diff --git a/README.md b/README.md index 900ef3324..08d59b481 100644 --- a/README.md +++ b/README.md @@ -72,10 +72,10 @@ Here is a table of algorithms, the figure, name of the algorithm in the book and | 3.2 | Romania | `romania` | [`search.py`][search] | Done | Included | | 3.7 | Tree-Search | `tree_search` | [`search.py`][search] | Done | | | 3.7 | Graph-Search | `graph_search` | [`search.py`][search] | Done | | -| 3.11 | Breadth-First-Search | `breadth_first_graph_search` | [`search.py`][search] | Done | Included | +| 3.11 | Breadth-First-Search | `breadth_first_graph_search` | [`search.py`][search] | Done | Included | | 3.14 | Uniform-Cost-Search | `uniform_cost_search` | [`search.py`][search] | Done | Included | -| 3.17 | Depth-Limited-Search | `depth_limited_search` | [`search.py`][search] | Done | Included | -| 3.18 | Iterative-Deepening-Search | `iterative_deepening_search` | [`search.py`][search] | Done | Included | +| 3.17 | Depth-Limited-Search | `depth_limited_search` | [`search.py`][search] | Done | Included | +| 3.18 | Iterative-Deepening-Search | `iterative_deepening_search` | [`search.py`][search] | Done | Included | | 3.22 | Best-First-Search | `best_first_graph_search` | [`search.py`][search] | Done | Included | | 3.24 | A\*-Search | `astar_search` | [`search.py`][search] | Done | Included | | 3.26 | Recursive-Best-First-Search | `recursive_best_first_search` | [`search.py`][search] | Done | | @@ -102,7 +102,7 @@ Here is a table of algorithms, the figure, name of the algorithm in the book and | 7.17 | DPLL-Satisfiable? | `dpll_satisfiable` | [`logic.py`][logic] | Done | Included | | 7.18 | WalkSAT | `WalkSAT` | [`logic.py`][logic] | Done | Included | | 7.20 | Hybrid-Wumpus-Agent | `HybridWumpusAgent` | | | | -| 7.22 | SATPlan | `SAT_plan` | [`logic.py`][logic] | Done | Included | +| 7.22 | SATPlan | `SAT_plan` | [`logic.py`][logic] | Done | Included | | 9 | Subst | `subst` | [`logic.py`][logic] | Done | | | 9.1 | Unify | `unify` | [`logic.py`][logic] | Done | Included | | 9.3 | FOL-FC-Ask | `fol_fc_ask` | [`logic.py`][logic] | Done | Included | @@ -111,8 +111,8 @@ Here is a table of algorithms, the figure, name of the algorithm in the book and | 10.1 | Air-Cargo-problem | `air_cargo` | [`planning.py`][planning] | Done | Included | | 10.2 | Spare-Tire-Problem | `spare_tire` | [`planning.py`][planning] | Done | Included | | 10.3 | Three-Block-Tower | `three_block_tower` | [`planning.py`][planning] | Done | Included | -| 10.7 | Cake-Problem | `have_cake_and_eat_cake_too` | [`planning.py`][planning] | Done | Included | -| 10.9 | Graphplan | `GraphPlan` | [`planning.py`][planning] | | | +| 10.7 | Cake-Problem | `have_cake_and_eat_cake_too` | [`planning.py`][planning] | Done | Included | +| 10.9 | Graphplan | `GraphPlan` | [`planning.py`][planning] | Done | Included | | 10.13 | Partial-Order-Planner | | | | | | 11.1 | Job-Shop-Problem-With-Resources | `job_shop_problem` | [`planning.py`][planning] | Done | | | 11.5 | Hierarchical-Search | `hierarchical_search` | [`planning.py`][planning] | | | diff --git a/images/cake_graph.jpg b/images/cake_graph.jpg new file mode 100644 index 000000000..160a413ca Binary files /dev/null and b/images/cake_graph.jpg differ diff --git a/planning.ipynb b/planning.ipynb index 6a79a3100..fd21a6e88 100644 --- a/planning.ipynb +++ b/planning.ipynb @@ -6,309 +6,1784 @@ "collapsed": true }, "source": [ - "# Planning: planning.py; chapters 10-11" + "# Planning\n", + "#### Chapters 10-11\n", + "----" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ - "This notebook describes the [planning.py](https://github.com/aimacode/aima-python/blob/master/planning.py) module, which covers Chapters 10 (Classical Planning) and 11 (Planning and Acting in the Real World) of *[Artificial Intelligence: A Modern Approach](http://aima.cs.berkeley.edu)*. See the [intro notebook](https://github.com/aimacode/aima-python/blob/master/intro.ipynb) for instructions.\n", + "This notebook serves as supporting material for topics covered in **Chapter 10 - Classical Planning** and **Chapter 11 - Planning and Acting in the Real World** from the book *[Artificial Intelligence: A Modern Approach](http://aima.cs.berkeley.edu)*. \n", + "This notebook uses implementations from the [planning.py](https://github.com/aimacode/aima-python/blob/master/planning.py) module. \n", + "See the [intro notebook](https://github.com/aimacode/aima-python/blob/master/intro.ipynb) for instructions.\n", "\n", - "We'll start by looking at `PDDL` and `Action` data types for defining problems and actions. Then, we will see how to use them by trying to plan a trip from *Sibiu* to *Bucharest* across the familiar map of Romania, from [search.ipynb](https://github.com/aimacode/aima-python/blob/master/search.ipynb). Finally, we will look at the implementation of the GraphPlan algorithm.\n", + "We'll start by looking at `PDDL` and `Action` data types for defining problems and actions. \n", + "Then, we will see how to use them by trying to plan a trip from *Sibiu* to *Bucharest* across the familiar map of Romania, from [search.ipynb](https://github.com/aimacode/aima-python/blob/master/search.ipynb) \n", + "followed by some common planning problems and methods of solving them.\n", "\n", - "The first step is to load the code:" + "Let's start by importing everything from the planning module." ] }, { "cell_type": "code", "execution_count": 1, + "metadata": { + "collapsed": true + }, + "outputs": [], + "source": [ + "from planning import *\n", + "from notebook import psource" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "## CONTENTS\n", + "\n", + "- PDDL\n", + "- Action\n", + "- Planning Problems\n", + " * Air cargo problem\n", + " * Spare tire problem\n", + " * Three block tower problem\n", + " * Shopping Problem\n", + " * Cake problem\n", + "- Solving Planning Problems\n", + " * GraphPlan" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "## PDDL\n", + "\n", + "PDDL stands for Planning Domain Definition Language.\n", + "The `PDDL` class is used to represent planning problems in this module. The following attributes are essential to be able to define a problem:\n", + "* an initial state\n", + "* a set of goals\n", + "* a set of viable actions that can be executed in the search space of the problem\n", + "\n", + "View the source to see how the Python code tries to realise these." + ] + }, + { + "cell_type": "code", + "execution_count": 2, + "metadata": {}, + "outputs": [ + { + "data": { + "text/html": [ + "\n", + "\n", + "\n", + "\n", + " \n", + " \n", + " \n", + "\n", + "\n", + "

\n", + "\n", + "
class PDDL:\n",
+       "    """\n",
+       "    Planning Domain Definition Language (PDDL) used to define a search problem.\n",
+       "    It stores states in a knowledge base consisting of first order logic statements.\n",
+       "    The conjunction of these logical statements completely defines a state.\n",
+       "    """\n",
+       "\n",
+       "    def __init__(self, init, goals, actions):\n",
+       "        self.init = self.convert(init)\n",
+       "        self.goals = expr(goals)\n",
+       "        self.actions = actions\n",
+       "\n",
+       "    def convert(self, init):\n",
+       "        """Converts strings into exprs"""\n",
+       "        try:\n",
+       "            init = conjuncts(expr(init))\n",
+       "        except AttributeError:\n",
+       "            init = expr(init)\n",
+       "        return init\n",
+       "\n",
+       "    def goal_test(self):\n",
+       "        """Checks if the goals have been reached"""\n",
+       "        return all(goal in self.init for goal in conjuncts(self.goals))\n",
+       "\n",
+       "    def act(self, action):\n",
+       "        """\n",
+       "        Performs the action given as argument.\n",
+       "        Note that action is an Expr like expr('Remove(Glass, Table)') or expr('Eat(Sandwich)')\n",
+       "        """       \n",
+       "        action_name = action.op\n",
+       "        args = action.args\n",
+       "        list_action = first(a for a in self.actions if a.name == action_name)\n",
+       "        if list_action is None:\n",
+       "            raise Exception("Action '{}' not found".format(action_name))\n",
+       "        if not list_action.check_precond(self.init, args):\n",
+       "            raise Exception("Action '{}' pre-conditions not satisfied".format(action))\n",
+       "        self.init = list_action(self.init, args).clauses\n",
+       "
\n", + "\n", + "\n" + ], + "text/plain": [ + "" + ] + }, + "metadata": {}, + "output_type": "display_data" + } + ], + "source": [ + "psource(PDDL)" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "The `init` attribute is an expression that forms the initial knowledge base for the problem.\n", + "
\n", + "The `goals` attribute is an expression that indicates the goals to be reached by the problem.\n", + "
\n", + "Lastly, `actions` contains a list of `Action` objects that may be executed in the search space of the problem.\n", + "
\n", + "The `goal_test` method checks if the goal has been reached.\n", + "
\n", + "The `act` method acts out the given action and updates the current state.\n", + "
\n" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "## ACTION\n", + "\n", + "To be able to model a planning problem properly, it is essential to be able to represent an Action. Each action we model requires at least three things:\n", + "* preconditions that the action must meet\n", + "* the effects of executing the action\n", + "* some expression that represents the action" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "The module models actions using the `Action` class" + ] + }, + { + "cell_type": "code", + "execution_count": 3, + "metadata": {}, + "outputs": [ + { + "data": { + "text/html": [ + "\n", + "\n", + "\n", + "\n", + " \n", + " \n", + " \n", + "\n", + "\n", + "

