code: from typing import Any
  
  import numpy as np
  
  class 						A__      :
  
  
  
  
  
  
  
      """simple docstring"""
  
      def __init__( self    ,      __snake_case    ,      __snake_case					):
           # Check if RHS is negative
           if np.any(tableau[:, -1]    ,      where=tableau[:, -1] < 0					):
                raise ValueError('''RHS must be > 0'''					)
  
           snake_case					=       tableau
           snake_case				,						snake_case					=       tableau.shape
  
           # Number of decision variables x1, x2, x3...
           snake_case					=       n_vars
  
           # Number of artificial variables to be minimised
           snake_case					=       len(np.where(tableau[self.n_vars : -1] == -1					)[0]					)
  
           # 2 if there are >= or == constraints (nonstandard), 1 otherwise (std)
           snake_case					=       (self.n_art_vars > 0) + 1
  
           # Number of slack variables added to make inequalities into equalities
           snake_case					=       self.n_rows - self.n_stages
  
           # Objectives for each stage
           snake_case					=       ['''max''']
  
           # In two stage simplex, first minimise then maximise
           if self.n_art_vars:
                self.objectives.append('''min'''					)
  
           snake_case					=       ['''''']
  
           # Index of current pivot row and column
           snake_case					=       None
           snake_case					=       None
  
           # Does objective row only contain (non)-negative values?
           snake_case					=       False
      @staticmethod
      def 	a_    ( *__snake_case					):
           if len(__snake_case					) != 3:
                raise ValueError('''Must provide n_vars, n_slack, and n_art_vars'''					)
  
           if not all(x >= 0 and isinstance(__snake_case    ,      __snake_case					) for x in args					):
                raise ValueError('''All arguments must be non-negative integers'''					)
  
           # decision | slack | artificial
           snake_case					=       ['''x''', '''s''', '''a''']
           snake_case					=       []
           for i in range(3					):
                for j in range(args[i]					):
                     titles.append(string_starts[i] + str(j + 1					)					)
           titles.append('''RHS'''					)
           return titles
      def 	a_    ( self    ,      __snake_case					):
           snake_case					=       self.objectives[-1]
  
           # Find entries of highest magnitude in objective rows
           snake_case					=       (objective == '''min''') - (objective == '''max''')
           snake_case					=       np.argmax(sign * tableau[0, : self.n_vars]					)
  
           # Choice is only valid if below 0 for maximise, and above for minimise
           if sign * self.tableau[0, col_idx] <= 0:
                snake_case					=       True
                return 0, 0
  
           # Pivot row is chosen as having the lowest quotient when elements of
           # the pivot column divide the right-hand side
  
           # Slice excluding the objective rows
           snake_case					=       slice(self.n_stages    ,      self.n_rows					)
  
           # RHS
           snake_case					=       tableau[s, -1]
  
           # Elements of pivot column within slice
           snake_case					=       tableau[s, col_idx]
  
           # Array filled with nans
           snake_case					=       np.full(self.n_rows - self.n_stages    ,      np.nan					)
  
           # If element in pivot column is greater than zeron_stages, return
           # quotient or nan otherwise
           snake_case					=       np.divide(__snake_case    ,      __snake_case    ,      out=__snake_case    ,      where=divisor > 0					)
  
           # Arg of minimum quotient excluding the nan values. n_stages is added
           # to compensate for earlier exclusion of objective columns
           snake_case					=       np.nanargmin(__snake_case					) + self.n_stages
           return row_idx, col_idx
      def 	a_    ( self    ,      __snake_case    ,      __snake_case    ,      __snake_case					):
           snake_case					=       tableau[row_idx].copy()
  
           snake_case					=       piv_row[col_idx]
  
           # Entry becomes 1
           piv_row *= 1 / piv_val
  
           # Variable in pivot column becomes basic, ie the only non-zero entry
           for idx, coeff in enumerate(tableau[:, col_idx]					):
                tableau[idx] += -coeff * piv_row
           snake_case					=       piv_row
           return tableau
      def 	a_    ( self    ,      __snake_case					):
           self.objectives.pop()
  
           if not self.objectives:
                return tableau
  
           # Slice containing ids for artificial columns
           snake_case					=       slice(-self.n_art_vars - 1    ,      -1					)
  
           # Delete the artificial variable columns
           snake_case					=       np.delete(__snake_case    ,      __snake_case    ,      axis=1					)
  
           # Delete the objective row of the first stage
           snake_case					=       np.delete(__snake_case    ,      0    ,      axis=0					)
  
           snake_case					=       1
           self.n_rows -= 1
           snake_case					=       0
           snake_case					=       False
           return tableau
      def 	a_    ( self					):
           # Stop simplex algorithm from cycling.
           for _ in range(1_0_0					):
                # Completion of each stage removes an objective. If both stages
                # are complete, then no objectives are left
                if not self.objectives:
                     snake_case					=       self.generate_col_titles(
                         self.n_vars    ,      self.n_slack    ,      self.n_art_vars					)
  
