planning.py

"""Planning (Chapters 10-11)
"""

import copy
import itertools
from collections import deque, defaultdict
from functools import reduce as _reduce

import search
from csp import sat_up, NaryCSP, Constraint, ac_search_solver, is_
from logic import FolKB, conjuncts, unify, associate, SAT_plan, cdcl_satisfiable
from search import Node
from utils import Expr, expr, first


class PlanningProblem:
    """
    Planning Domain Definition Language (PlanningProblem) used to define a search problem.
    It stores states in a knowledge base consisting of first order logic statements.
    The conjunction of these logical statements completely defines a state.
    """

    def __init__(self, initial, goals, actions, domain=None):
        self.initial = self.convert(initial) if domain is None else self.convert(initial) + self.convert(domain)
        self.goals = self.convert(goals)
        self.actions = actions
        self.domain = domain

    def convert(self, clauses):
        """Converts strings into exprs"""
        if not isinstance(clauses, Expr):
            if len(clauses) > 0:
                clauses = expr(clauses)
            else:
                clauses = []
        try:
            clauses = conjuncts(clauses)
        except AttributeError:
            pass

        new_clauses = []
        for clause in clauses:
            if clause.op == '~':
                new_clauses.append(expr('Not' + str(clause.args[0])))
            else:
                new_clauses.append(clause)
        return new_clauses

    def expand_fluents(self, name=None):

        kb = None
        if self.domain:
            kb = FolKB(self.convert(self.domain))
            for action in self.actions:
                if action.precond:
                    for fests in set(action.precond).union(action.effect).difference(self.convert(action.domain)):
                        if fests.op[:3] != 'Not':
                            kb.tell(expr(str(action.domain) + ' ==> ' + str(fests)))

        objects = set(arg for clause in set(self.initial + self.goals) for arg in clause.args)
        fluent_list = []
        if name is not None:
            for fluent in self.initial + self.goals:
                if str(fluent) == name:
                    fluent_list.append(fluent)
                    break
        else:
            fluent_list = list(map(lambda fluent: Expr(fluent[0], *fluent[1]),
                                   {fluent.op: fluent.args for fluent in self.initial + self.goals +
                                    [clause for action in self.actions for clause in action.effect if
                                     clause.op[:3] != 'Not']}.items()))

        expansions = []
        for fluent in fluent_list:
            for permutation in itertools.permutations(objects, len(fluent.args)):
                new_fluent = Expr(fluent.op, *permutation)
                if (self.domain and kb.ask(new_fluent) is not False) or not self.domain:
                    expansions.append(new_fluent)

        return expansions

    def expand_actions(self, name=None):
        """Generate all possible actions with variable bindings for precondition selection heuristic"""

        has_domains = all(action.domain for action in self.actions if action.precond)
        kb = None
        if has_domains:
            kb = FolKB(self.initial)
            for action in self.actions:
                if action.precond:
                    kb.tell(expr(str(action.domain) + ' ==> ' + str(action)))

        objects = set(arg for clause in self.initial for arg in clause.args)
        expansions = []
        action_list = []
        if name is not None:
            for action in self.actions:
                if str(action.name) == name:
                    action_list.append(action)
                    break
        else:
            action_list = self.actions

        for action in action_list:
            for permutation in itertools.permutations(objects, len(action.args)):
                bindings = unify(Expr(action.name, *action.args), Expr(action.name, *permutation))
                if bindings is not None:
                    new_args = []
                    for arg in action.args:
                        if arg in bindings:
                            new_args.append(bindings[arg])
                        else:
                            new_args.append(arg)
                    new_expr = Expr(str(action.name), *new_args)
                    if (has_domains and kb.ask(new_expr) is not False) or (
                            has_domains and not action.precond) or not has_domains:
                        new_preconds = []
                        for precond in action.precond:
                            new_precond_args = []
                            for arg in precond.args:
                                if arg in bindings:
                                    new_precond_args.append(bindings[arg])
                                else:
                                    new_precond_args.append(arg)
                            new_precond = Expr(str(precond.op), *new_precond_args)
                            new_preconds.append(new_precond)
                        new_effects = []
                        for effect in action.effect:
                            new_effect_args = []
                            for arg in effect.args:
                                if arg in bindings:
                                    new_effect_args.append(bindings[arg])
                                else:
                                    new_effect_args.append(arg)
                            new_effect = Expr(str(effect.op), *new_effect_args)
                            new_effects.append(new_effect)
                        expansions.append(Action(new_expr, new_preconds, new_effects))