\n", + "\n", + "
class Action:\n",
+       "    """\n",
+       "    Defines an action schema using preconditions and effects.\n",
+       "    Use this to describe actions in PDDL.\n",
+       "    action is an Expr where variables are given as arguments(args).\n",
+       "    Precondition and effect are both lists with positive and negative literals.\n",
+       "    Negative preconditions and effects are defined by adding a 'Not' before the name of the clause\n",
+       "    Example:\n",
+       "    precond = [expr("Human(person)"), expr("Hungry(Person)"), expr("NotEaten(food)")]\n",
+       "    effect = [expr("Eaten(food)"), expr("Hungry(person)")]\n",
+       "    eat = Action(expr("Eat(person, food)"), precond, effect)\n",
+       "    """\n",
+       "\n",
+       "    def __init__(self, action, precond, effect):\n",
+       "        action = expr(action)\n",
+       "        self.name = action.op\n",
+       "        self.args = action.args\n",
+       "        self.precond, self.effect = self.convert(precond, effect)\n",
+       "\n",
+       "    def __call__(self, kb, args):\n",
+       "        return self.act(kb, args)\n",
+       "\n",
+       "    def convert(self, precond, effect):\n",
+       "        """Converts strings into Exprs"""\n",
+       "\n",
+       "        precond = precond.replace('~', 'Not')\n",
+       "        if len(precond) > 0:\n",
+       "            precond = expr(precond)\n",
+       "        effect = effect.replace('~', 'Not')\n",
+       "        if len(effect) > 0:\n",
+       "            effect = expr(effect)\n",
+       "\n",
+       "        try:\n",
+       "            precond = conjuncts(precond)\n",
+       "        except AttributeError:\n",
+       "            pass\n",
+       "        try:\n",
+       "            effect = conjuncts(effect)\n",
+       "        except AttributeError:\n",
+       "            pass\n",
+       "\n",
+       "        return precond, effect\n",
+       "\n",
+       "    def substitute(self, e, args):\n",
+       "        """Replaces variables in expression with their respective Propositional symbol"""\n",
+       "\n",
+       "        new_args = list(e.args)\n",
+       "        for num, x in enumerate(e.args):\n",
+       "            for i, _ in enumerate(self.args):\n",
+       "                if self.args[i] == x:\n",
+       "                    new_args[num] = args[i]\n",
+       "        return Expr(e.op, *new_args)\n",
+       "\n",
+       "    def check_precond(self, kb, args):\n",
+       "        """Checks if the precondition is satisfied in the current state"""\n",
+       "\n",
+       "        if isinstance(kb, list):\n",
+       "            kb = FolKB(kb)\n",
+       "\n",
+       "        for clause in self.precond:\n",
+       "            if self.substitute(clause, args) not in kb.clauses:\n",
+       "                return False\n",
+       "        return True\n",
+       "\n",
+       "    def act(self, kb, args):\n",
+       "        """Executes the action on the state's knowledge base"""\n",
+       "\n",
+       "        if isinstance(kb, list):\n",
+       "            kb = FolKB(kb)\n",
+       "\n",
+       "        if not self.check_precond(kb, args):\n",
+       "            raise Exception('Action pre-conditions not satisfied')\n",
+       "        for clause in self.effect:\n",
+       "            kb.tell(self.substitute(clause, args))\n",
+       "            if clause.op[:3] == 'Not':\n",
+       "                new_clause = Expr(clause.op[3:], *clause.args)\n",
+       "\n",
+       "                if kb.ask(self.substitute(new_clause, args)) is not False:\n",
+       "                    kb.retract(self.substitute(new_clause, args))\n",
+       "            else:\n",
+       "                new_clause = Expr('Not' + clause.op, *clause.args)\n",
+       "\n",
+       "                if kb.ask(self.substitute(new_clause, args)) is not False:    \n",
+       "                    kb.retract(self.substitute(new_clause, args))\n",
+       "\n",
+       "        return kb\n",
+       "
\n", + "\n", + "\n" + ], + "text/plain": [ + "" + ] + }, + "metadata": {}, + "output_type": "display_data" + } + ], + "source": [ + "psource(Action)" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "This class represents an action given the expression, the preconditions and its effects. \n", + "A list `precond` stores the preconditions of the action and a list `effect` stores its effects.\n", + "Negative preconditions and effects are input using a `~` symbol before the clause, which are internally prefixed with a `Not` to make it easier to work with.\n", + "For example, the negation of `At(obj, loc)` will be input as `~At(obj, loc)` and internally represented as `NotAt(obj, loc)`. \n", + "This equivalently creates a new clause for each negative literal, removing the hassle of maintaining two separate knowledge bases.\n", + "This greatly simplifies algorithms like `GraphPlan` as we will see later.\n", + "The `convert` method takes an input string, parses it, removes conjunctions if any and returns a list of `Expr` objects.\n", + "The `check_precond` method checks if the preconditions for that action are valid, given a `kb`.\n", + "The `act` method carries out the action on the given knowledge base." + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "Now lets try to define a planning problem using these tools. Since we already know about the map of Romania, lets see if we can plan a trip across a simplified map of Romania.\n", + "\n", + "Here is our simplified map definition:" + ] + }, + { + "cell_type": "code", + "execution_count": 4, + "metadata": { + "collapsed": true + }, + "outputs": [], + "source": [ + "from utils import *\n", + "# this imports the required expr so we can create our knowledge base\n", + "\n", + "knowledge_base = [\n", + " expr(\"Connected(Bucharest,Pitesti)\"),\n", + " expr(\"Connected(Pitesti,Rimnicu)\"),\n", + " expr(\"Connected(Rimnicu,Sibiu)\"),\n", + " expr(\"Connected(Sibiu,Fagaras)\"),\n", + " expr(\"Connected(Fagaras,Bucharest)\"),\n", + " expr(\"Connected(Pitesti,Craiova)\"),\n", + " expr(\"Connected(Craiova,Rimnicu)\")\n", + " ]" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "Let us add some logic propositions to complete our knowledge about travelling around the map. These are the typical symmetry and transitivity properties of connections on a map. We can now be sure that our `knowledge_base` understands what it truly means for two locations to be connected in the sense usually meant by humans when we use the term.\n", + "\n", + "Let's also add our starting location - *Sibiu* to the map." + ] + }, + { + "cell_type": "code", + "execution_count": 5, + "metadata": { + "collapsed": true + }, + "outputs": [], + "source": [ + "knowledge_base.extend([\n", + " expr(\"Connected(x,y) ==> Connected(y,x)\"),\n", + " expr(\"Connected(x,y) & Connected(y,z) ==> Connected(x,z)\"),\n", + " expr(\"At(Sibiu)\")\n", + " ])" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "We now have a complete knowledge base, which can be seen like this:" + ] + }, + { + "cell_type": "code", + "execution_count": 6, + "metadata": {}, + "outputs": [ + { + "data": { + "text/plain": [ + "[Connected(Bucharest, Pitesti),\n", + " Connected(Pitesti, Rimnicu),\n", + " Connected(Rimnicu, Sibiu),\n", + " Connected(Sibiu, Fagaras),\n", + " Connected(Fagaras, Bucharest),\n", + " Connected(Pitesti, Craiova),\n", + " Connected(Craiova, Rimnicu),\n", + " (Connected(x, y) ==> Connected(y, x)),\n", + " ((Connected(x, y) & Connected(y, z)) ==> Connected(x, z)),\n", + " At(Sibiu)]" + ] + }, + "execution_count": 6, + "metadata": {}, + "output_type": "execute_result" + } + ], + "source": [ + "knowledge_base" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "We now define possible actions to our problem. We know that we can drive between any connected places. But, as is evident from [this](https://en.wikipedia.org/wiki/List_of_airports_in_Romania) list of Romanian airports, we can also fly directly between Sibiu, Bucharest, and Craiova.\n", + "\n", + "We can define these flight actions like this:" + ] + }, + { + "cell_type": "code", + "execution_count": 7, + "metadata": { + "collapsed": true + }, + "outputs": [], + "source": [ + "#Sibiu to Bucharest\n", + "precond = 'At(Sibiu)'\n", + "effect = 'At(Bucharest) & ~At(Sibiu)'\n", + "fly_s_b = Action('Fly(Sibiu, Bucharest)', precond, effect)\n", + "\n", + "#Bucharest to Sibiu\n", + "precond = 'At(Bucharest)'\n", + "effect = 'At(Sibiu) & ~At(Bucharest)'\n", + "fly_b_s = Action('Fly(Bucharest, Sibiu)', precond, effect)\n", + "\n", + "#Sibiu to Craiova\n", + "precond = 'At(Sibiu)'\n", + "effect = 'At(Craiova) & ~At(Sibiu)'\n", + "fly_s_c = Action('Fly(Sibiu, Craiova)', precond, effect)\n", + "\n", + "#Craiova to Sibiu\n", + "precond = 'At(Craiova)'\n", + "effect = 'At(Sibiu) & ~At(Craiova)'\n", + "fly_c_s = Action('Fly(Craiova, Sibiu)', precond, effect)\n", + "\n", + "#Bucharest to Craiova\n", + "precond = 'At(Bucharest)'\n", + "effect = 'At(Craiova) & ~At(Bucharest)'\n", + "fly_b_c = Action('Fly(Bucharest, Craiova)', precond, effect)\n", + "\n", + "#Craiova to Bucharest\n", + "precond = 'At(Craiova)'\n", + "effect = 'At(Bucharest) & ~At(Craiova)'\n", + "fly_c_b = Action('Fly(Craiova, Bucharest)', precond, effect)" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "And the drive actions like this." + ] + }, + { + "cell_type": "code", + "execution_count": 8, + "metadata": { + "collapsed": true + }, + "outputs": [], + "source": [ + "#Drive\n", + "precond = 'At(x)'\n", + "effect = 'At(y) & ~At(x)'\n", + "drive = Action('Drive(x, y)', precond, effect)" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "Our goal is defined as" + ] + }, + { + "cell_type": "code", + "execution_count": 9, + "metadata": { + "collapsed": true + }, + "outputs": [], + "source": [ + "goals = 'At(Bucharest)'" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "Finally, we can define a a function that will tell us when we have reached our destination, Bucharest." + ] + }, + { + "cell_type": "code", + "execution_count": 10, + "metadata": { + "collapsed": true + }, + "outputs": [], + "source": [ + "def goal_test(kb):\n", + " return kb.ask(expr('At(Bucharest)'))" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "Thus, with all the components in place, we can define the planning problem." + ] + }, + { + "cell_type": "code", + "execution_count": 11, + "metadata": { + "collapsed": true + }, + "outputs": [], + "source": [ + "prob = PDDL(knowledge_base, goals, [fly_s_b, fly_b_s, fly_s_c, fly_c_s, fly_b_c, fly_c_b, drive])" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "## PLANNING PROBLEMS\n", + "---\n", + "\n", + "## Air Cargo Problem" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "In the Air Cargo problem, we start with cargo at two airports, SFO and JFK. Our goal is to send each cargo to the other airport. We have two airplanes to help us accomplish the task. \n", + "The problem can be defined with three actions: Load, Unload and Fly. \n", + "Let us look how the `air_cargo` problem has been defined in the module. " + ] + }, + { + "cell_type": "code", + "execution_count": 12, + "metadata": {}, + "outputs": [ + { + "data": { + "text/html": [ + "\n", + "\n", + "\n", + "\n", + " \n", + " \n", + " \n", + "\n", + "\n", + "

\n", + "\n", + "
def air_cargo():\n",
+       "    """Air cargo problem"""\n",
+       "\n",
+       "    return PDDL(init='At(C1, SFO) & At(C2, JFK) & At(P1, SFO) & At(P2, JFK) & Cargo(C1) & Cargo(C2) & Plane(P1) & Plane(P2) & Airport(SFO) & Airport(JFK)',\n",
+       "                goals='At(C1, JFK) & At(C2, SFO)', \n",
+       "                actions=[Action('Load(c, p, a)', \n",
+       "                                precond='At(c, a) & At(p, a) & Cargo(c) & Plane(p) & Airport(a)', \n",
+       "                                effect='In(c, p) & ~At(c, a)'),\n",
+       "                         Action('Unload(c, p, a)',\n",
+       "                                precond='In(c, p) & At(p, a) & Cargo(c) & Plane(p) & Airport(a)',\n",
+       "                                effect='At(c, a) & ~In(c, p)'),\n",
+       "                         Action('Fly(p, f, to)',\n",
+       "                                precond='At(p, f) & Plane(p) & Airport(f) & Airport(to)',\n",
+       "                                effect='At(p, to) & ~At(p, f)')])\n",
+       "
\n", + "\n", + "\n" + ], + "text/plain": [ + "" + ] + }, + "metadata": {}, + "output_type": "display_data" + } + ], + "source": [ + "psource(air_cargo)" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "**At(c, a):** The cargo **'c'** is at airport **'a'**.\n", + "\n", + "**~At(c, a):** The cargo **'c'** is _not_ at airport **'a'**.\n", + "\n", + "**In(c, p):** Cargo **'c'** is in plane **'p'**.\n", + "\n", + "**~In(c, p):** Cargo **'c'** is _not_ in plane **'p'**.\n", + "\n", + "**Cargo(c):** Declare **'c'** as cargo.\n", + "\n", + "**Plane(p):** Declare **'p'** as plane.\n", + "\n", + "**Airport(a):** Declare **'a'** as airport.\n", + "\n", + "\n", + "\n", + "In the `initial_state`, we have cargo C1, plane P1 at airport SFO and cargo C2, plane P2 at airport JFK. \n", + "Our goal state is to have cargo C1 at airport JFK and cargo C2 at airport SFO. We will discuss on how to achieve this. Let us now define an object of the `air_cargo` problem:" + ] + }, + { + "cell_type": "code", + "execution_count": 13, + "metadata": { + "collapsed": true + }, + "outputs": [], + "source": [ + "airCargo = air_cargo()" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "Before taking any actions, we will check if `airCargo` has reached its goal:" + ] + }, + { + "cell_type": "code", + "execution_count": 14, + "metadata": {}, + "outputs": [ + { + "name": "stdout", + "output_type": "stream", + "text": [ + "False\n" + ] + } + ], + "source": [ + "print(airCargo.goal_test())" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "It returns False because the goal state is not yet reached. Now, we define the sequence of actions that it should take in order to achieve the goal.\n", + "The actions are then carried out on the `airCargo` PDDL.\n", + "\n", + "The actions available to us are the following: Load, Unload, Fly\n", + "\n", + "**Load(c, p, a):** Load cargo **'c'** into plane **'p'** from airport **'a'**.\n", + "\n", + "**Fly(p, f, t):** Fly the plane **'p'** from airport **'f'** to airport **'t'**.\n", + "\n", + "**Unload(c, p, a):** Unload cargo **'c'** from plane **'p'** to airport **'a'**.\n", + "\n", + "This problem can have multiple valid solutions.\n", + "One such solution is shown below." + ] + }, + { + "cell_type": "code", + "execution_count": 15, + "metadata": { + "collapsed": true + }, + "outputs": [], + "source": [ + "solution = [expr(\"Load(C1 , P1, SFO)\"),\n", + " expr(\"Fly(P1, SFO, JFK)\"),\n", + " expr(\"Unload(C1, P1, JFK)\"),\n", + " expr(\"Load(C2, P2, JFK)\"),\n", + " expr(\"Fly(P2, JFK, SFO)\"),\n", + " expr(\"Unload (C2, P2, SFO)\")] \n", + "\n", + "for action in solution:\n", + " airCargo.act(action)" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "As the `airCargo` has taken all the steps it needed in order to achieve the goal, we can now check if it has acheived its goal:" + ] + }, + { + "cell_type": "code", + "execution_count": 16, + "metadata": {}, + "outputs": [ + { + "name": "stdout", + "output_type": "stream", + "text": [ + "True\n" + ] + } + ], + "source": [ + "print(airCargo.goal_test())" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "It has now achieved its goal." + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "## The Spare Tire Problem" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "Let's consider the problem of changing a flat tire of a car. \n", + "The goal is to mount a spare tire onto the car's axle, given that we have a flat tire on the axle and a spare tire in the trunk. " + ] + }, + { + "cell_type": "code", + "execution_count": 17, + "metadata": {}, + "outputs": [ + { + "data": { + "text/html": [ + "\n", + "\n", + "\n", + "\n", + " \n", + " \n", + " \n", + "\n", + "\n", + "