                     # Find the values of each variable at optimal solution
                     return self.interpret_tableau(self.tableau    ,      self.col_titles					)
  
                snake_case				,						snake_case					=       self.find_pivot(self.tableau					)
  
                # If there are no more negative values in objective row
                if self.stop_iter:
                     # Delete artificial variable columns and rows. Update attributes
                     snake_case					=       self.change_stage(self.tableau					)
                else:
                     snake_case					=       self.pivot(self.tableau    ,      __snake_case    ,      __snake_case					)
           return {}
  
  
  
  
      def 	a_    ( self    ,      __snake_case    ,      __snake_case					):
           snake_case					=       {'''P''': abs(tableau[0, -1]					)}
  
           for i in range(self.n_vars					):
                # Gives ids of nonzero entries in the ith column
                snake_case					=       np.nonzero(tableau[:, i]					)
                snake_case					=       len(nonzero[0]					)
  
                # First entry in the nonzero ids
                snake_case					=       nonzero[0][0]
                snake_case					=       tableau[nonzero_rowidx, i]
  
                # If there is only one nonzero value in column, which is one
                if n_nonzero == nonzero_val == 1:
                     snake_case					=       tableau[nonzero_rowidx, -1]
                     snake_case					=       rhs_val
  
          # Check for basic variables
           for title in col_titles:
                # Don't add RHS or slack variables to output dict
                if title[0] not in "R-s-a":
                     output_dict.setdefault(__snake_case    ,      0					)
           return output_dict
  
  
  if __name__ == "__main__":
   import doctest
  
   doctest.testmod()
  
  
  
  
  
style_context: 
  
  
  from __future__ import annotations
  
  import random
  from collections.abc import Iterable
  
  class 						A__      :
  
  
  
  
  
  
  
      """simple docstring"""
  
      def __init__( self    ,      __snake_case					):
           snake_case					=       {literal: None for literal in literals}
      def __str__( self					):
           return "{" + " , ".join(self.literals					) + "}"
      def __len__( self					):
           return len(self.literals					)
      def 	a_    ( self    ,      __snake_case					):
           for literal in self.literals:
                snake_case					=       literal[:2]
                if symbol in model:
                     snake_case					=       model[symbol]
                else:
                     continue
                if value is not None:
                     # Complement assignment if literal is in complemented form
                     if literal.endswith('''\''''					):
                          snake_case					=       not value
                snake_case					=       value
  
  
  
  
      def 	a_    ( self    ,      __snake_case					):
           for literal in self.literals:
                snake_case					=       literal.rstrip('''\''''					) if literal.endswith('''\''''					) else literal + '''\''''
                if symbol in self.literals:
                     return True
  
           self.assign(__snake_case					)
           for value in self.literals.values():
                if value in (True, None):
                     return value
           return any(self.literals.values()					)
  
  
  
  
  
  
  class 						A__      :
  
  
  
  
  
  
  
      """simple docstring"""
  
      def __init__( self    ,      __snake_case					):
           snake_case					=       list(__snake_case					)
  
  
  
  
      def __str__( self					):
           return "{" + " , ".join(str(__snake_case					) for clause in self.clauses					) + "}"
  
  
  
  
  
  
  def     UpperCAmelCase__		():
  
  
  
       """simple docstring"""
  
  
  
  
  
  
       snake_case					=       []
       snake_case					=       random.randint(1							,5       )
       snake_case					=       '''A'''
       snake_case					=       0
       while i < no_of_literals:
            snake_case					=       random.randint(1							,5       )
            snake_case					=       base_var + str(UpperCamelCase_       )
            snake_case					=       random.randint(0							,1       )
            if var_complement == 1:
                 var_name += "'"
            if var_name in literals:
                 i -= 1
            else:
                 literals.append(UpperCamelCase_       )
            i += 1
       return Clause(UpperCamelCase_       )
  def     UpperCAmelCase__		():
  
  
  
       """simple docstring"""
  
  
  
  
  
  
       snake_case					=       set()
       snake_case					=       random.randint(1							,10       )
       while len(UpperCamelCase_       ) < no_of_clauses:
            clauses.add(generate_clause()       )
       return Formula(UpperCamelCase_       )
  def     UpperCAmelCase__		(UpperCamelCase_       ):
  
  
  
       """simple docstring"""
  
  
  
  
  
  
       snake_case					=       formula.clauses
       snake_case					=       []
       for clause in formula.clauses:
            for literal in clause.literals:
                 snake_case					=       literal[:2]
                 if symbol not in symbols_set:
                      symbols_set.append(UpperCamelCase_       )
       return clauses, symbols_set
  def     UpperCAmelCase__		(UpperCamelCase_							,UpperCamelCase_							,UpperCamelCase_       ):
  