        return expansions

    def is_strips(self):
        """
        Returns True if the problem does not contain negative literals in preconditions and goals
        """
        return (all(clause.op[:3] != 'Not' for clause in self.goals) and
                all(clause.op[:3] != 'Not' for action in self.actions for clause in action.precond))

    def goal_test(self):
        """Checks if the goals have been reached"""
        return all(goal in self.initial for goal in self.goals)

    def act(self, action):
        """
        Performs the action given as argument.
        Note that action is an Expr like expr('Remove(Glass, Table)') or expr('Eat(Sandwich)')
        """
        action_name = action.op
        args = action.args
        list_action = first(a for a in self.actions if a.name == action_name)
        if list_action is None:
            raise Exception("Action '{}' not found".format(action_name))
        if not list_action.check_precond(self.initial, args):
            raise Exception("Action '{}' pre-conditions not satisfied".format(action))
        self.initial = list_action(self.initial, args).clauses


class Action:
    """
    Defines an action schema using preconditions and effects.
    Use this to describe actions in PlanningProblem.
    action is an Expr where variables are given as arguments(args).
    Precondition and effect are both lists with positive and negative literals.
    Negative preconditions and effects are defined by adding a 'Not' before the name of the clause
    Example:
    precond = [expr("Human(person)"), expr("Hungry(Person)"), expr("NotEaten(food)")]
    effect = [expr("Eaten(food)"), expr("Hungry(person)")]
    eat = Action(expr("Eat(person, food)"), precond, effect)
    """

    def __init__(self, action, precond, effect, domain=None):
        if isinstance(action, str):
            action = expr(action)
        self.name = action.op
        self.args = action.args
        self.precond = self.convert(precond) if domain is None else self.convert(precond) + self.convert(domain)
        self.effect = self.convert(effect)
        self.domain = domain

    def __call__(self, kb, args):
        return self.act(kb, args)

    def __repr__(self):
        return '{}'.format(Expr(self.name, *self.args))

    def convert(self, clauses):
        """Converts strings into Exprs"""
        if isinstance(clauses, Expr):
            clauses = conjuncts(clauses)
            for i in range(len(clauses)):
                if clauses[i].op == '~':
                    clauses[i] = expr('Not' + str(clauses[i].args[0]))

        elif isinstance(clauses, str):
            clauses = clauses.replace('~', 'Not')
            if len(clauses) > 0:
                clauses = expr(clauses)

            try:
                clauses = conjuncts(clauses)
            except AttributeError:
                pass

        return clauses

    def relaxed(self):
        """
        Removes delete list from the action by removing all negative literals from action's effect
        """
        return Action(Expr(self.name, *self.args), self.precond,
                      list(filter(lambda effect: effect.op[:3] != 'Not', self.effect)))

    def substitute(self, e, args):
        """Replaces variables in expression with their respective Propositional symbol"""

        new_args = list(e.args)
        for num, x in enumerate(e.args):
            for i, _ in enumerate(self.args):
                if self.args[i] == x:
                    new_args[num] = args[i]
        return Expr(e.op, *new_args)

    def check_precond(self, kb, args):
        """Checks if the precondition is satisfied in the current state"""

        if isinstance(kb, list):
            kb = FolKB(kb)
        for clause in self.precond:
            if self.substitute(clause, args) not in kb.clauses:
                return False
        return True

    def act(self, kb, args):
        """Executes the action on the state's knowledge base"""

        if isinstance(kb, list):
            kb = FolKB(kb)

        if not self.check_precond(kb, args):
            raise Exception('Action pre-conditions not satisfied')
        for clause in self.effect:
            kb.tell(self.substitute(clause, args))
            if clause.op[:3] == 'Not':
                new_clause = Expr(clause.op[3:], *clause.args)

                if kb.ask(self.substitute(new_clause, args)) is not False:
                    kb.retract(self.substitute(new_clause, args))
            else:
                new_clause = Expr('Not' + clause.op, *clause.args)

                if kb.ask(self.substitute(new_clause, args)) is not False:
                    kb.retract(self.substitute(new_clause, args))

        return kb


def goal_test(goals, state):
    """Generic goal testing helper function"""

    if isinstance(state, list):
        kb = FolKB(state)
    else:
        kb = state
    return all(kb.ask(q) is not False for q in goals)


def air_cargo():
    """
    [Figure 10.1] AIR-CARGO-PROBLEM

    An air-cargo shipment problem for delivering cargo to different locations,
    given the starting location and airplanes.