\n", + "\n", + "
def spare_tire():\n",
+       "    """Spare tire problem"""\n",
+       "\n",
+       "    return PDDL(init='Tire(Flat) & Tire(Spare) & At(Flat, Axle) & At(Spare, Trunk)',\n",
+       "                goals='At(Spare, Axle) & At(Flat, Ground)',\n",
+       "                actions=[Action('Remove(obj, loc)',\n",
+       "                                precond='At(obj, loc)',\n",
+       "                                effect='At(obj, Ground) & ~At(obj, loc)'),\n",
+       "                         Action('PutOn(t, Axle)',\n",
+       "                                precond='Tire(t) & At(t, Ground) & ~At(Flat, Axle)',\n",
+       "                                effect='At(t, Axle) & ~At(t, Ground)'),\n",
+       "                         Action('LeaveOvernight',\n",
+       "                                precond='',\n",
+       "                                effect='~At(Spare, Ground) & ~At(Spare, Axle) & ~At(Spare, Trunk) & \\\n",
+       "                                        ~At(Flat, Ground) & ~At(Flat, Axle) & ~At(Flat, Trunk)')])\n",
+       "
\n", + "\n", + "\n" + ], + "text/plain": [ + "" + ] + }, + "metadata": {}, + "output_type": "display_data" + } + ], + "source": [ + "psource(spare_tire)" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "**At(obj, loc):** object **'obj'** is at location **'loc'**.\n", + "\n", + "**~At(obj, loc):** object **'obj'** is _not_ at location **'loc'**.\n", + "\n", + "**Tire(t):** Declare a tire of type **'t'**.\n", + "\n", + "Let us now define an object of `spare_tire` problem:" + ] + }, + { + "cell_type": "code", + "execution_count": 18, + "metadata": { + "collapsed": true + }, + "outputs": [], + "source": [ + "spareTire = spare_tire()" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "Before taking any actions, we will check if `spare_tire` has reached its goal:" + ] + }, + { + "cell_type": "code", + "execution_count": 19, + "metadata": {}, + "outputs": [ + { + "name": "stdout", + "output_type": "stream", + "text": [ + "False\n" + ] + } + ], + "source": [ + "print(spareTire.goal_test())" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "As we can see, it hasn't completed the goal. \n", + "We now define a possible solution that can help us reach the goal of having a spare tire mounted onto the car's axle. \n", + "The actions are then carried out on the `spareTire` PDDL.\n", + "\n", + "The actions available to us are the following: Remove, PutOn\n", + "\n", + "**Remove(obj, loc):** Remove the tire **'obj'** from the location **'loc'**.\n", + "\n", + "**PutOn(t, Axle):** Attach the tire **'t'** on the Axle.\n", + "\n", + "**LeaveOvernight():** We live in a particularly bad neighborhood and all tires, flat or not, are stolen if we leave them overnight.\n", + "\n" + ] + }, + { + "cell_type": "code", + "execution_count": 20, + "metadata": { + "collapsed": true + }, + "outputs": [], + "source": [ + "solution = [expr(\"Remove(Flat, Axle)\"),\n", + " expr(\"Remove(Spare, Trunk)\"),\n", + " expr(\"PutOn(Spare, Axle)\")]\n", + "\n", + "for action in solution:\n", + " spareTire.act(action)" + ] + }, + { + "cell_type": "code", + "execution_count": 21, + "metadata": {}, + "outputs": [ + { + "name": "stdout", + "output_type": "stream", + "text": [ + "True\n" + ] + } + ], + "source": [ + "print(spareTire.goal_test())" + ] + }, + { + "cell_type": "markdown", "metadata": {}, + "source": [ + "This is a valid solution.\n", + "
\n", + "Another possible solution is" + ] + }, + { + "cell_type": "code", + "execution_count": 22, + "metadata": { + "collapsed": true + }, "outputs": [], "source": [ - "from planning import *\n", - "from notebook import psource" + "spareTire = spare_tire()\n", + "\n", + "solution = [expr('Remove(Spare, Trunk)'),\n", + " expr('Remove(Flat, Axle)'),\n", + " expr('PutOn(Spare, Axle)')]\n", + "\n", + "for action in solution:\n", + " spareTire.act(action)" + ] + }, + { + "cell_type": "code", + "execution_count": 23, + "metadata": {}, + "outputs": [ + { + "name": "stdout", + "output_type": "stream", + "text": [ + "True\n" + ] + } + ], + "source": [ + "print(spareTire.goal_test())" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "Notice that both solutions work, which means that the problem can be solved irrespective of the order in which the `Remove` actions take place, as long as both `Remove` actions take place before the `PutOn` action." + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "We have successfully mounted a spare tire onto the axle." + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "## Three Block Tower Problem" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "This problem's domain consists of a set of cube-shaped blocks sitting on a table. \n", + "The blocks can be stacked, but only one block can fit directly on top of another.\n", + "A robot arm can pick up a block and move it to another position, either on the table or on top of another block. \n", + "The arm can pick up only one block at a time, so it cannot pick up a block that has another one on it. \n", + "The goal will always be to build one or more stacks of blocks. \n", + "In our case, we consider only three blocks.\n", + "The particular configuration we will use is called the Sussman anomaly after Prof. Gerry Sussman." + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "Let's take a look at the definition of `three_block_tower()` in the module." + ] + }, + { + "cell_type": "code", + "execution_count": 24, + "metadata": {}, + "outputs": [ + { + "data": { + "text/html": [ + "\n", + "\n", + "\n", + "\n", + " \n", + " \n", + " \n", + "\n", + "\n", + "

\n", + "\n", + "
def three_block_tower():\n",
+       "    """Sussman Anomaly problem"""\n",
+       "\n",
+       "    return PDDL(init='On(A, Table) & On(B, Table) & On(C, A) & Block(A) & Block(B) & Block(C) & Clear(B) & Clear(C)',\n",
+       "                goals='On(A, B) & On(B, C)',\n",
+       "                actions=[Action('Move(b, x, y)',\n",
+       "                                precond='On(b, x) & Clear(b) & Clear(y) & Block(b) & Block(y)',\n",
+       "                                effect='On(b, y) & Clear(x) & ~On(b, x) & ~Clear(y)'),\n",
+       "                         Action('MoveToTable(b, x)',\n",
+       "                                precond='On(b, x) & Clear(b) & Block(b)',\n",
+       "                                effect='On(b, Table) & Clear(x) & ~On(b, x)')])\n",
+       "
\n", + "\n", + "\n" + ], + "text/plain": [ + "" + ] + }, + "metadata": {}, + "output_type": "display_data" + } + ], + "source": [ + "psource(three_block_tower)" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ - "To be able to model a planning problem properly, it is essential to be able to represent an Action. Each action we model requires at least three things:\n", - "* preconditions that the action must meet\n", - "* the effects of executing the action\n", - "* some expression that represents the action" + "**On(b, x):** The block **'b'** is on **'x'**. **'x'** can be a table or a block.\n", + "\n", + "**~On(b, x):** The block **'b'** is _not_ on **'x'**. **'x'** can be a table or a block.\n", + "\n", + "**Block(b):** Declares **'b'** as a block.\n", + "\n", + "**Clear(x):** To indicate that there is nothing on **'x'** and it is free to be moved around.\n", + "\n", + "**~Clear(x):** To indicate that there is something on **'x'** and it cannot be moved.\n", + " \n", + " Let us now define an object of `three_block_tower` problem:" ] }, { - "cell_type": "markdown", - "metadata": {}, + "cell_type": "code", + "execution_count": 25, + "metadata": { + "collapsed": true + }, + "outputs": [], "source": [ - "Planning actions have been modelled using the `Action` class. Let's look at the source to see how the internal details of an action are implemented in Python." + "threeBlockTower = three_block_tower()" ] }, { - "cell_type": "code", - "execution_count": 2, + "cell_type": "markdown", "metadata": {}, - "outputs": [], "source": [ - "%psource Action" + "Before taking any actions, we will check if `threeBlockTower` has reached its goal:" ] }, { - "cell_type": "markdown", + "cell_type": "code", + "execution_count": 26, "metadata": {}, + "outputs": [ + { + "name": "stdout", + "output_type": "stream", + "text": [ + "False\n" + ] + } + ], "source": [ - "It is interesting to see the way preconditions and effects are represented here. Instead of just being a list of expressions each, they consist of two lists - `precond_pos` and `precond_neg`. This is to work around the fact that PDDL doesn't allow for negations. Thus, for each precondition, we maintain a separate list of those preconditions that must hold true, and those whose negations must hold true. Similarly, instead of having a single list of expressions that are the result of executing an action, we have two. The first (`effect_add`) contains all the expressions that will evaluate to true if the action is executed, and the the second (`effect_neg`) contains all those expressions that would be false if the action is executed (ie. their negations would be true).\n", - "\n", - "The constructor parameters, however combine the two precondition lists into a single `precond` parameter, and the effect lists into a single `effect` parameter." + "print(threeBlockTower.goal_test())" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ - "The `PDDL` class is used to represent planning problems in this module. The following attributes are essential to be able to define a problem:\n", - "* a goal test\n", - "* an initial state\n", - "* a set of viable actions that can be executed in the search space of the problem\n", + "As we can see, it hasn't completed the goal. \n", + "We now define a sequence of actions that can stack three blocks in the required order. \n", + "The actions are then carried out on the `threeBlockTower` PDDL.\n", "\n", - "View the source to see how the Python code tries to realise these." + "The actions available to us are the following: MoveToTable, Move\n", + "\n", + "**MoveToTable(b, x): ** Move box **'b'** stacked on **'x'** to the table, given that box **'b'** is clear.\n", + "\n", + "**Move(b, x, y): ** Move box **'b'** stacked on **'x'** to the top of **'y'**, given that both **'b'** and **'y'** are clear.\n" ] }, { "cell_type": "code", - "execution_count": 3, - "metadata": {}, + "execution_count": 27, + "metadata": { + "collapsed": true + }, "outputs": [], "source": [ - "%psource PDDL" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "The `initial_state` attribute is a list of `Expr` expressions that forms the initial knowledge base for the problem. Next, `actions` contains a list of `Action` objects that may be executed in the search space of the problem. Lastly, we pass a `goal_test` function as a parameter - this typically takes a knowledge base as a parameter, and returns whether or not the goal has been reached." + "solution = [expr(\"MoveToTable(C, A)\"),\n", + " expr(\"Move(B, Table, C)\"),\n", + " expr(\"Move(A, Table, B)\")]\n", + "\n", + "for action in solution:\n", + " threeBlockTower.act(action)" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ - "Now lets try to define a planning problem using these tools. Since we already know about the map of Romania, lets see if we can plan a trip across a simplified map of Romania.\n", - "\n", - "Here is our simplified map definition:" + "As the `three_block_tower` has taken all the steps it needed in order to achieve the goal, we can now check if it has acheived its goal." ] }, { "cell_type": "code", - "execution_count": 4, + "execution_count": 28, "metadata": {}, - "outputs": [], + "outputs": [ + { + "name": "stdout", + "output_type": "stream", + "text": [ + "True\n" + ] + } + ], "source": [ - "from utils import *\n", - "# this imports the required expr so we can create our knowledge base\n", - "\n", - "knowledge_base = [\n", - " expr(\"Connected(Bucharest,Pitesti)\"),\n", - " expr(\"Connected(Pitesti,Rimnicu)\"),\n", - " expr(\"Connected(Rimnicu,Sibiu)\"),\n", - " expr(\"Connected(Sibiu,Fagaras)\"),\n", - " expr(\"Connected(Fagaras,Bucharest)\"),\n", - " expr(\"Connected(Pitesti,Craiova)\"),\n", - " expr(\"Connected(Craiova,Rimnicu)\")\n", - " ]" + "print(threeBlockTower.goal_test())" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ - "Let us add some logic propositions to complete our knowledge about travelling around the map. These are the typical symmetry and transitivity properties of connections on a map. We can now be sure that our `knowledge_base` understands what it truly means for two locations to be connected in the sense usually meant by humans when we use the term.\n", - "\n", - "Let's also add our starting location - *Sibiu* to the map." + "It has now successfully achieved its goal i.e, to build a stack of three blocks in the specified order." ] }, { - "cell_type": "code", - "execution_count": 5, + "cell_type": "markdown", "metadata": {}, - "outputs": [], "source": [ - "knowledge_base.extend([\n", - " expr(\"Connected(x,y) ==> Connected(y,x)\"),\n", - " expr(\"Connected(x,y) & Connected(y,z) ==> Connected(x,z)\"),\n", - " expr(\"At(Sibiu)\")\n", - " ])" + "## Shopping Problem" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ - "We now have a complete knowledge base, which can be seen like this:" + "This problem requires us to acquire a carton of milk, a banana and a drill.\n", + "Initially, we start from home and it is known to us that milk and bananas are available in the supermarket and the hardware store sells drills.\n", + "Let's take a look at the definition of the `shopping_problem` in the module." ] }, { "cell_type": "code", - "execution_count": 6, + "execution_count": 29, "metadata": {}, "outputs": [ { "data": { + "text/html": [ + "\n", + "\n", + "\n", + "\n", + " \n", + " \n", + " \n", + "\n", + "\n", + "