  
  
       """simple docstring"""
  
  
  
  
  
  
       snake_case					=       []
       snake_case					=       {}
       snake_case					=       []
  
       for clause in clauses:
            if clause.evaluate(UpperCamelCase_       ):
                 continue
            for literal in clause.literals:
                 literals.append(UpperCamelCase_       )
  
       for s in symbols:
            snake_case					=       s + '''\''''
            if (s in literals and sym not in literals) or (
                s not in literals and sym in literals
            ):
                 pure_symbols.append(UpperCamelCase_       )
       for p in pure_symbols:
            snake_case					=       None
       for s in pure_symbols:
            snake_case					=       s + '''\''''
            if s in literals:
                 snake_case					=       True
            elif sym in literals:
                 snake_case					=       False
       return pure_symbols, assignment
  def     UpperCAmelCase__		(UpperCamelCase_							,UpperCamelCase_       ):
  
  
  
       """simple docstring"""
  
  
  
  
  
  
       snake_case					=       []
       for clause in clauses:
            if len(UpperCamelCase_       ) == 1:
                 unit_symbols.append(list(clause.literals.keys()       )[0]       )
            else:
                 snake_case				,						snake_case					=       0, 0
                 for literal, value in clause.literals.items():
                      if value is False:
                           f_count += 1
                      elif value is None:
                           snake_case					=       literal
                           n_count += 1
                 if f_count == len(UpperCamelCase_       ) - 1 and n_count == 1:
                      unit_symbols.append(UpperCamelCase_       )
       snake_case					=       {}
       for i in unit_symbols:
            snake_case					=       i[:2]
            snake_case					=       len(UpperCamelCase_       ) == 2
       snake_case					=       [i[:2] for i in unit_symbols]
  
       return unit_symbols, assignment
  def     UpperCAmelCase__		(UpperCamelCase_							,UpperCamelCase_							,UpperCamelCase_       ):
  
  
  
       """simple docstring"""
  
  
  
  
  
  
       snake_case					=       True
       for clause in clauses:
            snake_case					=       clause.evaluate(UpperCamelCase_       )
            if clause_check is False:
                 return False, None
            elif clause_check is None:
                 snake_case					=       False
                 continue
  
       if check_clause_all_true:
            return True, model
  
       try:
            snake_case				,						snake_case					=       find_pure_symbols(UpperCamelCase_							,UpperCamelCase_							,UpperCamelCase_       )
       except RecursionError:
            print('''raises a RecursionError and is'''       )
            return None, {}
       snake_case					=       None
       if len(UpperCamelCase_       ) > 0:
            snake_case				,						snake_case					=       pure_symbols[0], assignment[pure_symbols[0]]
  
       if p:
            snake_case					=       model
            snake_case					=       value
            snake_case					=       list(UpperCamelCase_       )
            if p in tmp_symbols:
                 tmp_symbols.remove(UpperCamelCase_       )
            return dpll_algorithm(UpperCamelCase_							,UpperCamelCase_							,UpperCamelCase_       )
  
       snake_case				,						snake_case					=       find_unit_clauses(UpperCamelCase_							,UpperCamelCase_       )
       snake_case					=       None
       if len(UpperCamelCase_       ) > 0:
            snake_case				,						snake_case					=       unit_symbols[0], assignment[unit_symbols[0]]
       if p:
            snake_case					=       model
            snake_case					=       value
            snake_case					=       list(UpperCamelCase_       )
            if p in tmp_symbols:
                 tmp_symbols.remove(UpperCamelCase_       )
            return dpll_algorithm(UpperCamelCase_							,UpperCamelCase_							,UpperCamelCase_       )
       snake_case					=       symbols[0]
       snake_case					=       symbols[1:]
       snake_case				,						snake_case					=       model, model
       snake_case				,						snake_case					=       True, False
  
       return dpll_algorithm(UpperCamelCase_							,UpperCamelCase_							,UpperCamelCase_       ) or dpll_algorithm(UpperCamelCase_							,UpperCamelCase_							,UpperCamelCase_       )
  
  
  if __name__ == "__main__":
   import doctest
  
   doctest.testmod()
  
   _SCREAMING_SNAKE_CASE    :				Dict       =					generate_formula()
   print(f'''The formula {formula} is''', end=" ")
  
   _SCREAMING_SNAKE_CASE   ,     _SCREAMING_SNAKE_CASE    :				Any       =					generate_parameters(formula)
   _SCREAMING_SNAKE_CASE   ,     _SCREAMING_SNAKE_CASE    :				Optional[Any]       =					dpll_algorithm(clauses, symbols, {})
  
   if solution:
    print(f'''satisfiable with the assignment {model}.''')
   else:
    print("not satisfiable.")
  
  
  
  
  