text: """Representations and Inference for Logic (Chapters 7-9, 12)
  
  Covers both Propositional and First-Order Logic. First we have four
  important data types:
  
      KB            Abstract class holds a knowledge base of logical expressions
      KB_Agent      Abstract class subclasses agents.Agent
      Expr          A logical expression, imported from utils.py
      substitution  Implemented as a dictionary of var:value pairs, {x:1, y:x}
  
  Be careful: some functions take an Expr as argument, and some take a KB.
  
  Logical expressions can be created with Expr or expr, imported from utils, TODO
  or with expr, which adds the capability to write a string that uses
  the connectives ==>, <==, <=>, or <=/=>. But be careful: these have the
  operator precedence of commas; you may need to add parens to make precedence work.
  See logic.ipynb for examples.
  
  Then we implement various functions for doing logical inference:
  
      pl_true          Evaluate a propositional logical sentence in a model
      tt_entails       Say if a statement is entailed by a KB
      pl_resolution    Do resolution on propositional sentences
      dpll_satisfiable See if a propositional sentence is satisfiable
      WalkSAT          Try to find a solution for a set of clauses
  
  And a few other functions:
  
      to_cnf           Convert to conjunctive normal form
      unify            Do unification of two FOL sentences
      diff, simp       Symbolic differentiation and simplification
  """
  
  from utils import (
      removeall, unique, first, argmax, probability,
      isnumber, issequence, Expr, expr, subexpressions
  )
  import agents
  
  import itertools
  import random
  from collections import defaultdict
  
  # ______________________________________________________________________________
  
  
  class KB:
  
      """A knowledge base to which you can tell and ask sentences.
      To create a KB, first subclass this class and implement
      tell, ask_generator, and retract.  Why ask_generator instead of ask?
      The book is a bit vague on what ask means --
      For a Propositional Logic KB, ask(P & Q) returns True or False, but for an
      FOL KB, something like ask(Brother(x, y)) might return many substitutions
      such as {x: Cain, y: Abel}, {x: Abel, y: Cain}, {x: George, y: Jeb}, etc.
      So ask_generator generates these one at a time, and ask either returns the
      first one or returns False."""
  
      def __init__(self, sentence=None):
          raise NotImplementedError
  
      def tell(self, sentence):
          """Add the sentence to the KB."""
          raise NotImplementedError
  
      def ask(self, query):
          """Return a substitution that makes the query true, or, failing that, return False."""
          return first(self.ask_generator(query), default=False)
  
      def ask_generator(self, query):
          """Yield all the substitutions that make query true."""
          raise NotImplementedError
  
      def retract(self, sentence):
          """Remove sentence from the KB."""
          raise NotImplementedError
  
  
  class PropKB(KB):
      """A KB for propositional logic. Inefficient, with no indexing."""
  
      def __init__(self, sentence=None):
          self.clauses = []
          if sentence:
              self.tell(sentence)
  
      def tell(self, sentence):
          """Add the sentence's clauses to the KB."""
          self.clauses.extend(conjuncts(to_cnf(sentence)))
  
      def ask_generator(self, query):
          """Yield the empty substitution {} if KB entails query; else no results."""
          if tt_entails(Expr('&', *self.clauses), query):
              yield {}
  
      def ask_if_true(self, query):
          """Return True if the KB entails query, else return False."""
          for _ in self.ask_generator(query):
              return True
          return False
  
      def retract(self, sentence):
          """Remove the sentence's clauses from the KB."""
          for c in conjuncts(to_cnf(sentence)):
              if c in self.clauses:
                  self.clauses.remove(c)
  
  # ______________________________________________________________________________
  
  
  def KB_AgentProgram(KB):
      """A generic logical knowledge-based agent program. [Figure 7.1]"""
      steps = itertools.count()
  
      def program(percept):
          t = next(steps)
          KB.tell(make_percept_sentence(percept, t))
          action = KB.ask(make_action_query(t))
          KB.tell(make_action_sentence(action, t))
          return action
  
      def make_percept_sentence(percept, t):
          return Expr("Percept")(percept, t)
  
      def make_action_query(t):
          return expr("ShouldDo(action, {})".format(t))
  
      def make_action_sentence(action, t):
          return Expr("Did")(action[expr('action')], t)
  
      return program
  
  
  def is_symbol(s):
      """A string s is a symbol if it starts with an alphabetic char."""
      return isinstance(s, str) and s[:1].isalpha()
  
  
  def is_var_symbol(s):
      """A logic variable symbol is an initial-lowercase string."""
      return is_symbol(s) and s[0].islower()
  
  
  def is_prop_symbol(s):
      """A proposition logic symbol is an initial-uppercase string."""
      return is_symbol(s) and s[0].isupper()
  
  
  def variables(s):
      """Return a set of the variables in expression s.
      >>> variables(expr('F(x, x) & G(x, y) & H(y, z) & R(A, z, 2)')) == {x, y, z}
      True
      """
      return {x for x in subexpressions(s) if is_variable(x)}
  
  
  def is_definite_clause(s):
      """Returns True for exprs s of the form A & B & ... & C ==> D,
      where all literals are positive.  In clause form, this is
      ~A | ~B | ... | ~C | D, where exactly one clause is positive.
      >>> is_definite_clause(expr('Farmer(Mac)'))
      True
      """
      if is_symbol(s.op):
          return True
      elif s.op == '==>':
          antecedent, consequent = s.args
          return (is_symbol(consequent.op) and
                  all(is_symbol(arg.op) for arg in conjuncts(antecedent)))
      else:
          return False
  
  
  def parse_definite_clause(s):
      """Return the antecedents and the consequent of a definite clause."""
      assert is_definite_clause(s)
      if is_symbol(s.op):
          return [], s
      else:
          antecedent, consequent = s.args
          return conjuncts(antecedent), consequent
  