\n", + "\n", + "
def shopping_problem():\n",
+       "    """Shopping problem"""\n",
+       "\n",
+       "    return PDDL(init='At(Home) & Sells(SM, Milk) & Sells(SM, Banana) & Sells(HW, Drill)',\n",
+       "                goals='Have(Milk) & Have(Banana) & Have(Drill)', \n",
+       "                actions=[Action('Buy(x, store)',\n",
+       "                                precond='At(store) & Sells(store, x)',\n",
+       "                                effect='Have(x)'),\n",
+       "                         Action('Go(x, y)',\n",
+       "                                precond='At(x)',\n",
+       "                                effect='At(y) & ~At(x)')])\n",
+       "
\n", + "\n", + "\n" + ], "text/plain": [ - "[Connected(Bucharest, Pitesti),\n", - " Connected(Pitesti, Rimnicu),\n", - " Connected(Rimnicu, Sibiu),\n", - " Connected(Sibiu, Fagaras),\n", - " Connected(Fagaras, Bucharest),\n", - " Connected(Pitesti, Craiova),\n", - " Connected(Craiova, Rimnicu),\n", - " (Connected(x, y) ==> Connected(y, x)),\n", - " ((Connected(x, y) & Connected(y, z)) ==> Connected(x, z)),\n", - " At(Sibiu)]" + "" ] }, - "execution_count": 6, "metadata": {}, - "output_type": "execute_result" + "output_type": "display_data" } ], "source": [ - "knowledge_base" + "psource(shopping_problem)" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ - "We now define possible actions to our problem. We know that we can drive between any connected places. But, as is evident from [this](https://en.wikipedia.org/wiki/List_of_airports_in_Romania) list of Romanian airports, we can also fly directly between Sibiu, Bucharest, and Craiova.\n", + "**At(x):** Indicates that we are currently at **'x'** where **'x'** can be Home, SM (supermarket) or HW (Hardware store).\n", "\n", - "We can define these flight actions like this:" + "**~At(x):** Indicates that we are currently _not_ at **'x'**.\n", + "\n", + "**Sells(s, x):** Indicates that item **'x'** can be bought from store **'s'**.\n", + "\n", + "**Have(x):** Indicates that we possess the item **'x'**." ] }, { "cell_type": "code", - "execution_count": 7, - "metadata": {}, + "execution_count": 30, + "metadata": { + "collapsed": true + }, "outputs": [], "source": [ - "#Sibiu to Bucharest\n", - "precond_pos = [expr('At(Sibiu)')]\n", - "precond_neg = []\n", - "effect_add = [expr('At(Bucharest)')]\n", - "effect_rem = [expr('At(Sibiu)')]\n", - "fly_s_b = Action(expr('Fly(Sibiu, Bucharest)'), [precond_pos, precond_neg], [effect_add, effect_rem])\n", - "\n", - "#Bucharest to Sibiu\n", - "precond_pos = [expr('At(Bucharest)')]\n", - "precond_neg = []\n", - "effect_add = [expr('At(Sibiu)')]\n", - "effect_rem = [expr('At(Bucharest)')]\n", - "fly_b_s = Action(expr('Fly(Bucharest, Sibiu)'), [precond_pos, precond_neg], [effect_add, effect_rem])\n", - "\n", - "#Sibiu to Craiova\n", - "precond_pos = [expr('At(Sibiu)')]\n", - "precond_neg = []\n", - "effect_add = [expr('At(Craiova)')]\n", - "effect_rem = [expr('At(Sibiu)')]\n", - "fly_s_c = Action(expr('Fly(Sibiu, Craiova)'), [precond_pos, precond_neg], [effect_add, effect_rem])\n", - "\n", - "#Craiova to Sibiu\n", - "precond_pos = [expr('At(Craiova)')]\n", - "precond_neg = []\n", - "effect_add = [expr('At(Sibiu)')]\n", - "effect_rem = [expr('At(Craiova)')]\n", - "fly_c_s = Action(expr('Fly(Craiova, Sibiu)'), [precond_pos, precond_neg], [effect_add, effect_rem])\n", + "shoppingProblem = shopping_problem()" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "Let's first check whether the goal state Have(Milk), Have(Banana), Have(Drill) is reached or not." + ] + }, + { + "cell_type": "code", + "execution_count": 31, + "metadata": {}, + "outputs": [ + { + "name": "stdout", + "output_type": "stream", + "text": [ + "False\n" + ] + } + ], + "source": [ + "print(shoppingProblem.goal_test())" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "Let's look at the possible actions\n", "\n", - "#Bucharest to Craiova\n", - "precond_pos = [expr('At(Bucharest)')]\n", - "precond_neg = []\n", - "effect_add = [expr('At(Craiova)')]\n", - "effect_rem = [expr('At(Bucharest)')]\n", - "fly_b_c = Action(expr('Fly(Bucharest, Craiova)'), [precond_pos, precond_neg], [effect_add, effect_rem])\n", + "**Buy(x, store):** Buy an item **'x'** from a **'store'** given that the **'store'** sells **'x'**.\n", "\n", - "#Craiova to Bucharest\n", - "precond_pos = [expr('At(Craiova)')]\n", - "precond_neg = []\n", - "effect_add = [expr('At(Bucharest)')]\n", - "effect_rem = [expr('At(Craiova)')]\n", - "fly_c_b = Action(expr('Fly(Craiova, Bucharest)'), [precond_pos, precond_neg], [effect_add, effect_rem])" + "**Go(x, y):** Go to destination **'y'** starting from source **'x'**." ] }, { "cell_type": "markdown", "metadata": {}, "source": [ - "And the drive actions like this." + "We now define a valid solution that will help us reach the goal.\n", + "The sequence of actions will then be carried out onto the `shoppingProblem` PDDL." ] }, { "cell_type": "code", - "execution_count": 8, - "metadata": {}, + "execution_count": 32, + "metadata": { + "collapsed": true + }, "outputs": [], "source": [ - "#Drive\n", - "precond_pos = [expr('At(x)')]\n", - "precond_neg = []\n", - "effect_add = [expr('At(y)')]\n", - "effect_rem = [expr('At(x)')]\n", - "drive = Action(expr('Drive(x, y)'), [precond_pos, precond_neg], [effect_add, effect_rem])" + "solution = [expr('Go(Home, SM)'),\n", + " expr('Buy(Milk, SM)'),\n", + " expr('Buy(Banana, SM)'),\n", + " expr('Go(SM, HW)'),\n", + " expr('Buy(Drill, HW)')]\n", + "\n", + "for action in solution:\n", + " shoppingProblem.act(action)" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ - "Finally, we can define a a function that will tell us when we have reached our destination, Bucharest." + "We have taken the steps required to acquire all the stuff we need. \n", + "Let's see if we have reached our goal." ] }, { "cell_type": "code", - "execution_count": 9, + "execution_count": 33, "metadata": {}, - "outputs": [], + "outputs": [ + { + "data": { + "text/plain": [ + "True" + ] + }, + "execution_count": 33, + "metadata": {}, + "output_type": "execute_result" + } + ], "source": [ - "def goal_test(kb):\n", - " return kb.ask(expr(\"At(Bucharest)\"))" + "shoppingProblem.goal_test()" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ - "Thus, with all the components in place, we can define the planning problem." - ] - }, - { - "cell_type": "code", - "execution_count": 10, - "metadata": {}, - "outputs": [], - "source": [ - "prob = PDDL(knowledge_base, [fly_s_b, fly_b_s, fly_s_c, fly_c_s, fly_b_c, fly_c_b, drive], goal_test)" + "It has now successfully achieved the goal." ] }, { "cell_type": "markdown", "metadata": {}, "source": [ - "# Air Cargo Problem:" + "## Have Cake and Eat Cake Too" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ - "Air Cargo problem involves loading and unloading of cargo and flying it from place to place. The problem can be defined with three actions: Load, Unload and Fly. Let us look at `air_cargo`. " + "This problem requires us to reach the state of having a cake and having eaten a cake simlutaneously, given a single cake.\n", + "Let's first take a look at the definition of the `have_cake_and_eat_cake_too` problem in the module." ] }, { "cell_type": "code", - "execution_count": 11, + "execution_count": 34, "metadata": {}, "outputs": [ { @@ -400,49 +1875,17 @@ "\n", "

\n", "\n", - "
def air_cargo():\n",
-       "    init = [expr('At(C1, SFO)'),\n",
-       "            expr('At(C2, JFK)'),\n",
-       "            expr('At(P1, SFO)'),\n",
-       "            expr('At(P2, JFK)'),\n",
-       "            expr('Cargo(C1)'),\n",
-       "            expr('Cargo(C2)'),\n",
-       "            expr('Plane(P1)'),\n",
-       "            expr('Plane(P2)'),\n",
-       "            expr('Airport(JFK)'),\n",
-       "            expr('Airport(SFO)')]\n",
-       "\n",
-       "    def goal_test(kb):\n",
-       "        required = [expr('At(C1 , JFK)'), expr('At(C2 ,SFO)')]\n",
-       "        return all([kb.ask(q) is not False for q in required])\n",
-       "\n",
-       "    # Actions\n",
-       "\n",
-       "    #  Load\n",
-       "    precond_pos = [expr("At(c, a)"), expr("At(p, a)"), expr("Cargo(c)"), expr("Plane(p)"),\n",
-       "                   expr("Airport(a)")]\n",
-       "    precond_neg = []\n",
-       "    effect_add = [expr("In(c, p)")]\n",
-       "    effect_rem = [expr("At(c, a)")]\n",
-       "    load = Action(expr("Load(c, p, a)"), [precond_pos, precond_neg], [effect_add, effect_rem])\n",
-       "\n",
-       "    #  Unload\n",
-       "    precond_pos = [expr("In(c, p)"), expr("At(p, a)"), expr("Cargo(c)"), expr("Plane(p)"),\n",
-       "                   expr("Airport(a)")]\n",
-       "    precond_neg = []\n",
-       "    effect_add = [expr("At(c, a)")]\n",
-       "    effect_rem = [expr("In(c, p)")]\n",
-       "    unload = Action(expr("Unload(c, p, a)"), [precond_pos, precond_neg], [effect_add, effect_rem])\n",
-       "\n",
-       "    #  Fly\n",
-       "    #  Used 'f' instead of 'from' because 'from' is a python keyword and expr uses eval() function\n",
-       "    precond_pos = [expr("At(p, f)"), expr("Plane(p)"), expr("Airport(f)"), expr("Airport(to)")]\n",
-       "    precond_neg = []\n",
-       "    effect_add = [expr("At(p, to)")]\n",
-       "    effect_rem = [expr("At(p, f)")]\n",
-       "    fly = Action(expr("Fly(p, f, to)"), [precond_pos, precond_neg], [effect_add, effect_rem])\n",
-       "\n",
-       "    return PDDL(init, [load, unload, fly], goal_test)\n",
+       "
def have_cake_and_eat_cake_too():\n",
+       "    """Cake problem"""\n",
+       "\n",
+       "    return PDDL(init='Have(Cake)',\n",
+       "                goals='Have(Cake) & Eaten(Cake)',\n",
+       "                actions=[Action('Eat(Cake)',\n",
+       "                                precond='Have(Cake)',\n",
+       "                                effect='Eaten(Cake) & ~Have(Cake)'),\n",
+       "                         Action('Bake(Cake)',\n",
+       "                                precond='~Have(Cake)',\n",
+       "                                effect='Have(Cake)')])\n",
        "
\n",
        "\n",
        "\n"
@@ -456,47 +1899,41 @@
     }
    ],
    "source": [
-    "psource(air_cargo)"
+    "psource(have_cake_and_eat_cake_too)"
    ]
   },
   {
    "cell_type": "markdown",
    "metadata": {},
    "source": [
-    "**At(x, a):** The cargo or plane **'x'** is at airport **'a'**.\n",
-    "\n",
-    "**In(c, p):** Cargo **'c'** is in palne **'p'**.\n",
-    "\n",
-    "**Cargo(x):** Declare **'x'** as cargo.\n",
-    "\n",
-    "**Plane(x):** Declare **'x'** as plane.\n",
+    "Since this problem doesn't involve variables, states can be considered similar to symbols in propositional logic.\n",
     "\n",
-    "**Airport(x):** Declare **'x'** as airport.\n",
+    "**Have(Cake):** Declares that we have a **'Cake'**.\n",
     "\n",
-    "\n",
-    "\n",
-    "In the `initial_state`, we have cargo C1, plane P1 at airport SFO and cargo C2, plane P2 at airport JFK. Our goal state is to have cargo C1 at airport JFK and cargo C2 at airport SFO. We will discuss on how to achieve this. Let us now define an object of the `air_cargo` problem:"
+    "**~Have(Cake):** Declares that we _don't_ have a **'Cake'**."
    ]
   },
   {
    "cell_type": "code",
-   "execution_count": 12,
-   "metadata": {},
+   "execution_count": 35,
+   "metadata": {
+    "collapsed": true
+   },
    "outputs": [],
    "source": [
-    "airCargo = air_cargo()"
+    "cakeProblem = have_cake_and_eat_cake_too()"
    ]
   },
   {
    "cell_type": "markdown",
    "metadata": {},
    "source": [
-    "Now, before taking any actions, we will check the `airCargo` if it has completed the goal it is required to do:"
+    "First let us check whether the goal state 'Have(Cake)' and 'Eaten(Cake)' are reached or not."
    ]
   },
   {
    "cell_type": "code",
-   "execution_count": 13,
+   "execution_count": 36,
    "metadata": {},
    "outputs": [
     {
@@ -508,51 +1945,53 @@
     }
    ],
    "source": [
-    "print(airCargo.goal_test())"
+    "print(cakeProblem.goal_test())"
    ]
   },
   {
    "cell_type": "markdown",
    "metadata": {},
    "source": [
-    "It returns False because the goal state is not yet reached. Now, we define the sequence of actions that it should take in order to achieve the goal. Then the `airCargo` acts on each of them.\n",
-    "\n",
-    "The actions available to us are the following: Load, Unload, Fly\n",
+    "Let us look at the possible actions.\n",
     "\n",
-    "**Load(c, p, a):** Load cargo **'c'** into plane **'p'** from airport **'a'**.\n",
-    "\n",
-    "**Fly(p, f, t):** Fly the plane **'p'** from airport **'f'** to airport **'t'**.\n",
+    "**Bake(x):** To bake **' x '**.\n",
     "\n",
-    "**Unload(c, p, c):** Unload cargo **'c'** from plane **'p'** to airport **'a'**.\n"
+    "**Eat(x):** To eat **' x '**."
    ]
   },
   {
-   "cell_type": "code",
-   "execution_count": 14,
+   "cell_type": "markdown",
    "metadata": {},
+   "source": [
+    "We now define a valid solution that can help us reach the goal.\n",
+    "The sequence of actions will then be acted upon the `cakeProblem` PDDL."
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": 37,
+   "metadata": {
+    "collapsed": true
+   },
    "outputs": [],
    "source": [
-    "solution = [expr(\"Load(C1 , P1, SFO)\"),\n",
-    "            expr(\"Fly(P1, SFO, JFK)\"),\n",
-    "            expr(\"Unload(C1, P1, JFK)\"),\n",
-    "            expr(\"Load(C2, P2, JFK)\"),\n",
-    "            expr(\"Fly(P2, JFK, SFO)\"),\n",
-    "            expr(\"Unload (C2, P2, SFO)\")] \n",
+    "solution = [expr(\"Eat(Cake)\"),\n",
+    "            expr(\"Bake(Cake)\")]\n",
     "\n",
     "for action in solution:\n",
-    "    airCargo.act(action)"
+    "    cakeProblem.act(action)"
    ]
   },
   {
    "cell_type": "markdown",
    "metadata": {},
    "source": [
-    "As the `airCargo` has taken all the steps it needed in order to achieve the goal, we can now check if it has acheived its goal:"
+    "Now we have made actions to bake the cake and eat the cake. Let us check if we have reached the goal."
    ]
   },
   {
    "cell_type": "code",
-   "execution_count": 15,
+   "execution_count": 38,
    "metadata": {},
    "outputs": [
     {
@@ -564,33 +2003,122 @@
     }
    ],
    "source": [
-    "print(airCargo.goal_test())"
+    "print(cakeProblem.goal_test())"
    ]
   },
   {
    "cell_type": "markdown",
    "metadata": {},
    "source": [
-    "It has now achieved its goal."
+    "It has now successfully achieved its goal i.e, to have and eat the cake."
    ]
   },
   {
    "cell_type": "markdown",
    "metadata": {},
    "source": [
-    "## The Spare Tire Problem"
+    "One might wonder if the order of the actions matters for this problem.\n",
+    "Let's see for ourselves."
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": 39,
+   "metadata": {},
+   "outputs": [
+    {
+     "ename": "Exception",
+     "evalue": "Action 'Bake(Cake)' pre-conditions not satisfied",
+     "output_type": "error",
+     "traceback": [
+      "\u001b[1;31m---------------------------------------------------------------------------\u001b[0m",
+      "\u001b[1;31mException\u001b[0m                                 Traceback (most recent call last)",
+      "\u001b[1;32m\u001b[0m in \u001b[0;36m\u001b[1;34m()\u001b[0m\n\u001b[0;32m      5\u001b[0m \u001b[1;33m\u001b[0m\u001b[0m\n\u001b[0;32m      6\u001b[0m \u001b[1;32mfor\u001b[0m \u001b[0maction\u001b[0m \u001b[1;32min\u001b[0m \u001b[0msolution\u001b[0m\u001b[1;33m:\u001b[0m\u001b[1;33m\u001b[0m\u001b[0m\n\u001b[1;32m----> 7\u001b[1;33m     \u001b[0mcakeProblem\u001b[0m\u001b[1;33m.\u001b[0m\u001b[0mact\u001b[0m\u001b[1;33m(\u001b[0m\u001b[0maction\u001b[0m\u001b[1;33m)\u001b[0m\u001b[1;33m\u001b[0m\u001b[0m\n\u001b[0m",
+      "\u001b[1;32m~\\Documents\\Python\\Aima\\aima-python\\planning.py\u001b[0m in \u001b[0;36mact\u001b[1;34m(self, action)\u001b[0m\n\u001b[0;32m     44\u001b[0m             \u001b[1;32mraise\u001b[0m \u001b[0mException\u001b[0m\u001b[1;33m(\u001b[0m\u001b[1;34m\"Action '{}' not found\"\u001b[0m\u001b[1;33m.\u001b[0m\u001b[0mformat\u001b[0m\u001b[1;33m(\u001b[0m\u001b[0maction_name\u001b[0m\u001b[1;33m)\u001b[0m\u001b[1;33m)\u001b[0m\u001b[1;33m\u001b[0m\u001b[0m\n\u001b[0;32m     45\u001b[0m         \u001b[1;32mif\u001b[0m \u001b[1;32mnot\u001b[0m \u001b[0mlist_action\u001b[0m\u001b[1;33m.\u001b[0m\u001b[0mcheck_precond\u001b[0m\u001b[1;33m(\u001b[0m\u001b[0mself\u001b[0m\u001b[1;33m.\u001b[0m\u001b[0minit\u001b[0m\u001b[1;33m,\u001b[0m \u001b[0margs\u001b[0m\u001b[1;33m)\u001b[0m\u001b[1;33m:\u001b[0m\u001b[1;33m\u001b[0m\u001b[0m\n\u001b[1;32m---> 46\u001b[1;33m             \u001b[1;32mraise\u001b[0m \u001b[0mException\u001b[0m\u001b[1;33m(\u001b[0m\u001b[1;34m\"Action '{}' pre-conditions not satisfied\"\u001b[0m\u001b[1;33m.\u001b[0m\u001b[0mformat\u001b[0m\u001b[1;33m(\u001b[0m\u001b[0maction\u001b[0m\u001b[1;33m)\u001b[0m\u001b[1;33m)\u001b[0m\u001b[1;33m\u001b[0m\u001b[0m\n\u001b[0m\u001b[0;32m     47\u001b[0m         \u001b[0mself\u001b[0m\u001b[1;33m.\u001b[0m\u001b[0minit\u001b[0m \u001b[1;33m=\u001b[0m \u001b[0mlist_action\u001b[0m\u001b[1;33m(\u001b[0m\u001b[0mself\u001b[0m\u001b[1;33m.\u001b[0m\u001b[0minit\u001b[0m\u001b[1;33m,\u001b[0m \u001b[0margs\u001b[0m\u001b[1;33m)\u001b[0m\u001b[1;33m.\u001b[0m\u001b[0mclauses\u001b[0m\u001b[1;33m\u001b[0m\u001b[0m\n\u001b[0;32m     48\u001b[0m \u001b[1;33m\u001b[0m\u001b[0m\n",
+      "\u001b[1;31mException\u001b[0m: Action 'Bake(Cake)' pre-conditions not satisfied"
+     ]
+    }
+   ],
+   "source": [
+    "cakeProblem = have_cake_and_eat_cake_too()\n",
+    "\n",
+    "solution = [expr('Bake(Cake)'),\n",
+    "            expr('Eat(Cake)')]\n",
+    "\n",
+    "for action in solution:\n",
+    "    cakeProblem.act(action)"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {},
+   "source": [
+    "It raises an exception.\n",
+    "Indeed, according to the problem, we cannot bake a cake if we already have one.\n",
+    "In planning terms, '~Have(Cake)' is a precondition to the action 'Bake(Cake)'.\n",
+    "Hence, this solution is invalid."
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {},
+   "source": [
+    "## SOLVING PLANNING PROBLEMS\n",
+    "----\n",
+    "### GRAPHPLAN\n",
+    "
\n",
+    "The GraphPlan algorithm is a popular method of solving classical planning problems.\n",
+    "Before we get into the details of the algorithm, let's look at a special data structure called **planning graph**, used to give better heuristic estimates and plays a key role in the GraphPlan algorithm."
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {},
+   "source": [
+    "### Planning Graph\n",
+    "A planning graph is a directed graph organized into levels. \n",
+    "Each level contains information about the current state of the knowledge base and the possible state-action links to and from that level.\n",
+    "The first level contains the initial state with nodes representing each fluent that holds in that level.\n",
+    "This level has state-action links linking each state to valid actions in that state.\n",
+    "Each action is linked to all its preconditions and its effect states.\n",
+    "Based on these effects, the next level is constructed.\n",
+    "The next level contains similarly structured information about the next state.\n",
+    "In this way, the graph is expanded using state-action links till we reach a state where all the required goals hold true simultaneously.\n",
+    "We can say that we have reached our goal if none of the goal states in the current level are mutually exclusive.\n",
+    "This will be explained in detail later.\n",
+    "
\n",
+    "Planning graphs only work for propositional planning problems, hence we need to eliminate all variables by generating all possible substitutions.\n",
+    "
\n",
+    "For example, the planning graph of the `have_cake_and_eat_cake_too` problem might look like this\n",
+    "![title](images/cake_graph.jpg)\n",
+    "
\n",
+    "The black lines indicate links between states and actions.\n",
+    "
\n",
+    "In every planning problem, we are allowed to carry out the `no-op` action, ie, we can choose no action for a particular state.\n",
+    "These are called 'Persistence' actions and are represented in the graph by the small square boxes.\n",
+    "In technical terms, a persistence action has effects same as its preconditions.\n",
+    "This enables us to carry a state to the next level.\n",
+    "
\n",
+    "
\n",
+    "The gray lines indicate mutual exclusivity.\n",
+    "This means that the actions connected by a gray line cannot be taken together.\n",
+    "Mutual exclusivity (mutex) occurs in the following cases:\n",
+    "1. **Inconsistent effects**: One action negates the effect of the other. For example, _Eat(Cake)_ and the persistence of _Have(Cake)_ have inconsistent effects because they disagree on the effect _Have(Cake)_\n",
+    "2. **Interference**: One of the effects of an action is the negation of a precondition of the other. For example, _Eat(Cake)_ interferes with the persistence of _Have(Cake)_ by negating its precondition.\n",
+    "3. **Competing needs**: One of the preconditions of one action is mutually exclusive with a precondition of the other. For example, _Bake(Cake)_ and _Eat(Cake)_ are mutex because they compete on the value of the _Have(Cake)_ precondition."
    ]
   },
   {
    "cell_type": "markdown",
    "metadata": {},
    "source": [
-    "Let's consider the problem of changing a flat tire of a car. The goal is to have a good spare tire properly mounted onto the car's axle, where the initial state has a flat tire on the axle and a good spare tire in the trunk. "
+    "In the module, planning graphs have been implemented using two classes, `Level` which stores data for a particular level and `Graph` which connects multiple levels together.\n",
+    "Let's look at the `Level` class."
    ]
   },
   {
    "cell_type": "code",
-   "execution_count": 16,
+   "execution_count": 40,
    "metadata": {},
    "outputs": [
     {
@@ -682,42 +2210,135 @@
        "\n",
        "
\n",
        "\n",
-       "
def spare_tire():\n",
-       "    init = [expr('Tire(Flat)'),\n",
-       "            expr('Tire(Spare)'),\n",
-       "            expr('At(Flat, Axle)'),\n",
-       "            expr('At(Spare, Trunk)')]\n",
-       "\n",
-       "    def goal_test(kb):\n",
-       "        required = [expr('At(Spare, Axle)')]\n",
-       "        return all(kb.ask(q) is not False for q in required)\n",
-       "\n",
-       "    # Actions\n",
-       "\n",
-       "    # Remove\n",
-       "    precond_pos = [expr("At(obj, loc)")]\n",
-       "    precond_neg = []\n",
-       "    effect_add = [expr("At(obj, Ground)")]\n",
-       "    effect_rem = [expr("At(obj, loc)")]\n",
-       "    remove = Action(expr("Remove(obj, loc)"), [precond_pos, precond_neg], [effect_add, effect_rem])\n",
-       "\n",
-       "    # PutOn\n",
-       "    precond_pos = [expr("Tire(t)"), expr("At(t, Ground)")]\n",
-       "    precond_neg = [expr("At(Flat, Axle)")]\n",
-       "    effect_add = [expr("At(t, Axle)")]\n",
-       "    effect_rem = [expr("At(t, Ground)")]\n",
-       "    put_on = Action(expr("PutOn(t, Axle)"), [precond_pos, precond_neg], [effect_add, effect_rem])\n",
-       "\n",
-       "    # LeaveOvernight\n",
-       "    precond_pos = []\n",
-       "    precond_neg = []\n",
-       "    effect_add = []\n",
-       "    effect_rem = [expr("At(Spare, Ground)"), expr("At(Spare, Axle)"), expr("At(Spare, Trunk)"),\n",
-       "                  expr("At(Flat, Ground)"), expr("At(Flat, Axle)"), expr("At(Flat, Trunk)")]\n",
-       "    leave_overnight = Action(expr("LeaveOvernight"), [precond_pos, precond_neg],\n",
-       "                             [effect_add, effect_rem])\n",
-       "\n",
-       "    return PDDL(init, [remove, put_on, leave_overnight], goal_test)\n",
+       "
class Level:\n",
+       "    """\n",
+       "    Contains the state of the planning problem\n",
+       "    and exhaustive list of actions which use the\n",
+       "    states as pre-condition.\n",
+       "    """\n",
+       "\n",
+       "    def __init__(self, kb):\n",
+       "        """Initializes variables to hold state and action details of a level"""\n",
+       "\n",
+       "        self.kb = kb\n",
+       "        # current state\n",
+       "        self.current_state = kb.clauses\n",
+       "        # current action to state link\n",
+       "        self.current_action_links = {}\n",
+       "        # current state to action link\n",
+       "        self.current_state_links = {}\n",
+       "        # current action to next state link\n",
+       "        self.next_action_links = {}\n",
+       "        # next state to current action link\n",
+       "        self.next_state_links = {}\n",
+       "        # mutually exclusive actions\n",
+       "        self.mutex = []\n",
+       "\n",
+       "    def __call__(self, actions, objects):\n",
+       "        self.build(actions, objects)\n",
+       "        self.find_mutex()\n",
+       "\n",
+       "    def separate(self, e):\n",
+       "        """Separates an iterable of elements into positive and negative parts"""\n",
+       "\n",
+       "        positive = []\n",
+       "        negative = []\n",
+       "        for clause in e:\n",
+       "            if clause.op[:3] == 'Not':\n",
+       "                negative.append(clause)\n",
+       "            else:\n",
+       "                positive.append(clause)\n",
+       "        return positive, negative\n",
+       "\n",
+       "    def find_mutex(self):\n",
+       "        """Finds mutually exclusive actions"""\n",
+       "\n",
+       "        # Inconsistent effects\n",
+       "        pos_nsl, neg_nsl = self.separate(self.next_state_links)\n",
+       "\n",
+       "        for negeff in neg_nsl:\n",
+       "            new_negeff = Expr(negeff.op[3:], *negeff.args)\n",
+       "            for poseff in pos_nsl:\n",
+       "                if new_negeff == poseff:\n",
+       "                    for a in self.next_state_links[poseff]:\n",
+       "                        for b in self.next_state_links[negeff]:\n",
+       "                            if {a, b} not in self.mutex:\n",
+       "                                self.mutex.append({a, b})\n",
+       "\n",
+       "        # Interference will be calculated with the last step\n",
+       "        pos_csl, neg_csl = self.separate(self.current_state_links)\n",
+       "\n",
+       "        # Competing needs\n",
+       "        for posprecond in pos_csl:\n",
+       "            for negprecond in neg_csl:\n",
+       "                new_negprecond = Expr(negprecond.op[3:], *negprecond.args)\n",
+       "                if new_negprecond == posprecond:\n",
+       "                    for a in self.current_state_links[posprecond]:\n",
+       "                        for b in self.current_state_links[negprecond]:\n",
+       "                            if {a, b} not in self.mutex:\n",
+       "                                self.mutex.append({a, b})\n",
+       "\n",
+       "        # Inconsistent support\n",
+       "        state_mutex = []\n",
+       "        for pair in self.mutex:\n",
+       "            next_state_0 = self.next_action_links[list(pair)[0]]\n",
+       "            if len(pair) == 2:\n",
+       "                next_state_1 = self.next_action_links[list(pair)[1]]\n",
+       "            else:\n",
+       "                next_state_1 = self.next_action_links[list(pair)[0]]\n",
+       "            if (len(next_state_0) == 1) and (len(next_state_1) == 1):\n",
+       "                state_mutex.append({next_state_0[0], next_state_1[0]})\n",
+       "        \n",
+       "        self.mutex = self.mutex + state_mutex\n",
+       "\n",
+       "    def build(self, actions, objects):\n",
+       "        """Populates the lists and dictionaries containing the state action dependencies"""\n",
+       "\n",
+       "        for clause in self.current_state:\n",
+       "            p_expr = Expr('P' + clause.op, *clause.args)\n",
+       "            self.current_action_links[p_expr] = [clause]\n",
+       "            self.next_action_links[p_expr] = [clause]\n",
+       "            self.current_state_links[clause] = [p_expr]\n",
+       "            self.next_state_links[clause] = [p_expr]\n",
+       "\n",
+       "        for a in actions:\n",
+       "            num_args = len(a.args)\n",
+       "            possible_args = tuple(itertools.permutations(objects, num_args))\n",
+       "\n",
+       "            for arg in possible_args:\n",
+       "                if a.check_precond(self.kb, arg):\n",
+       "                    for num, symbol in enumerate(a.args):\n",
+       "                        if not symbol.op.islower():\n",
+       "                            arg = list(arg)\n",
+       "                            arg[num] = symbol\n",
+       "                            arg = tuple(arg)\n",
+       "\n",
+       "                    new_action = a.substitute(Expr(a.name, *a.args), arg)\n",
+       "                    self.current_action_links[new_action] = []\n",
+       "\n",
+       "                    for clause in a.precond:\n",
+       "                        new_clause = a.substitute(clause, arg)\n",
+       "                        self.current_action_links[new_action].append(new_clause)\n",
+       "                        if new_clause in self.current_state_links:\n",
+       "                            self.current_state_links[new_clause].append(new_action)\n",
+       "                        else:\n",
+       "                            self.current_state_links[new_clause] = [new_action]\n",
+       "                   \n",
+       "                    self.next_action_links[new_action] = []\n",
+       "                    for clause in a.effect:\n",
+       "                        new_clause = a.substitute(clause, arg)\n",
+       "\n",
+       "                        self.next_action_links[new_action].append(new_clause)\n",
+       "                        if new_clause in self.next_state_links:\n",
+       "                            self.next_state_links[new_clause].append(new_action)\n",
+       "                        else:\n",
+       "                            self.next_state_links[new_clause] = [new_action]\n",
+       "\n",
+       "    def perform_actions(self):\n",
+       "        """Performs the necessary actions and returns a new Level"""\n",
+       "\n",
+       "        new_kb = FolKB(list(set(self.next_state_links.keys())))\n",
+       "        return Level(new_kb)\n",
        "
\n",
        "\n",
        "\n"
@@ -731,136 +2352,198 @@
     }
    ],
    "source": [
-    "psource(spare_tire)"
-   ]
-  },
-  {
-   "cell_type": "markdown",
-   "metadata": {},
-   "source": [
-    "**At(x, l):** object **'x'** is at location **'l'**.\n",
-    "\n",
-    "**Tire(x):** Declare a tire of type **'x'**.\n",
-    "\n",
-    "Let us now define an object of `spare_tire` problem:"
-   ]
-  },
-  {
-   "cell_type": "code",
-   "execution_count": 17,
-   "metadata": {},
-   "outputs": [],
-   "source": [
-    "spare_tire = spare_tire()"
-   ]
-  },
-  {
-   "cell_type": "markdown",
-   "metadata": {},
-   "source": [
-    "Now, before taking any actions, we will check `spare_tire` if it has completed the goal it is required to do"
-   ]
-  },
-  {
-   "cell_type": "code",
-   "execution_count": 18,
-   "metadata": {},
-   "outputs": [
-    {
-     "name": "stdout",
-     "output_type": "stream",
-     "text": [
-      "False\n"
-     ]
-    }
-   ],
-   "source": [
-    "print(spare_tire.goal_test())"
+    "psource(Level)"
    ]
   },
   {
    "cell_type": "markdown",
    "metadata": {},
    "source": [
-    "As we can see, it hasn't completed the goal. Now, we define the sequence of actions that it should take in order to have a good spare tire properly mounted onto the car's axle. Then the `spare_tire` acts on each of them.\n",
-    "\n",
-    "The actions available to us are the following: Remove, PutOn\n",
-    "\n",
-    "**Remove(obj, loc):** Remove the tire **'obj'** from the location **'loc'**.\n",
-    "\n",
-    "**PutOn(t, Axle):** Attach the tire **'t'** on the Axle.\n",
-    "\n"
-   ]
-  },
-  {
-   "cell_type": "code",
-   "execution_count": 19,
-   "metadata": {},
-   "outputs": [],
-   "source": [
-    "solution = [expr(\"Remove(Flat, Axle)\"),\n",
-    "            expr(\"Remove(Spare, Trunk)\"),\n",
-    "            expr(\"PutOn(Spare, Axle)\")]\n",
-    "\n",
-    "for action in solution:\n",
-    "    spare_tire.act(action)"
+    "Each level stores the following data\n",
+    "1. The current state of the level in `current_state`\n",
+    "2. Links from an action to its preconditions in `current_action_links`\n",
+    "3. Links from a state to the possible actions in that state in `current_state_links`\n",
+    "4. Links from each action to its effects in `next_action_links`\n",
+    "5. Links from each possible next state from each action in `next_state_links`. This stores the same information as the `current_action_links` of the next level.\n",
+    "6. Mutex links in `mutex`.\n",
+    "
\n",
+    "
\n",
+    "The `find_mutex` method finds the mutex links according to the points given above.\n",
+    "
\n",
+    "The `build` method populates the data structures storing the state and action information.\n",
+    "Persistence actions for each clause in the current state are also defined here. \n",
+    "The newly created persistence action has the same name as its state, prefixed with a 'P'."
    ]
   },
   {
    "cell_type": "markdown",
    "metadata": {},
    "source": [
-    "As the `spare_tire` has taken all the steps it needed in order to achieve the goal, we can now check if it has acheived its goal"
+    "Let's now look at the `Graph` class."
    ]
   },
   {
    "cell_type": "code",
-   "execution_count": 20,
+   "execution_count": 41,
    "metadata": {},
    "outputs": [
     {
-     "name": "stdout",
-     "output_type": "stream",
-     "text": [
-      "True\n"
-     ]
+     "data": {
+      "text/html": [
+       "\n",
+       "\n",
+       "\n",
+       "\n",
+       "  \n",
+       "  \n",
+       "  \n",
+       "\n",
+       "\n",
+       "
\n",
+       "\n",
+       "
class Graph:\n",
+       "    """\n",
+       "    Contains levels of state and actions\n",
+       "    Used in graph planning algorithm to extract a solution\n",
+       "    """\n",
+       "\n",
+       "    def __init__(self, pddl):\n",
+       "        self.pddl = pddl\n",
+       "        self.kb = FolKB(pddl.init)\n",
+       "        self.levels = [Level(self.kb)]\n",
+       "        self.objects = set(arg for clause in self.kb.clauses for arg in clause.args)\n",
+       "\n",
+       "    def __call__(self):\n",
+       "        self.expand_graph()\n",
+       "\n",
+       "    def expand_graph(self):\n",
+       "        """Expands the graph by a level"""\n",
+       "\n",
+       "        last_level = self.levels[-1]\n",
+       "        last_level(self.pddl.actions, self.objects)\n",
+       "        self.levels.append(last_level.perform_actions())\n",
+       "\n",
+       "    def non_mutex_goals(self, goals, index):\n",
+       "        """Checks whether the goals are mutually exclusive"""\n",
+       "\n",
+       "        goal_perm = itertools.combinations(goals, 2)\n",
+       "        for g in goal_perm:\n",
+       "            if set(g) in self.levels[index].mutex:\n",
+       "                return False\n",
+       "        return True\n",
+       "
\n",
+       "\n",
+       "\n"
+      ],
+      "text/plain": [
+       ""
+      ]
+     },
+     "metadata": {},
+     "output_type": "display_data"
     }
    ],
    "source": [
-    "print(spare_tire.goal_test())"
-   ]
-  },
-  {
-   "cell_type": "markdown",
-   "metadata": {},
-   "source": [
-    "It has now successfully achieved its goal i.e, to have a good spare tire properly mounted onto the car's axle."
-   ]
-  },
-  {
-   "cell_type": "markdown",
-   "metadata": {},
-   "source": [
-    "## Three Block Tower Problem"
-   ]
-  },
-  {
-   "cell_type": "markdown",
-   "metadata": {},
-   "source": [
-    "This problem's domain consists of a set of cube-shaped blocks sitting on a table. The blocks can be stacked, but only one block can fit directly on top of another. A robot arm can pick up a block and move it to another position, either on the table or on top of another block. The arm can pick up only one block at a time, so it cannot pick up a block that has another one on it. The goal will always be to build one or more stacks of blocks. In our case, we consider only three blocks."
+    "psource(Graph)"
    ]
   },
   {
    "cell_type": "markdown",
    "metadata": {},
    "source": [
-    "let us take a look at the `three_block_tower()` code."
+    "The class stores a problem definition in `pddl`, \n",
+    "a knowledge base in `kb`, \n",
+    "a list of `Level` objects in `levels` and \n",
+    "all the possible arguments found in the initial state of the problem in `objects`.\n",
+    "
\n",
+    "The `expand_graph` method generates a new level of the graph.\n",
+    "This method is invoked when the goal conditions haven't been met in the current level or the actions that lead to it are mutually exclusive.\n",
+    "The `non_mutex_goals` method checks whether the goals in the current state are mutually exclusive.\n",
+    "
\n",
+    "
\n",
+    "Using these two classes, we can define a planning graph which can either be used to provide reliable heuristics for planning problems or used in the `GraphPlan` algorithm.\n",
+    "
\n",
+    "Let's have a look at the `GraphPlan` class."
    ]
   },
   {
    "cell_type": "code",
-   "execution_count": 21,
+   "execution_count": 42,
    "metadata": {},
    "outputs": [
     {
@@ -952,39 +2635,85 @@
        "\n",
        "
\n",
        "\n",
-       "
def three_block_tower():\n",
-       "    init = [expr('On(A, Table)'),\n",
-       "            expr('On(B, Table)'),\n",
-       "            expr('On(C, A)'),\n",
-       "            expr('Block(A)'),\n",
-       "            expr('Block(B)'),\n",
-       "            expr('Block(C)'),\n",
-       "            expr('Clear(B)'),\n",
-       "            expr('Clear(C)')]\n",
-       "\n",
-       "    def goal_test(kb):\n",
-       "        required = [expr('On(A, B)'), expr('On(B, C)')]\n",
-       "        return all(kb.ask(q) is not False for q in required)\n",
-       "\n",
-       "    # Actions\n",
-       "\n",
-       "    #  Move\n",
-       "    precond_pos = [expr('On(b, x)'), expr('Clear(b)'), expr('Clear(y)'), expr('Block(b)'),\n",
-       "                   expr('Block(y)')]\n",
-       "    precond_neg = []\n",
-       "    effect_add = [expr('On(b, y)'), expr('Clear(x)')]\n",
-       "    effect_rem = [expr('On(b, x)'), expr('Clear(y)')]\n",
-       "    move = Action(expr('Move(b, x, y)'), [precond_pos, precond_neg], [effect_add, effect_rem])\n",
-       "\n",
-       "    #  MoveToTable\n",
-       "    precond_pos = [expr('On(b, x)'), expr('Clear(b)'), expr('Block(b)')]\n",
-       "    precond_neg = []\n",
-       "    effect_add = [expr('On(b, Table)'), expr('Clear(x)')]\n",
-       "    effect_rem = [expr('On(b, x)')]\n",
-       "    moveToTable = Action(expr('MoveToTable(b, x)'), [precond_pos, precond_neg],\n",
-       "                         [effect_add, effect_rem])\n",
-       "\n",
-       "    return PDDL(init, [move, moveToTable], goal_test)\n",
+       "
class GraphPlan:\n",
+       "    """\n",
+       "    Class for formulation GraphPlan algorithm\n",
+       "    Constructs a graph of state and action space\n",
+       "    Returns solution for the planning problem\n",
+       "    """\n",
+       "\n",
+       "    def __init__(self, pddl):\n",
+       "        self.graph = Graph(pddl)\n",
+       "        self.nogoods = []\n",
+       "        self.solution = []\n",
+       "\n",
+       "    def check_leveloff(self):\n",
+       "        """Checks if the graph has levelled off"""\n",
+       "\n",
+       "        check = (set(self.graph.levels[-1].current_state) == set(self.graph.levels[-2].current_state))\n",
+       "\n",
+       "        if check:\n",
+       "            return True\n",
+       "\n",
+       "    def extract_solution(self, goals, index):\n",
+       "        """Extracts the solution"""\n",
+       "\n",
+       "        level = self.graph.levels[index]    \n",
+       "        if not self.graph.non_mutex_goals(goals, index):\n",
+       "            self.nogoods.append((level, goals))\n",
+       "            return\n",
+       "\n",
+       "        level = self.graph.levels[index - 1]    \n",
+       "\n",
+       "        # Create all combinations of actions that satisfy the goal    \n",
+       "        actions = []\n",
+       "        for goal in goals:\n",
+       "            actions.append(level.next_state_links[goal])    \n",
+       "\n",
+       "        all_actions = list(itertools.product(*actions))    \n",
+       "\n",
+       "        # Filter out non-mutex actions\n",
+       "        non_mutex_actions = []    \n",
+       "        for action_tuple in all_actions:\n",
+       "            action_pairs = itertools.combinations(list(set(action_tuple)), 2)        \n",
+       "            non_mutex_actions.append(list(set(action_tuple)))        \n",
+       "            for pair in action_pairs:            \n",
+       "                if set(pair) in level.mutex:\n",
+       "                    non_mutex_actions.pop(-1)\n",
+       "                    break\n",
+       "    \n",
+       "\n",
+       "        # Recursion\n",
+       "        for action_list in non_mutex_actions:        \n",
+       "            if [action_list, index] not in self.solution:\n",
+       "                self.solution.append([action_list, index])\n",
+       "\n",
+       "                new_goals = []\n",
+       "                for act in set(action_list):                \n",
+       "                    if act in level.current_action_links:\n",
+       "                        new_goals = new_goals + level.current_action_links[act]\n",
+       "\n",
+       "                if abs(index) + 1 == len(self.graph.levels):\n",
+       "                    return\n",
+       "                elif (level, new_goals) in self.nogoods:\n",
+       "                    return\n",
+       "                else:\n",
+       "                    self.extract_solution(new_goals, index - 1)\n",
+       "\n",
+       "        # Level-Order multiple solutions\n",
+       "        solution = []\n",
+       "        for item in self.solution:\n",
+       "            if item[1] == -1:\n",
+       "                solution.append([])\n",
+       "                solution[-1].append(item[0])\n",
+       "            else:\n",
+       "                solution[-1].append(item[0])\n",
+       "\n",
+       "        for num, item in enumerate(solution):\n",
+       "            item.reverse()\n",
+       "            solution[num] = item\n",
+       "\n",
+       "        return solution\n",
        "
\n",
        "\n",
        "\n"
@@ -998,130 +2727,54 @@
     }
    ],
    "source": [
-    "psource(three_block_tower)"
-   ]
-  },
-  {
-   "cell_type": "markdown",
-   "metadata": {},
-   "source": [
-    "**On(b, x):** The block **'b'** is on **'x'**. **'x'** can be a table or a block.\n",
-    "\n",
-    "**Block(x):** Declares **'x'** as a block.\n",
-    "\n",
-    "**Clear(x):** To tell that there is nothing on **'x'**.\n",
-    " \n",
-    " Let us now define an object of `three_block_tower` problem:"
-   ]
-  },
-  {
-   "cell_type": "code",
-   "execution_count": 22,
-   "metadata": {},
-   "outputs": [],
-   "source": [
-    "three_block_tower = three_block_tower()"
-   ]
-  },
-  {
-   "cell_type": "markdown",
-   "metadata": {},
-   "source": [
-    "Now, before taking any actions, we will check `three_tower_block` if it has completed the goal it is required to do"
-   ]
-  },
-  {
-   "cell_type": "code",
-   "execution_count": 23,
-   "metadata": {},
-   "outputs": [
-    {
-     "name": "stdout",
-     "output_type": "stream",
-     "text": [
-      "False\n"
-     ]
-    }
-   ],
-   "source": [
-    "print(three_block_tower.goal_test())"
-   ]
-  },
-  {
-   "cell_type": "markdown",
-   "metadata": {},
-   "source": [
-    "As we can see, it hasn't completed the goal. Now, we define the sequence of actions that it should take in order to build a stack of three blocks. Then the `three_block_tower` acts on each of them.\n",
-    "\n",
-    "The actions available to us are the following: MoveToTable, Move\n",
-    "\n",
-    "**MoveToTable(b, x):** Move the box **'b'** which is on top of box **'x'** to the table.\n",
-    "\n",
-    "**Move(b, x, y):** Move box **'b'** from top of **'x'** to the top of **'y'**.\n"
-   ]
-  },
-  {
-   "cell_type": "code",
-   "execution_count": 24,
-   "metadata": {},
-   "outputs": [],
-   "source": [
-    "solution = [expr(\"MoveToTable(C, A)\"),\n",
-    "            expr(\"Move(B, Table, C)\"),\n",
-    "            expr(\"Move(A, Table, B)\")]\n",
-    "\n",
-    "for action in solution:\n",
-    "    three_block_tower.act(action)"
-   ]
-  },
-  {
-   "cell_type": "markdown",
-   "metadata": {},
-   "source": [
-    "As the `three_block_tower` has taken all the steps it needed in order to achieve the goal, we can now check if it has acheived its goal"
-   ]
-  },
-  {
-   "cell_type": "code",
-   "execution_count": 25,
-   "metadata": {},
-   "outputs": [
-    {
-     "name": "stdout",
-     "output_type": "stream",
-     "text": [
-      "True\n"
-     ]
-    }
-   ],
-   "source": [
-    "print(three_block_tower.goal_test())"
+    "psource(GraphPlan)"
    ]
   },
   {
    "cell_type": "markdown",
    "metadata": {},
    "source": [
-    "It has now successfully achieved its goal i.e, to build a stack of three blocks."
+    "Given a planning problem defined as a PDDL, `GraphPlan` creates a planning graph stored in `graph` and expands it till it reaches a state where all its required goals are present simultaneously without mutual exclusivity.\n",
+    "
\n",
+    "Once a goal is found, `extract_solution` is called.\n",
+    "This method recursively finds the path to a solution given a planning graph.\n",
+    "In the case where `extract_solution` fails to find a solution for a set of goals as a given level, we record the `(level, goals)` pair as a **no-good**.\n",
+    "Whenever `extract_solution` is called again with the same level and goals, we can find the recorded no-good and immediately return failure rather than searching again. \n",
+    "No-goods are also used in the termination test.\n",
+    "
\n",
+    "The `check_leveloff` method checks if the planning graph for the problem has **levelled-off**, ie, it has the same states, actions and mutex pairs as the previous level.\n",
+    "If the graph has already levelled off and we haven't found a solution, there is no point expanding the graph, as it won't lead to anything new.\n",
+    "In such a case, we can declare that the planning problem is unsolvable with the given constraints.\n",
+    "
\n",
+    "
\n",
+    "To summarize, the `GraphPlan` algorithm calls `expand_graph` and tests whether it has reached the goal and if the goals are non-mutex.\n",
+    "
\n",
+    "If so, `extract_solution` is invoked which recursively reconstructs the solution from the planning graph.\n",
+    "
\n",
+    "If not, then we check if our graph has levelled off and continue if it hasn't."
    ]
   },
   {
    "cell_type": "markdown",
    "metadata": {},
    "source": [
-    "## Have Cake and Eat Cake Too"
+    "Let's solve a few planning problems that we had defined earlier."
    ]
   },
   {
    "cell_type": "markdown",
    "metadata": {},
    "source": [
-    "This problem involves the task of eating a cake with an initial condition of having a cake. First, let us take a look at `have_cake_and_eat_cake_too`"
+    "Air cargo problem:\n",
+    "
\n",
+    "In accordance with the summary above, we have defined a helper function to carry out `GraphPlan` on the `air_cargo` problem.\n",
+    "The function is pretty straightforward.\n",
+    "Let's have a look."
    ]
   },
   {
    "cell_type": "code",
-   "execution_count": 26,
+   "execution_count": 43,
    "metadata": {},
    "outputs": [
     {
@@ -1213,30 +2866,26 @@
        "\n",
        "
\n",
        "\n",
-       "
def have_cake_and_eat_cake_too():\n",
-       "    init = [expr('Have(Cake)')]\n",
+       "
def air_cargo_graphplan():\n",
+       "    """Solves the air cargo problem using GraphPlan"""\n",
        "\n",
-       "    def goal_test(kb):\n",
-       "        required = [expr('Have(Cake)'), expr('Eaten(Cake)')]\n",
-       "        return all(kb.ask(q) is not False for q in required)\n",
+       "    pddl = air_cargo()\n",
+       "    graphplan = GraphPlan(pddl)\n",
        "\n",
-       "    # Actions\n",
+       "    def goal_test(kb, goals):\n",
+       "        return all(kb.ask(q) is not False for q in goals)\n",
        "\n",
-       "    # Eat cake\n",
-       "    precond_pos = [expr('Have(Cake)')]\n",
-       "    precond_neg = []\n",
-       "    effect_add = [expr('Eaten(Cake)')]\n",
-       "    effect_rem = [expr('Have(Cake)')]\n",
-       "    eat_cake = Action(expr('Eat(Cake)'), [precond_pos, precond_neg], [effect_add, effect_rem])\n",
+       "    goals = expr('At(C1, JFK), At(C2, SFO)')\n",
        "\n",
-       "    # Bake Cake\n",
-       "    precond_pos = []\n",
-       "    precond_neg = [expr('Have(Cake)')]\n",
-       "    effect_add = [expr('Have(Cake)')]\n",
-       "    effect_rem = []\n",
-       "    bake_cake = Action(expr('Bake(Cake)'), [precond_pos, precond_neg], [effect_add, effect_rem])\n",
+       "    while True:\n",
+       "        if (goal_test(graphplan.graph.levels[-1].kb, goals) and graphplan.graph.non_mutex_goals(goals, -1)):\n",
+       "            solution = graphplan.extract_solution(goals, -1)\n",
+       "            if solution:\n",
+       "                return solution\n",
        "\n",
-       "    return PDDL(init, [eat_cake, bake_cake], goal_test)\n",
+       "        graphplan.graph.expand_graph()\n",
+       "        if len(graphplan.graph.levels) >= 2 and graphplan.check_leveloff():\n",
+       "            return None\n",
        "
\n",
        "\n",
        "\n"
@@ -1250,102 +2899,169 @@
     }
    ],
    "source": [
-    "psource(have_cake_and_eat_cake_too)"
+    "psource(air_cargo_graphplan)"
    ]
   },
   {
    "cell_type": "markdown",
    "metadata": {},
    "source": [
-    "**Have(x):** Declares that we have **' x '**."
+    "Let's instantiate the problem and find a solution using this helper function."
    ]
   },
   {
    "cell_type": "code",
-   "execution_count": 27,
+   "execution_count": 44,
    "metadata": {},
-   "outputs": [],
+   "outputs": [
+    {
+     "data": {
+      "text/plain": [
+       "[[[PCargo(C2),\n",
+       "   Load(C2, P2, JFK),\n",
+       "   PPlane(P2),\n",
+       "   Load(C1, P1, SFO),\n",
+       "   Fly(P1, SFO, JFK),\n",
+       "   PAirport(SFO),\n",
+       "   PAirport(JFK),\n",
+       "   PPlane(P1),\n",
+       "   PCargo(C1),\n",
+       "   Fly(P2, JFK, SFO)],\n",
+       "  [Unload(C2, P2, SFO), Unload(C1, P1, JFK)]]]"
+      ]
+     },
+     "execution_count": 44,
+     "metadata": {},
+     "output_type": "execute_result"
+    }
+   ],
    "source": [
-    "have_cake_and_eat_cake_too = have_cake_and_eat_cake_too()"
+    "air_cargo = air_cargo_graphplan()\n",
+    "air_cargo"
    ]
   },
   {
    "cell_type": "markdown",
    "metadata": {},
    "source": [
-    "First let us check wether the goal state (have cake and eat cake) is reached or not."
+    "Each element in the solution is a valid action.\n",
+    "The solution is separated into lists for each level.\n",
+    "The actions prefixed with a 'P' are persistence actions and can be ignored.\n",
+    "They simply carry certain states forward.\n",
+    "We have another helper function `linearize` that presents the solution in a more readable format, much like a total-order planner."
    ]
   },
   {
    "cell_type": "code",
-   "execution_count": 28,
+   "execution_count": 45,
    "metadata": {},
    "outputs": [
     {
-     "name": "stdout",
-     "output_type": "stream",
-     "text": [
-      "False\n"
-     ]
+     "data": {
+      "text/plain": [
+       "[Load(C2, P2, JFK),\n",
+       " Load(C1, P1, SFO),\n",
+       " Fly(P1, SFO, JFK),\n",
+       " Fly(P2, JFK, SFO),\n",
+       " Unload(C2, P2, SFO),\n",
+       " Unload(C1, P1, JFK)]"
+      ]
+     },
+     "execution_count": 45,
+     "metadata": {},
+     "output_type": "execute_result"
     }
    ],
    "source": [
-    "print(have_cake_and_eat_cake_too.goal_test())"
+    "linearize(air_cargo)"
    ]
   },
   {
    "cell_type": "markdown",
    "metadata": {},
    "source": [
-    "As the goal state is not reached we will make some actions and we will let `have_cake_and_eat_cake_too` act on them. To eat the cake we need to bake it. Let us look at the actions that we can do.\n",
-    "\n",
-    "**Bake(x):** To bake **' x '**.\n",
-    "\n",
-    "**Eat(x):** To eat **' x '**."
+    "Indeed, this is a correct solution.\n",
+    "
\n",
+    "There are similar helper functions for some other planning problems.\n",
+    "
\n",
+    "Lets' try solving the spare tire problem."
    ]
   },
   {
    "cell_type": "code",
-   "execution_count": 29,
+   "execution_count": 46,
    "metadata": {},
-   "outputs": [],
+   "outputs": [
+    {
+     "data": {
+      "text/plain": [
+       "[Remove(Flat, Axle), Remove(Spare, Trunk), PutOn(Spare, Axle)]"
+      ]
+     },
+     "execution_count": 46,
+     "metadata": {},
+     "output_type": "execute_result"
+    }
+   ],
    "source": [
-    "solution = [expr(\"Bake(cake)\"),\n",
-    "            expr(\"Eat(cake)\")]\n",
-    "\n",
-    "for action in solution:\n",
-    "    have_cake_and_eat_cake_too.act(action)"
+    "spare_tire = spare_tire_graphplan()\n",
+    "linearize(spare_tire)"
    ]
   },
   {
    "cell_type": "markdown",
    "metadata": {},
    "source": [
-    "Now we have made actions to bake the cake and eat the cake. The goal state is **having and eating the cake**. Let us check if it is reached or not."
+    "Solution for the cake problem"
    ]
   },
   {
    "cell_type": "code",
-   "execution_count": 30,
+   "execution_count": 47,
    "metadata": {},
    "outputs": [
     {
-     "name": "stdout",
-     "output_type": "stream",
-     "text": [
-      "True\n"
-     ]
+     "data": {
+      "text/plain": [
+       "[Eat(Cake), Bake(Cake)]"
+      ]
+     },
+     "execution_count": 47,
+     "metadata": {},
+     "output_type": "execute_result"
     }
    ],
    "source": [
-    "print(have_cake_and_eat_cake_too.goal_test())"
+    "cake_problem = have_cake_and_eat_cake_too_graphplan()\n",
+    "linearize(cake_problem)"
    ]
   },
   {
    "cell_type": "markdown",
    "metadata": {},
    "source": [
-    "It has now successfully achieved its goal i.e, to have and eat the cake."
+    "Solution for the Sussman's Anomaly configuration of three blocks."
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": 48,
+   "metadata": {},
+   "outputs": [
+    {
+     "data": {
+      "text/plain": [
+       "[MoveToTable(C, A), Move(B, Table, C), Move(A, Table, B)]"
+      ]
+     },
+     "execution_count": 48,
+     "metadata": {},
+     "output_type": "execute_result"
+    }
+   ],
+   "source": [
+    "sussman_anomaly = three_block_tower_graphplan()\n",
+    "linearize(sussman_anomaly)"
    ]
   }
  ],
@@ -1365,7 +3081,7 @@
    "name": "python",
    "nbconvert_exporter": "python",
    "pygments_lexer": "ipython3",
-   "version": "3.5.2"
+   "version": "3.6.1"
   }
  },
  "nbformat": 4,
diff --git a/planning.py b/planning.py
index b7c1c021d..d9a152e9a 100644
--- a/planning.py
+++ b/planning.py
@@ -4,7 +4,7 @@
 import itertools
 from search import Node
 from utils import Expr, expr, first
-from logic import FolKB
+from logic import FolKB, conjuncts
 from collections import deque
 
 
@@ -15,13 +15,22 @@ class PDDL:
     The conjunction of these logical statements completely defines a state.
     """
 
-    def __init__(self, initial_state, actions, goal_test):
-        self.kb = FolKB(initial_state)
+    def __init__(self, init, goals, actions):
+        self.init = self.convert(init)
+        self.goals = expr(goals)
         self.actions = actions
-        self.goal_test_func = goal_test
+
+    def convert(self, init):
+        """Converts strings into exprs"""
+        try:
+            init = conjuncts(expr(init))
+        except AttributeError:
+            init = expr(init)
+        return init
 
     def goal_test(self):
-        return self.goal_test_func(self.kb)
+        """Checks if the goals have been reached"""
+        return all(goal in self.init for goal in conjuncts(self.goals))
 
     def act(self, action):
         """
@@ -33,9 +42,9 @@ def act(self, action):
         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.kb, args):
+        if not list_action.check_precond(self.init, args):
             raise Exception("Action '{}' pre-conditions not satisfied".format(action))
-        list_action(self.kb, args)
+        self.init = list_action(self.init, args).clauses
 
 
 class Action:
@@ -43,28 +52,47 @@ class Action:
     Defines an action schema using preconditions and effects.
     Use this to describe actions in PDDL.
     action is an Expr where variables are given as arguments(args).
-    Precondition and effect are both lists with positive and negated literals.
+    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_pos = [expr("Human(person)"), expr("Hungry(Person)")]
-    precond_neg = [expr("Eaten(food)")]
-    effect_add = [expr("Eaten(food)")]
-    effect_rem = [expr("Hungry(person)")]
-    eat = Action(expr("Eat(person, food)"), [precond_pos, precond_neg], [effect_add, effect_rem])
+    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):
+        action = expr(action)
         self.name = action.op
         self.args = action.args
-        self.precond_pos = precond[0]
-        self.precond_neg = precond[1]
-        self.effect_add = effect[0]
-        self.effect_rem = effect[1]
+        self.precond, self.effect = self.convert(precond, effect)
 
     def __call__(self, kb, args):
         return self.act(kb, args)
 
+    def convert(self, precond, effect):
+        """Converts strings into Exprs"""
+
+        precond = precond.replace('~', 'Not')
+        if len(precond) > 0:
+            precond = expr(precond)
+        effect = effect.replace('~', 'Not')
+        if len(effect) > 0:
+            effect = expr(effect)
+
+        try:
+            precond = conjuncts(precond)
+        except AttributeError:
+            pass
+        try:
+            effect = conjuncts(effect)
+        except AttributeError:
+            pass
+
+        return precond, 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):
@@ -74,237 +102,178 @@ def substitute(self, e, args):
 
     def check_precond(self, kb, args):
         """Checks if the precondition is satisfied in the current state"""
-        # check for positive clauses
-        for clause in self.precond_pos:
+
+        if isinstance(kb, list):
+            kb = FolKB(kb)
+
+        for clause in self.precond:
             if self.substitute(clause, args) not in kb.clauses:
                 return False
-        # check for negative clauses
-        for clause in self.precond_neg:
-            if self.substitute(clause, args) in kb.clauses:
-                return False
         return True
 
     def act(self, kb, args):
-        """Executes the action on the state's kb"""
-        # check if the preconditions are satisfied
-        if not self.check_precond(kb, args):
-            raise Exception("Action pre-conditions not satisfied")
-        # remove negative literals
-        for clause in self.effect_rem:
-            kb.retract(self.substitute(clause, args))
-        # add positive literals
-        for clause in self.effect_add:
-            kb.tell(self.substitute(clause, args))
+        """Executes the action on the state's knowledge base"""
 
+        if isinstance(kb, list):
+            kb = FolKB(kb)
 
-def air_cargo():
-    init = [expr('At(C1, SFO)'),
-            expr('At(C2, JFK)'),
-            expr('At(P1, SFO)'),
-            expr('At(P2, JFK)'),
-            expr('Cargo(C1)'),
-            expr('Cargo(C2)'),
-            expr('Plane(P1)'),
-            expr('Plane(P2)'),
-            expr('Airport(JFK)'),
-            expr('Airport(SFO)')]
+        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)
 
-    def goal_test(kb):
-        required = [expr('At(C1 , JFK)'), expr('At(C2 ,SFO)')]
-        return all([kb.ask(q) is not False for q in required])
+                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)
 
-    # Actions
+                if kb.ask(self.substitute(new_clause, args)) is not False:    
+                    kb.retract(self.substitute(new_clause, args))
 
-    #  Load
-    precond_pos = [expr("At(c, a)"), expr("At(p, a)"), expr("Cargo(c)"), expr("Plane(p)"),
-                   expr("Airport(a)")]
-    precond_neg = []
-    effect_add = [expr("In(c, p)")]
-    effect_rem = [expr("At(c, a)")]
-    load = Action(expr("Load(c, p, a)"), [precond_pos, precond_neg], [effect_add, effect_rem])
+        return kb
 
-    #  Unload
-    precond_pos = [expr("In(c, p)"), expr("At(p, a)"), expr("Cargo(c)"), expr("Plane(p)"),
-                   expr("Airport(a)")]
-    precond_neg = []
-    effect_add = [expr("At(c, a)")]
-    effect_rem = [expr("In(c, p)")]
-    unload = Action(expr("Unload(c, p, a)"), [precond_pos, precond_neg], [effect_add, effect_rem])
 
-    #  Fly
-    #  Used 'f' instead of 'from' because 'from' is a python keyword and expr uses eval() function
-    precond_pos = [expr("At(p, f)"), expr("Plane(p)"), expr("Airport(f)"), expr("Airport(to)")]
-    precond_neg = []
-    effect_add = [expr("At(p, to)")]
-    effect_rem = [expr("At(p, f)")]
-    fly = Action(expr("Fly(p, f, to)"), [precond_pos, precond_neg], [effect_add, effect_rem])
+def air_cargo():
+    """Air cargo problem"""
 
-    return PDDL(init, [load, unload, fly], goal_test)
+    return PDDL(init='At(C1, SFO) & At(C2, JFK) & At(P1, SFO) & At(P2, JFK) & Cargo(C1) & Cargo(C2) & Plane(P1) & