowerCAmelCase__      )      ):
																								SCREAMING_SNAKE_CASE_   :	List[str]													=   0
												temp.append(prime_implicants[i]      )
				while True:
								SCREAMING_SNAKE_CASE_   :	Optional[Any]													=   0
								SCREAMING_SNAKE_CASE_   :	List[str]													=   -1
								SCREAMING_SNAKE_CASE_   :	Tuple													=   0
								for i in range(len(lowerCAmelCase__      )      ):
												SCREAMING_SNAKE_CASE_   :	Optional[int]													=   chart[i].count(1      )
												if count_n > max_n:
																SCREAMING_SNAKE_CASE_   :	str													=   count_n
																SCREAMING_SNAKE_CASE_   :	Optional[Any]													=   i

								if max_n == 0:
												return temp

								temp.append(prime_implicants[rem]      )

								for i in range(len(chart[0]      )      ):
												if chart[rem][i] == 1:
																for j in range(len(lowerCAmelCase__      )      ):
																				SCREAMING_SNAKE_CASE_   :	str													=   0



def 					_lowerCamelCase(			lowerCAmelCase__				:		list[str]    ,    lowerCAmelCase__				:		list[str]      ):





				'''simple docstring'''







				SCREAMING_SNAKE_CASE_   :	Dict													=   [[0 for x in range(len(lowerCAmelCase__      )      )] for x in range(len(lowerCAmelCase__      )      )]
				for i in range(len(lowerCAmelCase__      )      ):
								SCREAMING_SNAKE_CASE_   :	List[Any]													=   prime_implicants[i].count('_'      )
								for j in range(len(lowerCAmelCase__      )      ):
												if is_for_table(prime_implicants[i]    ,    binary[j]    ,    lowerCAmelCase__      ):
																SCREAMING_SNAKE_CASE_   :	Optional[int]													=   1

				return chart



def 					_lowerCamelCase(			):





				'''simple docstring'''







				SCREAMING_SNAKE_CASE_   :	Any													=   int(input('Enter the no. of variables\n'      )      )
				SCREAMING_SNAKE_CASE_   :	str													=   [
				    float(lowerCAmelCase__      )
				    for x in input(
				        'Enter the decimal representation of Minterms \'Spaces Separated\'\n'      ).split()
				]
				SCREAMING_SNAKE_CASE_   :	str													=   decimal_to_binary(lowerCAmelCase__    ,    lowerCAmelCase__      )

				SCREAMING_SNAKE_CASE_   :	List[Any]													=   check(lowerCAmelCase__      )
				print('Prime Implicants are:'      )
				print(lowerCAmelCase__      )
				SCREAMING_SNAKE_CASE_   :	Dict													=   prime_implicant_chart(lowerCAmelCase__    ,    lowerCAmelCase__      )

				SCREAMING_SNAKE_CASE_   :	Union[str, Any]													=   selection(lowerCAmelCase__    ,    lowerCAmelCase__      )
				print('Essential Prime Implicants are:'      )
				print(lowerCAmelCase__      )


if __name__ == "__main__":
				import doctest

				doctest.testmod()
				main()







from __future__ import annotations

from collections import Counter
from random import random


class 							__a							:





		'''simple docstring'''



		def __init__(						self       ):
						SCREAMING_SNAKE_CASE_   :	List[str]													=   {}


		def   __snake_case			(						self ,    UpperCamelCase__       ):
						SCREAMING_SNAKE_CASE_   :	int													=   {}


		def   __snake_case			(						self ,    UpperCamelCase__ ,    UpperCamelCase__ ,    UpperCamelCase__       ):
						if nodea not in self.connections:
										self.add_node(UpperCamelCase__       )
						if nodea not in self.connections:
										self.add_node(UpperCamelCase__       )
						SCREAMING_SNAKE_CASE_   :	Tuple													=   probability


		def   __snake_case			(						self       ):
						return list(self.connections       )






		def   __snake_case			(						self ,    UpperCamelCase__       ):
						SCREAMING_SNAKE_CASE_   :	Dict													=   0
						SCREAMING_SNAKE_CASE_   :	List[Any]													=   random()

						for dest in self.connections[node]:
										current_probability += self.connections[node][dest]
										if current_probability > random_value:
														return dest
						return ""






def 					_lowerCamelCase(			lowerCAmelCase__				:		str    ,    lowerCAmelCase__				:		list[tuple[str, str, float]]    ,    lowerCAmelCase__				:		int      ):





				'''simple docstring'''







				SCREAMING_SNAKE_CASE_   :	Union[str, Any]													=   MarkovChainGraphUndirectedUnweighted()

				for nodea, nodea, probability in transitions:
								graph.add_transition_probability(lowerCAmelCase__    ,    lowerCAmelCase__    ,    lowerCAmelCase__      )

				SCREAMING_SNAKE_CASE_   :	List[str]													=   Counter(graph.get_nodes()      )
				SCREAMING_SNAKE_CASE_   :	Optional[Any]													=   start

				for _ in range(lowerCAmelCase__      ):
								SCREAMING_SNAKE_CASE_   :	Dict													=   graph.transition(lowerCAmelCase__      )
								visited[node] += 1

				return visited


if __name__ == "__main__":
				import doctest

				doctest.testmod()