    Example:
    >>> from planning import *
    >>> ac = air_cargo()
    >>> ac.goal_test()
    False
    >>> ac.act(expr('Load(C2, P2, JFK)'))
    >>> ac.act(expr('Load(C1, P1, SFO)'))
    >>> ac.act(expr('Fly(P1, SFO, JFK)'))
    >>> ac.act(expr('Fly(P2, JFK, SFO)'))
    >>> ac.act(expr('Unload(C2, P2, SFO)'))
    >>> ac.goal_test()
    False
    >>> ac.act(expr('Unload(C1, P1, JFK)'))
    >>> ac.goal_test()
    True
    >>>
    """

    return PlanningProblem(initial='At(C1, SFO) & At(C2, JFK) & At(P1, SFO) & At(P2, JFK)',
                           goals='At(C1, JFK) & At(C2, SFO)',
                           actions=[Action('Load(c, p, a)',
                                           precond='At(c, a) & At(p, a)',
                                           effect='In(c, p) & ~At(c, a)',
                                           domain='Cargo(c) & Plane(p) & Airport(a)'),
                                    Action('Unload(c, p, a)',
                                           precond='In(c, p) & At(p, a)',
                                           effect='At(c, a) & ~In(c, p)',
                                           domain='Cargo(c) & Plane(p) & Airport(a)'),
                                    Action('Fly(p, f, to)',
                                           precond='At(p, f)',
                                           effect='At(p, to) & ~At(p, f)',
                                           domain='Plane(p) & Airport(f) & Airport(to)')],
                           domain='Cargo(C1) & Cargo(C2) & Plane(P1) & Plane(P2) & Airport(SFO) & Airport(JFK)')


def spare_tire():
    """[Figure 10.2] SPARE-TIRE-PROBLEM

    A problem involving changing the flat tire of a car
    with a spare tire from the trunk.

    Example:
    >>> from planning import *
    >>> st = spare_tire()
    >>> st.goal_test()
    False
    >>> st.act(expr('Remove(Spare, Trunk)'))
    >>> st.act(expr('Remove(Flat, Axle)'))
    >>> st.goal_test()
    False
    >>> st.act(expr('PutOn(Spare, Axle)'))
    >>> st.goal_test()
    True
    >>>
    """

    return PlanningProblem(initial='At(Flat, Axle) & At(Spare, Trunk)',
                           goals='At(Spare, Axle) & At(Flat, Ground)',
                           actions=[Action('Remove(obj, loc)',
                                           precond='At(obj, loc)',
                                           effect='At(obj, Ground) & ~At(obj, loc)',
                                           domain='Tire(obj)'),
                                    Action('PutOn(t, Axle)',
                                           precond='At(t, Ground) & ~At(Flat, Axle)',
                                           effect='At(t, Axle) & ~At(t, Ground)',
                                           domain='Tire(t)'),
                                    Action('LeaveOvernight',
                                           precond='',
                                           effect='~At(Spare, Ground) & ~At(Spare, Axle) & ~At(Spare, Trunk) & \
                                        ~At(Flat, Ground) & ~At(Flat, Axle) & ~At(Flat, Trunk)')],
                           domain='Tire(Flat) & Tire(Spare)')


def three_block_tower():
    """
    [Figure 10.3] THREE-BLOCK-TOWER

    A blocks-world problem of stacking three blocks in a certain configuration,
    also known as the Sussman Anomaly.

    Example:
    >>> from planning import *
    >>> tbt = three_block_tower()
    >>> tbt.goal_test()
    False
    >>> tbt.act(expr('MoveToTable(C, A)'))
    >>> tbt.act(expr('Move(B, Table, C)'))
    >>> tbt.goal_test()
    False
    >>> tbt.act(expr('Move(A, Table, B)'))
    >>> tbt.goal_test()
    True
    >>>
    """
    return PlanningProblem(initial='On(A, Table) & On(B, Table) & On(C, A) & Clear(B) & Clear(C)',
                           goals='On(A, B) & On(B, C)',
                           actions=[Action('Move(b, x, y)',
                                           precond='On(b, x) & Clear(b) & Clear(y)',
                                           effect='On(b, y) & Clear(x) & ~On(b, x) & ~Clear(y)',
                                           domain='Block(b) & Block(y)'),
                                    Action('MoveToTable(b, x)',
                                           precond='On(b, x) & Clear(b)',
                                           effect='On(b, Table) & Clear(x) & ~On(b, x)',
                                           domain='Block(b) & Block(x)')],
                           domain='Block(A) & Block(B) & Block(C)')


def simple_blocks_world():
    """
    SIMPLE-BLOCKS-WORLD

    A simplified definition of the Sussman Anomaly problem.

    Example:
    >>> from planning import *
    >>> sbw = simple_blocks_world()
    >>> sbw.goal_test()
    False
    >>> sbw.act(expr('ToTable(A, B)'))
    >>> sbw.act(expr('FromTable(B, A)'))
    >>> sbw.goal_test()
    False
    >>> sbw.act(expr('FromTable(C, B)'))
    >>> sbw.goal_test()
    True
    >>>
    """

    return PlanningProblem(initial='On(A, B) & Clear(A) & OnTable(B) & OnTable(C) & Clear(C)',
                           goals='On(B, A) & On(C, B)',
                           actions=[Action('ToTable(x, y)',
                                           precond='On(x, y) & Clear(x)',
                                           effect='~On(x, y) & Clear(y) & OnTable(x)'),
                                    Action('FromTable(y, x)',
                                           precond='OnTable(y) & Clear(y) & Clear(x)',
                                           effect='~OnTable(y) & ~Clear(x) & On(y, x)')])


def have_cake_and_eat_cake_too():
    """
    [Figure 10.7] CAKE-PROBLEM

    A problem where we begin with a cake and want to
    reach the state of having a cake and having eaten a cake.
    The possible actions include baking a cake and eating a cake.

    Example:
    >>> from planning import *
    >>> cp = have_cake_and_eat_cake_too()
    >>> cp.goal_test()
    False
    >>> cp.act(expr('Eat(Cake)'))
    >>> cp.goal_test()
    False
    >>> cp.act(expr('Bake(Cake)'))
    >>> cp.goal_test()
    True
    >>>
    """

    return PlanningProblem(initial='Have(Cake)',
                           goals='Have(Cake) & Eaten(Cake)',
                           actions=[Action('Eat(Cake)',
                                           precond='Have(Cake)',
                                           effect='Eaten(Cake) & ~Have(Cake)'),
                                    Action('Bake(Cake)',
                                           precond='~Have(Cake)',
                                           effect='Have(Cake)')])


def shopping_problem():
    """
    SHOPPING-PROBLEM

    A problem of acquiring some items given their availability at certain stores.

    Example:
    >>> from planning import *
    >>> sp = shopping_problem()
    >>> sp.goal_test()
    False
    >>> sp.act(expr('Go(Home, HW)'))
    >>> sp.act(expr('Buy(Drill, HW)'))
    >>> sp.act(expr('Go(HW, SM)'))
    >>> sp.act(expr('Buy(Banana, SM)'))
    >>> sp.goal_test()
    False
    >>> sp.act(expr('Buy(Milk, SM)'))
    >>> sp.goal_test()
    True
    >>>
    """

    return PlanningProblem(initial='At(Home) & Sells(SM, Milk) & Sells(SM, Banana) & Sells(HW, Drill)',
                           goals='Have(Milk) & Have(Banana) & Have(Drill)',
                           actions=[Action('Buy(x, store)',
                                           precond='At(store) & Sells(store, x)',
                                           effect='Have(x)',
                                           domain='Store(store) & Item(x)'),
                                    Action('Go(x, y)',
                                           precond='At(x)',
                                           effect='At(y) & ~At(x)',
                                           domain='Place(x) & Place(y)')],
                           domain='Place(Home) & Place(SM) & Place(HW) & Store(SM) & Store(HW) & '
                                  'Item(Milk) & Item(Banana) & Item(Drill)')


def socks_and_shoes():
    """
    SOCKS-AND-SHOES-PROBLEM

    A task of wearing socks and shoes on both feet

    Example:
    >>> from planning import *
    >>> ss = socks_and_shoes()
    >>> ss.goal_test()
    False
    >>> ss.act(expr('RightSock'))
    >>> ss.act(expr('RightShoe'))
    >>> ss.act(expr('LeftSock'))
    >>> ss.goal_test()
    False
    >>> ss.act(expr('LeftShoe'))
    >>> ss.goal_test()
    True
    >>>
    """

    return PlanningProblem(initial='',
                           goals='RightShoeOn & LeftShoeOn',
                           actions=[Action('RightShoe',
                                           precond='RightSockOn',
                                           effect='RightShoeOn'),
                                    Action('RightSock',
                                           precond='',
                                           effect='RightSockOn'),
                                    Action('LeftShoe',
                                           precond='LeftSockOn',
                                           effect='LeftShoeOn'),
                                    Action('LeftSock',
                                           precond='',
                                           effect='LeftSockOn')])


def double_tennis_problem():
    """
    [Figure 11.10] DOUBLE-TENNIS-PROBLEM

    A multiagent planning problem involving two partner tennis players
    trying to return an approaching ball and repositioning around in the court.

    Example:
    >>> from planning import *
    >>> dtp = double_tennis_problem()
    >>> goal_test(dtp.goals, dtp.initial)
    False
    >>> dtp.act(expr('Go(A, RightBaseLine, LeftBaseLine)'))
    >>> dtp.act(expr('Hit(A, Ball, RightBaseLine)'))
    >>> goal_test(dtp.goals, dtp.initial)
    False
    >>> dtp.act(expr('Go(A, LeftNet, RightBaseLine)'))
    >>> goal_test(dtp.goals, dtp.initial)
    True
    >>>
    """

    return PlanningProblem(
        initial='At(A, LeftBaseLine) & At(B, RightNet) & Approaching(Ball, RightBaseLine) & Partner(A, B) & Partner(B, A)',
        goals='Returned(Ball) & At(a, LeftNet) & At(a, RightNet)',
        actions=[Action('Hit(actor, Ball, loc)',
                        precond='Approaching(Ball, loc) & At(actor, loc)',
                        effect='Returned(Ball)'),
                 Action('Go(actor, to, loc)',
                        precond='At(actor, loc)',
                        effect='At(actor, to) & ~At(actor, loc)')])


class ForwardPlan(search.Problem):
    """
    Forward state-space search [Section 10.2.1]
    """

    def __init__(self, planning_problem):
        super().__init__(associate('&', planning_problem.initial), associate('&', planning_problem.goals))
        self.planning_problem = planning_problem
        self.expanded_actions = self.planning_problem.expand_actions()

    def actions(self, state):
        return [action for action in self.expanded_actions if all(pre in conjuncts(state) for pre in action.precond)]

    def result(self, state, action):
        return associate('&', action(conjuncts(state), action.args).clauses)

    def goal_test(self, state):
        return all(goal in conjuncts(state) for goal in self.planning_problem.goals)

    def h(self, state):
        """
        Computes ignore delete lists heuristic by creating a relaxed version of the original problem (we can do that
        by removing the delete lists from all actions, ie. removing all negative literals from effects) that will be
        easier to solve through GraphPlan and where the length of the solution will serve as a good heuristic.
        """
        relaxed_planning_problem = PlanningProblem(initial=state.state,
                                                   goals=self.goal,
                                                   actions=[action.relaxed() for action in
                                                            self.planning_problem.actions])
        try:
            return len(linearize(GraphPlan(relaxed_planning_problem).execute()))
        except:
            return float('inf')


class BackwardPlan(search.Problem):
    """
    Backward relevant-states search [Section 10.2.2]
    """

    def __init__(self, planning_problem):
        super().__init__(associate('&', planning_problem.goals), associate('&', planning_problem.initial))
        self.planning_problem = planning_problem
        self.expanded_actions = self.planning_problem.expand_actions()

    def actions(self, subgoal):
        """
        Returns True if the action is relevant to the subgoal, ie.:
        - the action achieves an element of the effects
        - the action doesn't delete something that needs to be achieved
        - the preconditions are consistent with other subgoals that need to be achieved
        """

        def negate_clause(clause):
            return Expr(clause.op.replace('Not', ''), *clause.args) if clause.op[:3] == 'Not' else Expr(
                'Not' + clause.op, *clause.args)

        subgoal = conjuncts(subgoal)
        return [action for action in self.expanded_actions if
                (any(prop in action.effect for prop in subgoal) and
                 not any(negate_clause(prop) in subgoal for prop in action.effect) and
                 not any(negate_clause(prop) in subgoal and negate_clause(prop) not in action.effect
                         for prop in action.precond))]

    def result(self, subgoal, action):
        # g' = (g - effects(a)) + preconds(a)
        return associate('&', set(set(conjuncts(subgoal)).difference(action.effect)).union(action.precond))

    def goal_test(self, subgoal):
        return all(goal in conjuncts(self.goal) for goal in conjuncts(subgoal))

    def h(self, subgoal):
        """
        Computes ignore delete lists heuristic by creating a relaxed version of the original problem (we can do that
        by removing the delete lists from all actions, ie. removing all negative literals from effects) that will be
        easier to solve through GraphPlan and where the length of the solution will serve as a good heuristic.
        """
        relaxed_planning_problem = PlanningProblem(initial=self.goal,
                                                   goals=subgoal.state,
                                                   actions=[action.relaxed() for action in
                                                            self.planning_problem.actions])
        try:
            return len(linearize(GraphPlan(relaxed_planning_problem).execute()))
        except:
            return float('inf')


def CSPlan(planning_problem, solution_length, CSP_solver=ac_search_solver, arc_heuristic=sat_up):
    """
        Planning as Constraint Satisfaction Problem [Section 10.4.3]
    """

    def st(var, stage):
        """Returns a string for the var-stage pair that can be used as a variable"""
        return str(var) + "_" + str(stage)

    def if_(v1, v2):
        """If the second argument is v2, the first argument must be v1"""

        def if_fun(x1, x2):
            return x1 == v1 if x2 == v2 else True

        if_fun.__name__ = "if the second argument is " + str(v2) + " then the first argument is " + str(v1) + " "
        return if_fun

    def eq_if_not_in_(actset):
        """First and third arguments are equal if action is not in actset"""

        def eq_if_not_in(x1, a, x2):
            return x1 == x2 if a not in actset else True

        eq_if_not_in.__name__ = "first and third arguments are equal if action is not in " + str(actset) + " "
        return eq_if_not_in

    expanded_actions = planning_problem.expand_actions()
    fluent_values = planning_problem.expand_fluents()
    for horizon in range(solution_length):
        act_vars = [st('action', stage) for stage in range(horizon + 1)]
        domains = {av: list(map(lambda action: expr(str(action)), expanded_actions)) for av in act_vars}
        domains.update({st(var, stage): {True, False} for var in fluent_values for stage in range(horizon + 2)})
        # initial state constraints
        constraints = [Constraint((st(var, 0),), is_(val))
                       for (var, val) in {expr(str(fluent).replace('Not', '')):
                                              True if fluent.op[:3] != 'Not' else False
                                          for fluent in planning_problem.initial}.items()]
        constraints += [Constraint((st(var, 0),), is_(False))
                        for var in {expr(str(fluent).replace('Not', ''))
                                    for fluent in fluent_values if fluent not in planning_problem.initial}]
        # goal state constraints
        constraints += [Constraint((st(var, horizon + 1),), is_(val))
                        for (var, val) in {expr(str(fluent).replace('Not', '')):
                                               True if fluent.op[:3] != 'Not' else False
                                           for fluent in planning_problem.goals}.items()]
        # precondition constraints
        constraints += [Constraint((st(var, stage), st('action', stage)), if_(val, act))
                        # st(var, stage) == val if st('action', stage) == act
                        for act, strps in {expr(str(action)): action for action in expanded_actions}.items()
                        for var, val in {expr(str(fluent).replace('Not', '')):
                                             True if fluent.op[:3] != 'Not' else False
                                         for fluent in strps.precond}.items()
                        for stage in range(horizon + 1)]
        # effect constraints
        constraints += [Constraint((st(var, stage + 1), st('action', stage)), if_(val, act))
                        # st(var, stage + 1) == val if st('action', stage) == act
                        for act, strps in {expr(str(action)): action for action in expanded_actions}.items()
                        for var, val in {expr(str(fluent).replace('Not', '')): True if fluent.op[:3] != 'Not' else False
                                         for fluent in strps.effect}.items()
                        for stage in range(horizon + 1)]
        # frame constraints
        constraints += [Constraint((st(var, stage), st('action', stage), st(var, stage + 1)),
                                   eq_if_not_in_(set(map(lambda action: expr(str(action)),
                                                         {act for act in expanded_actions if var in act.effect
                                                          or Expr('Not' + var.op, *var.args) in act.effect}))))
                        for var in fluent_values for stage in range(horizon + 1)]
        csp = NaryCSP(domains, constraints)
        sol = CSP_solver(csp, arc_heuristic=arc_heuristic)
        if sol:
            return [sol[a] for a in act_vars]


def SATPlan(planning_problem, solution_length, SAT_solver=cdcl_satisfiable):
    """
    Planning as Boolean satisfiability [Section 10.4.1]
    """

    def expand_transitions(state, actions):
        state = sorted(conjuncts(state))
        for action in filter(lambda act: act.check_precond(state, act.args), actions):
            transition[associate('&', state)].update(
                {Expr(action.name, *action.args):
                     associate('&', sorted(set(filter(lambda clause: clause.op[:3] != 'Not',
                                                      action(state, action.args).clauses))))
                     if planning_problem.is_strips()
                     else associate('&', sorted(set(action(state, action.args).clauses)))})
        for state in transition[associate('&', state)].values():
            if state not in transition:
                expand_transitions(expr(state), actions)

    transition = defaultdict(dict)
    expand_transitions(associate('&', planning_problem.initial), planning_problem.expand_actions())

    return SAT_plan(associate('&', sorted(planning_problem.initial)), transition,
                    associate('&', sorted(planning_problem.goals)), solution_length, SAT_solver=SAT_solver)


class Level:
    """
    Contains the state of the planning problem
    and exhaustive list of actions which use the
    states as pre-condition.
    """

    def __init__(self, kb):
        """Initializes variables to hold state and action details of a level"""

        self.kb = kb
        # current state
        self.current_state = kb.clauses
        # current action to state link
        self.current_action_links = {}
        # current state to action link
        self.current_state_links = {}
        # current action to next state link
        self.next_action_links = {}
        # next state to current action link
        self.next_state_links = {}
        # mutually exclusive actions
        self.mutex = []

    def __call__(self, actions, objects):
        self.build(actions, objects)
        self.find_mutex()

    def separate(self, e):
        """Separates an iterable of elements into positive and negative parts"""

        positive = []
        negative = []
        for clause in e:
            if clause.op[:3] == 'Not':
                negative.append(clause)
            else:
                positive.append(clause)
        return positive, negative

    def find_mutex(self):
        """Finds mutually exclusive actions"""

        # Inconsistent effects
        pos_nsl, neg_nsl = self.separate(self.next_state_links)

        for negeff in neg_nsl:
            new_negeff = Expr(negeff.op[3:], *negeff.args)
            for poseff in pos_nsl:
                if new_negeff == poseff:
                    for a in self.next_state_links[poseff]:
                        for b in self.next_state_links[negeff]:
                            if {a, b} not in self.mutex:
                                self.mutex.append({a, b})

        # Interference will be calculated with the last step
        pos_csl, neg_csl = self.separate(self.current_state_links)

        # Competing needs
        for pos_precond in pos_csl:
            for neg_precond in neg_csl:
                new_neg_precond = Expr(neg_precond.op[3:], *neg_precond.args)
                if new_neg_precond == pos_precond:
                    for a in self.current_state_links[pos_precond]:
                        for b in self.current_state_links[neg_precond]:
                            if {a, b} not in self.mutex:
                                self.mutex.append({a, b})

        # Inconsistent support
        state_mutex = []
        for pair in self.mutex:
            next_state_0 = self.next_action_links[list(pair)[0]]
            if len(pair) == 2:
                next_state_1 = self.next_action_links[list(pair)[1]]
            else:
                next_state_1 = self.next_action_links[list(pair)[0]]
            if (len(next_state_0) == 1) and (len(next_state_1) == 1):
                state_mutex.append({next_state_0[0], next_state_1[0]})

        self.mutex = self.mutex + state_mutex

    def build(self, actions, objects):
        """Populates the lists and dictionaries containing the state action dependencies"""

        for clause in self.current_state:
            p_expr = Expr('P' + clause.op, *clause.args)
            self.current_action_links[p_expr] = [clause]
            self.next_action_links[p_expr] = [clause]
            self.current_state_links[clause] = [p_expr]
            self.next_state_links[clause] = [p_expr]

        for a in actions:
            num_args = len(a.args)
            possible_args = tuple(itertools.permutations(objects, num_args))

            for arg in possible_args:
                if a.check_precond(self.kb, arg):
                    for num, symbol in enumerate(a.args):
                        if not symbol.op.islower():
                            arg = list(arg)
                            arg[num] = symbol
                            arg = tuple(arg)

                    new_action = a.substitute(Expr(a.name, *a.args), arg)
                    self.current_action_links[new_action] = []

                    for clause in a.precond:
                        new_clause = a.substitute(clause, arg)
                        self.current_action_links[new_action].append(new_clause)
                        if new_clause in self.current_state_links:
                            self.current_state_links[new_clause].append(new_action)
                        else:
                            self.current_state_links[new_clause] = [new_action]

                    self.next_action_links[new_action] = []
                    for clause in a.effect:
                        new_clause = a.substitute(clause, arg)

                        self.next_action_links[new_action].append(new_clause)
                        if new_clause in self.next_state_links:
                            self.next_state_links[new_clause].append(new_action)
                        else:
                            self.next_state_links[new_clause] = [new_action]

    def perform_actions(self):
        """Performs the necessary actions and returns a new Level"""

        new_kb = FolKB(list(set(self.next_state_links.keys())))
        return Level(new_kb)


class Graph:
    """
    Contains levels of state and actions
    Used in graph planning algorithm to extract a solution
    """

    def __init__(self, planning_problem):
        self.planning_problem = planning_problem
        self.kb = FolKB(planning_problem.initial)
        self.levels = [Level(self.kb)]
        self.objects = set(arg for clause in self.kb.clauses for arg in clause.args)

    def __call__(self):
        self.expand_graph()

    def expand_graph(self):
        """Expands the graph by a level"""

        last_level = self.levels[-1]
        last_level(self.planning_problem.actions, self.objects)
        self.levels.append(last_level.perform_actions())

    def non_mutex_goals(self, goals, index):
        """Checks whether the goals are mutually exclusive"""

        goal_perm = itertools.combinations(goals, 2)
        for g in goal_perm:
            if set(g) in self.levels[index].mutex:
                return False
        return True


class GraphPlan:
    """
    Class for formulation GraphPlan algorithm
    Constructs a graph of state and action space
    Returns solution for the planning problem
    """

    def __init__(self, planning_problem):
        self.graph = Graph(planning_problem)
        self.no_goods = []
        self.solution = []

    def check_leveloff(self):
        """Checks if the graph has levelled off"""

        check = (set(self.graph.levels[-1].current_state) == set(self.graph.levels[-2].current_state))

        if check:
            return True

    def extract_solution(self, goals, index):
        """Extracts the solution"""

        level = self.graph.levels[index]
        if not self.graph.non_mutex_goals(goals, index):
            self.no_goods.append((level, goals))
            return

        level = self.graph.levels[index - 1]

        # Create all combinations of actions that satisfy the goal
        actions = []
        for goal in goals:
            actions.append(level.next_state_links[goal])

        all_actions = list(itertools.product(*actions))

        # Filter out non-mutex actions
        non_mutex_actions = []
        for action_tuple in all_actions:
            action_pairs = itertools.combinations(list(set(action_tuple)), 2)
            non_mutex_actions.append(list(set(action_tuple)))
            for pair in action_pairs:
                if set(pair) in level.mutex:
                    non_mutex_actions.pop(-1)
                    break

        # Recursion
        for action_list in non_mutex_actions:
            if [action_list, index] not in self.solution:
                self.solution.append([action_list, index])

                new_goals = []
                for act in set(action_list):
                    if act in level.current_action_links:
                        new_goals = new_goals + level.current_action_links[act]

                if abs(index) + 1 == len(self.graph.levels):
                    return
                elif (level, new_goals) in self.no_goods:
                    return
                else:
                    self.extract_solution(new_goals, index - 1)

        # Level-Order multiple solutions
        solution = []
        for item in self.solution:
            if item[1] == -1:
                solution.append([])
                solution[-1].append(item[0])
            else:
                solution[-1].append(item[0])

        for num, item in enumerate(solution):
            item.reverse()
            solution[num] = item

        return solution

    def goal_test(self, kb):
        return all(kb.ask(q) is not False for q in self.graph.planning_problem.goals)

    def execute(self):
        """Executes the GraphPlan algorithm for the given problem"""

        while True:
            self.graph.expand_graph()
            if (self.goal_test(self.graph.levels[-1].kb) and self.graph.non_mutex_goals(
                    self.graph.planning_problem.goals, -1)):
                solution = self.extract_solution(self.graph.planning_problem.goals, -1)
                if solution:
                    return solution