  
  # Useful constant Exprs used in examples and code:
  A, B, C, D, E, F, G, P, Q, x, y, z = map(Expr, 'ABCDEFGPQxyz')
  
  
  # ______________________________________________________________________________
  
  
  def tt_entails(kb, alpha):
      """Does kb entail the sentence alpha? Use truth tables. For propositional
      kb's and sentences. [Figure 7.10]. Note that the 'kb' should be an
      Expr which is a conjunction of clauses.
      >>> tt_entails(expr('P & Q'), expr('Q'))
      True
      """
      assert not variables(alpha)
      symbols = list(prop_symbols(kb & alpha))
      return tt_check_all(kb, alpha, symbols, {})
  
  
  def tt_check_all(kb, alpha, symbols, model):
      """Auxiliary routine to implement tt_entails."""
      if not symbols:
          if pl_true(kb, model):
              result = pl_true(alpha, model)
              assert result in (True, False)
              return result
          else:
              return True
      else:
          P, rest = symbols[0], symbols[1:]
          return (tt_check_all(kb, alpha, rest, extend(model, P, True)) and
                  tt_check_all(kb, alpha, rest, extend(model, P, False)))
  
  
  def prop_symbols(x):
      """Return the set of all propositional symbols in x."""
      if not isinstance(x, Expr):
          return set()
      elif is_prop_symbol(x.op):
          return {x}
      else:
          return {symbol for arg in x.args for symbol in prop_symbols(arg)}
  
  
  def constant_symbols(x):
      """Return the set of all constant symbols in x."""
      if not isinstance(x, Expr):
          return set()
      elif is_prop_symbol(x.op) and not x.args:
          return {x}
      else:
          return {symbol for arg in x.args for symbol in constant_symbols(arg)}
  
  
  def predicate_symbols(x):
      """Return a set of (symbol_name, arity) in x.
      All symbols (even functional) with arity > 0 are considered."""
      if not isinstance(x, Expr) or not x.args:
          return set()
      pred_set = {(x.op, len(x.args))} if is_prop_symbol(x.op) else set()
      pred_set.update({symbol for arg in x.args for symbol in predicate_symbols(arg)})
      return pred_set
  
  
  def tt_true(s):
      """Is a propositional sentence a tautology?
      >>> tt_true('P | ~P')
      True
      """
      s = expr(s)
      return tt_entails(True, s)
  
  
  def pl_true(exp, model={}):
      """Return True if the propositional logic expression is true in the model,
      and False if it is false. If the model does not specify the value for
      every proposition, this may return None to indicate 'not obvious';
      this may happen even when the expression is tautological."""
      if exp in (True, False):
          return exp
      op, args = exp.op, exp.args
      if is_prop_symbol(op):
          return model.get(exp)
      elif op == '~':
          p = pl_true(args[0], model)
          if p is None:
              return None
          else:
              return not p
      elif op == '|':
          result = False
          for arg in args:
              p = pl_true(arg, model)
              if p is True:
                  return True
              if p is None:
                  result = None
          return result
      elif op == '&':
          result = True
          for arg in args:
              p = pl_true(arg, model)
              if p is False:
                  return False
              if p is None:
                  result = None
          return result
      p, q = args
      if op == '==>':
          return pl_true(~p | q, model)
      elif op == '<==':
          return pl_true(p | ~q, model)
      pt = pl_true(p, model)
      if pt is None:
          return None
      qt = pl_true(q, model)
      if qt is None:
          return None
      if op == '<=>':
          return pt == qt
      elif op == '^':  # xor or 'not equivalent'
          return pt != qt
      else:
          raise ValueError("illegal operator in logic expression" + str(exp))
  
  # ______________________________________________________________________________
  
  # Convert to Conjunctive Normal Form (CNF)
  
  
  def to_cnf(s):
      """Convert a propositional logical sentence to conjunctive normal form.
      That is, to the form ((A | ~B | ...) & (B | C | ...) & ...) [p. 253]
      >>> to_cnf('~(B | C)')
      (~B & ~C)
      """
      s = expr(s)
      if isinstance(s, str):
          s = expr(s)
      s = eliminate_implications(s)  # Steps 1, 2 from p. 253
      s = move_not_inwards(s)  # Step 3
      return distribute_and_over_or(s)  # Step 4
  
  
  def eliminate_implications(s):
      """Change implications into equivalent form with only &, |, and ~ as logical operators."""
      s = expr(s)
      if not s.args or is_symbol(s.op):
          return s  # Atoms are unchanged.
      args = list(map(eliminate_implications, s.args))
      a, b = args[0], args[-1]
      if s.op == '==>':
          return b | ~a
      elif s.op == '<==':
          return a | ~b
      elif s.op == '<=>':
          return (a | ~b) & (b | ~a)
      elif s.op == '^':
          assert len(args) == 2  # TODO: relax this restriction
          return (a & ~b) | (~a & b)
      else:
          assert s.op in ('&', '|', '~')
          return Expr(s.op, *args)
  
  
  def move_not_inwards(s):
      """Rewrite sentence s by moving negation sign inward.
      >>> move_not_inwards(~(A | B))
      (~A & ~B)"""
      s = expr(s)
      if s.op == '~':
          def NOT(b):
              return move_not_inwards(~b)
          a = s.args[0]
          if a.op == '~':
              return move_not_inwards(a.args[0])  # ~~A ==> A
          if a.op == '&':
              return associate('|', list(map(NOT, a.args)))
          if a.op == '|':
              return associate('&', list(map(NOT, a.args)))
          return s
      elif is_symbol(s.op) or not s.args:
          return s
      else:
          return Expr(s.op, *list(map(move_not_inwards, s.args)))
  
  
  def distribute_and_over_or(s):
      """Given a sentence s consisting of conjunctions and disjunctions
      of literals, return an equivalent sentence in CNF.
      >>> distribute_and_over_or((A & B) | C)
      ((A | C) & (B | C))
      """
      s = expr(s)
      if s.op == '|':
          s = associate('|', s.args)
          if s.op != '|':
              return distribute_and_over_or(s)
          if len(s.args) == 0:
              return False
          if len(s.args) == 1:
              return distribute_and_over_or(s.args[0])
          conj = first(arg for arg in s.args if arg.op == '&')
          if not conj:
              return s
          others = [a for a in s.args if a is not conj]
          rest = associate('|', others)
          return associate('&', [distribute_and_over_or(c | rest)
                                 for c in conj.args])
      elif s.op == '&':
          return associate('&', list(map(distribute_and_over_or, s.args)))
      else:
          return s
  
  
  def associate(op, args):
      """Given an associative op, return an expression with the same
      meaning as Expr(op, *args), but flattened -- that is, with nested
      instances of the same op promoted to the top level.
      >>> associate('&', [(A&B),(B|C),(B&C)])
      (A & B & (B | C) & B & C)
      >>> associate('|', [A|(B|(C|(A&B)))])
      (A | B | C | (A & B))
      """
      args = dissociate(op, args)
      if len(args) == 0:
          return _op_identity[op]
      elif len(args) == 1:
          return args[0]
      else:
          return Expr(op, *args)
  
  
  _op_identity = {'&': True, '|': False, '+': 0, '*': 1}
  
  
  def dissociate(op, args):
      """Given an associative op, return a flattened list result such
      that Expr(op, *result) means the same as Expr(op, *args)."""
      result = []
  
      def collect(subargs):
          for arg in subargs:
              if arg.op == op:
                  collect(arg.args)
              else:
                  result.append(arg)
      collect(args)
      return result
  
  
  def conjuncts(s):
      """Return a list of the conjuncts in the sentence s.
      >>> conjuncts(A & B)
      [A, B]
      >>> conjuncts(A | B)
      [(A | B)]
      """
      return dissociate('&', [s])
  
  
  def disjuncts(s):
      """Return a list of the disjuncts in the sentence s.
      >>> disjuncts(A | B)
      [A, B]
      >>> disjuncts(A & B)
      [(A & B)]
      """
      return dissociate('|', [s])
  
  # ______________________________________________________________________________
  
  
  def pl_resolution(KB, alpha):
      """Propositional-logic resolution: say if alpha follows from KB. [Figure 7.12]"""
      clauses = KB.clauses + conjuncts(to_cnf(~alpha))
      new = set()
      while True:
          n = len(clauses)
          pairs = [(clauses[i], clauses[j])
                   for i in range(n) for j in range(i+1, n)]
          for (ci, cj) in pairs:
              resolvents = pl_resolve(ci, cj)
              if False in resolvents:
                  return True
              new = new.union(set(resolvents))
          if new.issubset(set(clauses)):
              return False
          for c in new:
              if c not in clauses:
                  clauses.append(c)
  
  
  def pl_resolve(ci, cj):
      """Return all clauses that can be obtained by resolving clauses ci and cj."""
      clauses = []
      for di in disjuncts(ci):
          for dj in disjuncts(cj):
              if di == ~dj or ~di == dj:
                  dnew = unique(removeall(di, disjuncts(ci)) +
                                removeall(dj, disjuncts(cj)))
                  clauses.append(associate('|', dnew))
      return clauses
  
  # ______________________________________________________________________________
  
  
  class PropDefiniteKB(PropKB):
      """A KB of propositional definite clauses."""
  
      def tell(self, sentence):
          """Add a definite clause to this KB."""
          assert is_definite_clause(sentence), "Must be definite clause"
          self.clauses.append(sentence)
  
      def ask_generator(self, query):
          """Yield the empty substitution if KB implies query; else nothing."""
          if pl_fc_entails(self.clauses, query):
              yield {}
  
      def retract(self, sentence):
          self.clauses.remove(sentence)
  
      def clauses_with_premise(self, p):
          """Return a list of the clauses in KB that have p in their premise.
          This could be cached away for O(1) speed, but we'll recompute it."""
          return [c for c in self.clauses
                  if c.op == '==>' and p in conjuncts(c.args[0])]
  
  
  def pl_fc_entails(KB, q):
      """Use forward chaining to see if a PropDefiniteKB entails symbol q.
      [Figure 7.15]
      >>> pl_fc_entails(horn_clauses_KB, expr('Q'))
      True
      """
      count = {c: len(conjuncts(c.args[0]))
               for c in KB.clauses
               if c.op == '==>'}
      inferred = defaultdict(bool)
      agenda = [s for s in KB.clauses if is_prop_symbol(s.op)]
      while agenda:
          p = agenda.pop()
          if p == q:
              return True
          if not inferred[p]:
              inferred[p] = True
              for c in KB.clauses_with_premise(p):
                  count[c] -= 1
                  if count[c] == 0:
                      agenda.append(c.args[1])
      return False
  
  
  """ [Figure 7.13]
  Simple inference in a wumpus world example
  """
  wumpus_world_inference = expr("(B11 <=> (P12 | P21))  &  ~B11")
  
  
  """ [Figure 7.16]
  Propositional Logic Forward Chaining example
  """
  horn_clauses_KB = PropDefiniteKB()
  for s in "P==>Q; (L&M)==>P; (B&L)==>M; (A&P)==>L; (A&B)==>L; A;B".split(';'):
      horn_clauses_KB.tell(expr(s))
  
  # ______________________________________________________________________________
  # DPLL-Satisfiable [Figure 7.17]
  
  
  def dpll_satisfiable(s):
      """Check satisfiability of a propositional sentence.
      This differs from the book code in two ways: (1) it returns a model
      rather than True when it succeeds; this is more useful. (2) The
      function find_pure_symbol is passed a list of unknown clauses, rather
      than a list of all clauses and the model; this is more efficient."""
      clauses = conjuncts(to_cnf(s))
      symbols = list(prop_symbols(s))
      return dpll(clauses, symbols, {})
  
  
  def dpll(clauses, symbols, model):
      """See if the clauses are true in a partial model."""
      unknown_clauses = []  # clauses with an unknown truth value
      for c in clauses:
          val = pl_true(c, model)
          if val is False:
              return False
          if val is not True:
              unknown_clauses.append(c)
      if not unknown_clauses:
          return model
      P, value = find_pure_symbol(symbols, unknown_clauses)
      if P:
          return dpll(clauses, removeall(P, symbols), extend(model, P, value))
      P, value = find_unit_clause(clauses, model)
      if P:
          return dpll(clauses, removeall(P, symbols), extend(model, P, value))
      if not symbols:
          raise TypeError("Argument should be of the type Expr.")
      P, symbols = symbols[0], symbols[1:]
      return (dpll(clauses, symbols, extend(model, P, True)) or
              dpll(clauses, symbols, extend(model, P, False)))
  
  
  def find_pure_symbol(symbols, clauses):
      """Find a symbol and its value if it appears only as a positive literal
      (or only as a negative) in clauses.
      >>> find_pure_symbol([A, B, C], [A|~B,~B|~C,C|A])
      (A, True)
      """
      for s in symbols:
          found_pos, found_neg = False, False
          for c in clauses:
              if not found_pos and s in disjuncts(c):
                  found_pos = True
              if not found_neg and ~s in disjuncts(c):
                  found_neg = True
          if found_pos != found_neg:
              return s, found_pos
      return None, None
  
  
  def find_unit_clause(clauses, model):
      """Find a forced assignment if possible from a clause with only 1
      variable not bound in the model.
      >>> find_unit_clause([A|B|C, B|~C, ~A|~B], {A:True})
      (B, False)
      """
      for clause in clauses:
          P, value = unit_clause_assign(clause, model)
          if P:
              return P, value
      return None, None
  
  
  def unit_clause_assign(clause, model):
      """Return a single variable/value pair that makes clause true in
      the model, if possible.
      >>> unit_clause_assign(A|B|C, {A:True})
      (None, None)
      >>> unit_clause_assign(B|~C, {A:True})
      (None, None)
      >>> unit_clause_assign(~A|~B, {A:True})
      (B, False)
      """
      P, value = None, None
      for literal in disjuncts(clause):
          sym, positive = inspect_literal(literal)
          if sym in model:
              if model[sym] == positive:
                  return None, None  # clause already True
          elif P:
              return None, None      # more than 1 unbound variable
          else:
              P, value = sym, positive
      return P, value
  
  
  def inspect_literal(literal):
      """The symbol in this literal, and the value it should take to
      make the literal true.
      >>> inspect_literal(P)
      (P, True)
      >>> inspect_literal(~P)
      (P, False)
      """
      if literal.op == '~':
          return literal.args[0], False
      else:
          return literal, True
  
  # ______________________________________________________________________________
  # Walk-SAT [Figure 7.18]
  
  
  def WalkSAT(clauses, p=0.5, max_flips=10000):
      """Checks for satisfiability of all clauses by randomly flipping values of variables
      """
      # Set of all symbols in all clauses
      symbols = {sym for clause in clauses for sym in prop_symbols(clause)}
      # model is a random assignment of true/false to the symbols in clauses
      model = {s: random.choice([True, False]) for s in symbols}
      for i in range(max_flips):
          satisfied, unsatisfied = [], []
          for clause in clauses:
              (satisfied if pl_true(clause, model) else unsatisfied).append(clause)
          if not unsatisfied:  # if model satisfies all the clauses
              return model
          clause = random.choice(unsatisfied)
          if probability(p):
              sym = random.choice(list(prop_symbols(clause)))
          else:
              # Flip the symbol in clause that maximizes number of sat. clauses
              def sat_count(sym):
                  # Return the the number of clauses satisfied after flipping the symbol.
                  model[sym] = not model[sym]
                  count = len([clause for clause in clauses if pl_true(clause, model)])
                  model[sym] = not model[sym]
                  return count
              sym = argmax(prop_symbols(clause), key=sat_count)
          model[sym] = not model[sym]
      # If no solution is found within the flip limit, we return failure
      return None
  
  # ______________________________________________________________________________
  
  
  # Expr functions for WumpusKB and HybridWumpusAgent
  
  def facing_east (time):
      return Expr('FacingEast', time)
  
  def facing_west (time):
      return Expr('FacingWest', time)
  
  def facing_north (time):
      return Expr('FacingNorth', time)
  
  def facing_south (time):
      return Expr('FacingSouth', time)
  
  def wumpus (x, y):
      return Expr('W', x, y)
  
  def pit(x, y):
      return Expr('P', x, y)
  
  def breeze(x, y):
      return Expr('B', x, y)
  
  def stench(x, y):
      return Expr('S', x, y)
  
  def wumpus_alive(time):
      return Expr('WumpusAlive', time)
  
  def have_arrow(time):
      return Expr('HaveArrow', time)
  
  def percept_stench(time):
      return Expr('Stench', time)
  
  def percept_breeze(time):
      return Expr('Breeze', time)
  
  def percept_glitter(time):
      return Expr('Glitter', time)
  
  def percept_bump(time):
      return Expr('Bump', time)
  
  def percept_scream(time):
      return Expr('Scream', time)
  
  def move_forward(time):
      return Expr('Forward', time)
  
  def shoot(time):
      return Expr('Shoot', time)
  
  def turn_left(time):
      return Expr('TurnLeft', time)
  
  def turn_right(time):
      return Expr('TurnRight', time)
  
  def ok_to_move(x, y, time):
      return Expr('OK', x, y, time)
  
  def location(x, y, time = None):
      if time is None:
          return Expr('L', x, y)
      else:
          return Expr('L', x, y, time)
  
  # Symbols
  
  def implies(lhs, rhs):
      return Expr('==>', lhs, rhs)
  
  def implies_and_implies(lhs, rhs):
      return Expr('<=>', lhs, rhs)
  
  # Helper Function
  
  def new_disjunction(sentences):
      t = sentences[0]
      for i in range(1,len(sentences)):
          t |= sentences[i]
      return t
  
  
  # ______________________________________________________________________________
  
  
  class WumpusKB(PropKB):
      """
      Create a Knowledge Base that contains the atemporal "Wumpus physics" and temporal rules with time zero.
      """
  
      def __init__(self,dimrow):
          super().__init__()
          self.dimrow = dimrow
          self.tell( ~wumpus(1, 1) )
          self.tell( ~pit(1, 1) )
  
          for y in range(1, dimrow+1):
              for x in range(1, dimrow+1):
  
                  pits_in = list()
                  wumpus_in = list()
  
                  if x > 1: # West room exists
                      pits_in.append(pit(x - 1, y))
                      wumpus_in.append(wumpus(x - 1, y))
  
                  if y < dimrow: # North room exists
                      pits_in.append(pit(x, y + 1))
                      wumpus_in.append(wumpus(x, y + 1))
  
                  if x < dimrow: # East room exists
                      pits_in.append(pit(x + 1, y))
                      wumpus_in.append(wumpus(x + 1, y))
  
                  if y > 1: # South room exists
                      pits_in.append(pit(x, y - 1))
                      wumpus_in.append(wumpus(x, y - 1))
  
                  self.tell(implies_and_implies(breeze(x, y), new_disjunction(pits_in)))
                  self.tell(implies_and_implies(stench(x, y), new_disjunction(wumpus_in)))
  
  
          ## Rule that describes existence of at least one Wumpus
          wumpus_at_least = list()
          for x in range(1, dimrow+1):
              for y in range(1, dimrow + 1):
                  wumps_at_least.append(wumpus(x, y))
  
          self.tell(new_disjunction(wumpus_at_least))
  
  
          ## Rule that describes existence of at most one Wumpus
          for i in range(1, dimrow+1):
              for j in range(1, dimrow+1):
                  for u in range(1, dimrow+1):
                      for v in range(1, dimrow+1):
                          if i!=u or j!=v:
                              self.tell(~wumpus(i, j) | ~wumpus(u, v))
  
  
          ## Temporal rules at time zero
          self.tell(location(1, 1, 0))
          for i in range(1, dimrow+1):
              for j in range(1, dimrow + 1):
                  self.tell(implies(location(i, j, 0), implies_and_implies(percept_breeze(0), breeze(i, j))))
                  self.tell(implies(location(i, j, 0), implies_and_implies(percept_stench(0), stench(i, j))))
                  if i != 1 or j != 1:
                      self.tell(~location(i, j, 0))
  
          self.tell(wumpus_alive(0))
          self.tell(have_arrow(0))
          self.tell(facing_east(0))
          self.tell(~facing_north(0))
          self.tell(~facing_south(0))
          self.tell(~facing_west(0))
  
  
      def make_action_sentence(self, action, time):
          actions = [move_forward(time), shoot(time), turn_left(time), turn_right(time)]
  
          for a in actions:
              if action is a:
                  self.tell(action)
              else:
                  self.tell(~a)
  
      def make_percept_sentence(self, percept, time):
          # Glitter, Bump, Stench, Breeze, Scream
          flags = [0, 0, 0, 0, 0]
  
          ## Things perceived
          if isinstance(percept, Glitter):
              flags[0] = 1
              self.tell(percept_glitter(time))
          elif isinstance(percept, Bump):
              flags[1] = 1
              self.tell(percept_bump(time))
          elif isinstance(percept, Stench):
              flags[2] = 1
              self.tell(percept_stench(time))
          elif isinstance(percept, Breeze):
              flags[3] = 1
              self.tell(percept_breeze(time))
          elif isinstance(percept, Scream):
              flags[4] = 1
              self.tell(percept_scream(time))
  
          ## Things not perceived
          for i in len(range(flags)):
              if flags[i] == 0:
                  if i == 0:
                      self.tell(~percept_glitter(time))
                  elif i == 1:
                      self.tell(~percept_bump(time))
                  elif i == 2:
                      self.tell(~percept_stench(time))
                  elif i == 3:
                      self.tell(~percept_breeze(time))
                  elif i == 4:
                      self.tell(~percept_scream(time))
  
  
      def add_temporal_sentences(self, time):
          if time == 0:
              return
          t = time - 1
  
          ## current location rules
          for i in range(1, self.dimrow+1):
              for j in range(1, self.dimrow+1):
                  self.tell(implies(location(i, j, time), implies_and_implies(percept_breeze(time), breeze(i, j))))
                  self.tell(implies(location(i, j, time), implies_and_implies(percept_stench(time), stench(i, j))))
  
                  s = list()
  
                  s.append(
                      implies_and_implies(
                          location(i, j, time), location(i, j, time) & ~move_forward(time) | percept_bump(time)))
  
                  if i != 1:
                      s.append(location(i - 1, j, t) & facing_east(t) & move_forward(t))
  
                  if i != self.dimrow:
                      s.append(location(i + 1, j, t) & facing_west(t) & move_forward(t))
  
                  if j != 1:
                      s.append(location(i, j - 1, t) & facing_north(t) & move_forward(t))
  
                  if j != self.dimrow:
                      s.append(location(i, j + 1, t) & facing_south(t) & move_forward(t))
  
                  ## add sentence about location i,j
                  self.tell(new_disjunction(s))
  
                  ## add sentence about safety of location i,j
                  self.tell(
                      implies_and_implies(ok_to_move(i, j, time), ~pit(i, j) & ~wumpus(i, j) & wumpus_alive(time))
                  )
  
          ## Rules about current orientation
  
          a = facing_north(t) & turn_right(t)
          b = facing_south(t) & turn_left(t)
          c = facing_east(t) & ~turn_left(t) & ~turn_right(t)
          s = implies_and_implies(facing_east(time), a | b | c)
          self.tell(s)
  
          a = facing_north(t) & turn_left(t)
          b = facing_south(t) & turn_right(t)
          c = facing_west(t) & ~turn_left(t) & ~turn_right(t)
          s = implies_and_implies(facing_west(time), a | b | c)
          self.tell(s)
  
          a = facing_east(t) & turn_left(t)
          b = facing_west(t) & turn_right(t)
          c = facing_north(t) & ~turn_left(t) & ~turn_right(t)
          s = implies_and_implies(facing_north(time), a | b | c)
          self.tell(s)
  
          a = facing_west(t) & turn_left(t)
          b = facing_east(t) & turn_right(t)
          c = facing_south(t) & ~turn_left(t) & ~turn_right(t)
          s = implies_and_implies(facing_south(time), a | b | c)
          self.tell(s)
  
          ## Rules about last action
          self.tell(implies_and_implies(move_forward(t), ~turn_right(t) & ~turn_left(t)))
  
          ##Rule about the arrow
          self.tell(implies_and_implies(have_arrow(time), have_arrow(t) & ~shoot(t)))
  
          ##Rule about Wumpus (dead or alive)
          self.tell(implies_and_implies(wumpus_alive(time), wumpus_alive(t) & ~percept_scream(time)))
  
  
      def ask_if_true(self, query):
          return pl_resolution(self, query)
        
            
  # ______________________________________________________________________________
  
  
  class WumpusPosition():
      def __init__(self, X, Y, orientation):
          self.X = X
          self.Y = Y
          self.orientation = orientation
  
  
      def get_location(self):
          return self.X, self.Y
  
      def get_orientation(self):
          return self.orientation
  
      def equals(self, wumpus_position):
          if wumpus_position.get_location() == self.get_location() and \
                          wumpus_position.get_orientation()==self.get_orientation():
              return True
          else:
              return False
          
  # ______________________________________________________________________________
  
  
  class HybridWumpusAgent(agents.Agent):
      """An agent for the wumpus world that does logical inference. [Figure 7.20]"""
  
      def __init__(self):
          super().__init__()
          self.dimrow = 4
          self.kb = WumpusKB(self.dimrow)
          self.t = 0
          self.plan = list()
          self.current_position = WumpusPosition(1, 1, 'UP')
  
  
      def execute(self, percept):
          self.kb.make_percept_sentence(percept, self.t)
          self.kb.add_temporal_sentences(self.t)
  
          temp = list()
  
          for i in range(1, self.dimrow+1):
              for j in range(1, self.dimrow+1):
                  if self.kb.ask_if_true(location(i, j, self.t)):
                      temp.append(i)
                      temp.append(j)
  
          if self.kb.ask_if_true(facing_north(self.t)):
              self.current_position = WumpusPosition(temp[0], temp[1], 'UP')
          elif self.kb.ask_if_true(facing_south(self.t)):
              self.current_position = WumpusPosition(temp[0], temp[1], 'DOWN')
          elif self.kb.ask_if_true(facing_west(self.t)):
              self.current_position = WumpusPosition(temp[0], temp[1], 'LEFT')
          elif self.kb.ask_if_true(facing_east(self.t)):
              self.current_position = WumpusPosition(temp[0], temp[1], 'RIGHT')
  
          safe_points = list()
          for i in range(1, self.dimrow+1):
              for j in range(1, self.dimrow+1):
                  if self.kb.ask_if_true(ok_to_move(i, j, self.t)):
                      safe_points.append([i, j])
  
          if self.kb.ask_if_true(percept_glitter(self.t)):
              goals = list()
              goals.append([1, 1])
              self.plan.append('Grab')
              actions = plan_route(self.current_position,goals,safe_points)
              for action in actions:
                  self.plan.append(action)
              self.plan.append('Climb')
  
          if len(self.plan) == 0:
              unvisited = list()
              for i in range(1, self.dimrow+1):
                  for j in range(1, self.dimrow+1):
                      for k in range(self.t):
                          if self.kb.ask_if_true(location(i, j, k)):
                              unvisited.append([i, j])
              unvisited_and_safe = list()
              for u in unvisited:
                  for s in safe_points:
                      if u not in unvisited_and_safe and s == u:
                          unvisited_and_safe.append(u)
  
              temp = plan_route(self.current_position,unvisited_and_safe,safe_points)
              for t in temp:
                  self.plan.append(t)
  
          if len(self.plan) == 0 and self.kb.ask_if_true(have_arrow(self.t)):
              possible_wumpus = list()
              for i in range(1, self.dimrow+1):
                  for j in range(1, self.dimrow+1):
                      if not self.kb.ask_if_true(wumpus(i, j)):
                          possible_wumpus.append([i, j])
  
              temp = plan_shot(self.current_position, possible_wumpus, safe_points)
              for t in temp:
                  self.plan.append(t)
  
          if len(self.plan) == 0:
              not_unsafe = list()
              for i in range(1, self.dimrow+1):
                  for j in range(1, self.dimrow+1):
                      if not self.kb.ask_if_true(ok_to_move(i, j, self.t)):
                          not_unsafe.append([i, j])
              temp = plan_route(self.current_position, not_unsafe, safe_points)
              for t in temp:
                  self.plan.append(t)
  
          if len(self.plan) == 0:
              start = list()
              start.append([1, 1])
              temp = plan_route(self.current_position, start, safe_points)
              for t in temp:
                  self.plan.append(t)
              self.plan.append('Climb')
  
          action = self.plan[0]
          self.plan = self.plan[1:]
          self.kb.make_action_sentence(action, self.t)
          self.t += 1
  
          return action
  
  
  def plan_route(current, goals, allowed):
      raise NotImplementedError
  
      
  def plan_shot(current, goals, allowed):
      raise NotImplementedError
  
  
  # ______________________________________________________________________________
  
  
  def SAT_plan(init, transition, goal, t_max, SAT_solver=dpll_satisfiable):
      """Converts a planning problem to Satisfaction problem by translating it to a cnf sentence.
      [Figure 7.22]"""
  
      # Functions used by SAT_plan
      def translate_to_SAT(init, transition, goal, time):
          clauses = []
          states = [state for state in transition]
  
          # Symbol claiming state s at time t
          state_counter = itertools.count()
          for s in states:
              for t in range(time+1):
                  state_sym[s, t] = Expr("State_{}".format(next(state_counter)))
  
          # Add initial state axiom
          clauses.append(state_sym[init, 0])
  
          # Add goal state axiom
          clauses.append(state_sym[goal, time])
  
          # All possible transitions
          transition_counter = itertools.count()
          for s in states:
              for action in transition[s]:
                  s_ = transition[s][action]
                  for t in range(time):
                      # Action 'action' taken from state 's' at time 't' to reach 's_'
                      action_sym[s, action, t] = Expr(
                          "Transition_{}".format(next(transition_counter)))
  
                      # Change the state from s to s_
                      clauses.append(action_sym[s, action, t] |'==>'| state_sym[s, t])
                      clauses.append(action_sym[s, action, t] |'==>'| state_sym[s_, t + 1])
  
          # Allow only one state at any time
          for t in range(time+1):
              # must be a state at any time
              clauses.append(associate('|', [state_sym[s, t] for s in states]))
  
              for s in states:
                  for s_ in states[states.index(s) + 1:]:
                      # for each pair of states s, s_ only one is possible at time t
                      clauses.append((~state_sym[s, t]) | (~state_sym[s_, t]))
  
          # Restrict to one transition per timestep
          for t in range(time):
              # list of possible transitions at time t
              transitions_t = [tr for tr in action_sym if tr[2] == t]
  
              # make sure at least one of the transitions happens
              clauses.append(associate('|', [action_sym[tr] for tr in transitions_t]))
  
              for tr in transitions_t:
                  for tr_ in transitions_t[transitions_t.index(tr) + 1:]:
                      # there cannot be two transitions tr and tr_ at time t
                      clauses.append(~action_sym[tr] | ~action_sym[tr_])
  
          # Combine the clauses to form the cnf
          return associate('&', clauses)
  
      def extract_solution(model):
          true_transitions = [t for t in action_sym if model[action_sym[t]]]
          # Sort transitions based on time, which is the 3rd element of the tuple
          true_transitions.sort(key=lambda x: x[2])
          return [action for s, action, time in true_transitions]
  
      # Body of SAT_plan algorithm
      for t in range(t_max):
          # dictionaries to help extract the solution from model
          state_sym = {}
          action_sym = {}
  
          cnf = translate_to_SAT(init, transition, goal, t)
          model = SAT_solver(cnf)
          if model is not False:
              return extract_solution(model)
      return None
  
  
  # ______________________________________________________________________________
  
  
  def unify(x, y, s={}):
      """Unify expressions x,y with substitution s; return a substitution that
      would make x,y equal, or None if x,y can not unify. x and y can be
      variables (e.g. Expr('x')), constants, lists, or Exprs. [Figure 9.1]"""
      if s is None:
          return None
      elif x == y:
          return s
      elif is_variable(x):
          return unify_var(x, y, s)
      elif is_variable(y):
          return unify_var(y, x, s)
      elif isinstance(x, Expr) and isinstance(y, Expr):
          return unify(x.args, y.args, unify(x.op, y.op, s))
      elif isinstance(x, str) or isinstance(y, str):
          return None
      elif issequence(x) and issequence(y) and len(x) == len(y):
          if not x:
              return s
          return unify(x[1:], y[1:], unify(x[0], y[0], s))
      else:
          return None
  
  
  def is_variable(x):
      """A variable is an Expr with no args and a lowercase symbol as the op."""
      return isinstance(x, Expr) and not x.args and x.op[0].islower()
  
  
  def unify_var(var, x, s):
      if var in s:
          return unify(s[var], x, s)
      elif x in s:
          return unify(var, s[x], s)
      elif occur_check(var, x, s):
          return None
      else:
          return extend(s, var, x)
  
  
  def occur_check(var, x, s):
      """Return true if variable var occurs anywhere in x
      (or in subst(s, x), if s has a binding for x)."""
      if var == x:
          return True
      elif is_variable(x) and x in s:
          return occur_check(var, s[x], s)
      elif isinstance(x, Expr):
          return (occur_check(var, x.op, s) or
                  occur_check(var, x.args, s))
      elif isinstance(x, (list, tuple)):
          return first(e for e in x if occur_check(var, e, s))
      else:
          return False
  
  
  def extend(s, var, val):
      """Copy the substitution s and extend it by setting var to val; return copy."""
      s2 = s.copy()
      s2[var] = val
      return s2
  
  
  def subst(s, x):
      """Substitute the substitution s into the expression x.
      >>> subst({x: 42, y:0}, F(x) + y)
      (F(42) + 0)
      """
      if isinstance(x, list):
          return [subst(s, xi) for xi in x]
      elif isinstance(x, tuple):
          return tuple([subst(s, xi) for xi in x])
      elif not isinstance(x, Expr):
          return x
      elif is_var_symbol(x.op):
          return s.get(x, x)
      else:
          return Expr(x.op, *[subst(s, arg) for arg in x.args])
  
  
  def standardize_variables(sentence, dic=None):
      """Replace all the variables in sentence with new variables."""
      if dic is None:
          dic = {}
      if not isinstance(sentence, Expr):
          return sentence
      elif is_var_symbol(sentence.op):
          if sentence in dic:
              return dic[sentence]
          else:
              v = Expr('v_{}'.format(next(standardize_variables.counter)))
              dic[sentence] = v
              return v
      else:
          return Expr(sentence.op,
                      *[standardize_variables(a, dic) for a in sentence.args])
  
  
  standardize_variables.counter = itertools.count()
  
  # ______________________________________________________________________________
  
  
  class FolKB(KB):
      """A knowledge base consisting of first-order definite clauses.
      >>> kb0 = FolKB([expr('Farmer(Mac)'), expr('Rabbit(Pete)'),
      ...              expr('(Rabbit(r) & Farmer(f)) ==> Hates(f, r)')])
      >>> kb0.tell(expr('Rabbit(Flopsie)'))
      >>> kb0.retract(expr('Rabbit(Pete)'))
      >>> kb0.ask(expr('Hates(Mac, x)'))[x]
      Flopsie
      >>> kb0.ask(expr('Wife(Pete, x)'))
      False
      """
  
      def __init__(self, initial_clauses=None):
          self.clauses = []  # inefficient: no indexing
          if initial_clauses:
              for clause in initial_clauses:
                  self.tell(clause)
  
      def tell(self, sentence):
          if is_definite_clause(sentence):
              self.clauses.append(sentence)
          else:
              raise Exception("Not a definite clause: {}".format(sentence))
  
      def ask_generator(self, query):
          return fol_bc_ask(self, query)
  
      def retract(self, sentence):
          self.clauses.remove(sentence)
  
      def fetch_rules_for_goal(self, goal):
          return self.clauses
  
  
  def fol_fc_ask(KB, alpha):
      """A simple forward-chaining algorithm. [Figure 9.3]"""
      # TODO: Improve efficiency
      kb_consts = list({c for clause in KB.clauses for c in constant_symbols(clause)})
      def enum_subst(p):
          query_vars = list({v for clause in p for v in variables(clause)})
          for assignment_list in itertools.product(kb_consts, repeat=len(query_vars)):
              theta = {x: y for x, y in zip(query_vars, assignment_list)}
              yield theta
  
      # check if we can answer without new inferences
      for q in KB.clauses:
          phi = unify(q, alpha, {})
          if phi is not None:
              yield phi
  
      while True:
          new = []
          for rule in KB.clauses:
              p, q = parse_definite_clause(rule)
              for theta in enum_subst(p):
                  if set(subst(theta, p)).issubset(set(KB.clauses)):
                      q_ = subst(theta, q)
                      if all([unify(x, q_, {}) is None for x in KB.clauses + new]):
                          new.append(q_)
                          phi = unify(q_, alpha, {})
                          if phi is not None:
                              yield phi
          if not new:
              break
          for clause in new:
              KB.tell(clause)
      return None
  
  
  def fol_bc_ask(KB, query):
      """A simple backward-chaining algorithm for first-order logic. [Figure 9.6]
      KB should be an instance of FolKB, and query an atomic sentence."""
      return fol_bc_or(KB, query, {})
  
  
  def fol_bc_or(KB, goal, theta):
      for rule in KB.fetch_rules_for_goal(goal):
          lhs, rhs = parse_definite_clause(standardize_variables(rule))
          for theta1 in fol_bc_and(KB, lhs, unify(rhs, goal, theta)):
              yield theta1
  
  
  def fol_bc_and(KB, goals, theta):
      if theta is None:
          pass
      elif not goals:
          yield theta
      else:
          first, rest = goals[0], goals[1:]
          for theta1 in fol_bc_or(KB, subst(theta, first), theta):
              for theta2 in fol_bc_and(KB, rest, theta1):
                  yield theta2
  
  
  # A simple KB that defines the relevant conditions of the Wumpus World as in Fig 7.4.
  # See Sec. 7.4.3
  wumpus_kb = PropKB()
  
  P11, P12, P21, P22, P31, B11, B21 = expr('P11, P12, P21, P22, P31, B11, B21')
  wumpus_kb.tell(~P11)
  wumpus_kb.tell(B11 | '<=>' | ((P12 | P21)))
  wumpus_kb.tell(B21 | '<=>' | ((P11 | P22 | P31)))
  wumpus_kb.tell(~B11)
  wumpus_kb.tell(B21)
  
  test_kb = FolKB(
      map(expr, ['Farmer(Mac)',
                 'Rabbit(Pete)',
                 'Mother(MrsMac, Mac)',
                 'Mother(MrsRabbit, Pete)',
                 '(Rabbit(r) & Farmer(f)) ==> Hates(f, r)',
                 '(Mother(m, c)) ==> Loves(m, c)',
                 '(Mother(m, r) & Rabbit(r)) ==> Rabbit(m)',
                 '(Farmer(f)) ==> Human(f)',
                 # Note that this order of conjuncts
                 # would result in infinite recursion:
                 # '(Human(h) & Mother(m, h)) ==> Human(m)'
                 '(Mother(m, h) & Human(h)) ==> Human(m)'
                 ]))
  
  crime_kb = FolKB(
      map(expr, ['(American(x) & Weapon(y) & Sells(x, y, z) & Hostile(z)) ==> Criminal(x)',
                 'Owns(Nono, M1)',
                 'Missile(M1)',
                 '(Missile(x) & Owns(Nono, x)) ==> Sells(West, x, Nono)',
                 'Missile(x) ==> Weapon(x)',
                 'Enemy(x, America) ==> Hostile(x)',
                 'American(West)',
                 'Enemy(Nono, America)'
                 ]))
  
  # ______________________________________________________________________________
  
  # Example application (not in the book).
  # You can use the Expr class to do symbolic differentiation.  This used to be
  # a part of AI; now it is considered a separate field, Symbolic Algebra.
  
  
  def diff(y, x):
      """Return the symbolic derivative, dy/dx, as an Expr.
      However, you probably want to simplify the results with simp.
      >>> diff(x * x, x)
      ((x * 1) + (x * 1))
      """
      if y == x:
          return 1
      elif not y.args:
          return 0
      else:
          u, op, v = y.args[0], y.op, y.args[-1]
          if op == '+':
              return diff(u, x) + diff(v, x)
          elif op == '-' and len(y.args) == 1:
              return -diff(u, x)
          elif op == '-':
              return diff(u, x) - diff(v, x)
          elif op == '*':
              return u * diff(v, x) + v * diff(u, x)
          elif op == '/':
              return (v * diff(u, x) - u * diff(v, x)) / (v * v)
          elif op == '**' and isnumber(x.op):
              return (v * u ** (v - 1) * diff(u, x))
          elif op == '**':
              return (v * u ** (v - 1) * diff(u, x) +
                      u ** v * Expr('log')(u) * diff(v, x))
          elif op == 'log':
              return diff(u, x) / u
          else:
              raise ValueError("Unknown op: {} in diff({}, {})".format(op, y, x))
  
  
  def simp(x):
      """Simplify the expression x."""
      if isnumber(x) or not x.args:
          return x
      args = list(map(simp, x.args))
      u, op, v = args[0], x.op, args[-1]
      if op == '+':
          if v == 0:
              return u
          if u == 0:
              return v
          if u == v:
              return 2 * u
          if u == -v or v == -u:
              return 0
      elif op == '-' and len(args) == 1:
          if u.op == '-' and len(u.args) == 1:
              return u.args[0]  # --y ==> y
      elif op == '-':
          if v == 0:
              return u
          if u == 0:
              return -v
          if u == v:
              return 0
          if u == -v or v == -u:
              return 0
      elif op == '*':
          if u == 0 or v == 0:
              return 0
          if u == 1:
              return v
          if v == 1:
              return u
          if u == v:
              return u ** 2
      elif op == '/':
          if u == 0:
              return 0
          if v == 0:
              return Expr('Undefined')
          if u == v:
              return 1
          if u == -v or v == -u:
              return 0
      elif op == '**':
          if u == 0:
              return 0
          if v == 0:
              return 1
          if u == 1:
              return 1
          if v == 1:
              return u
      elif op == 'log':
          if u == 1:
              return 0
      else:
          raise ValueError("Unknown op: " + op)
      # If we fall through to here, we can not simplify further
      return Expr(op, *args)
  
  
  def d(y, x):
      """Differentiate and then simplify."""
      return simp(diff(y, x))
  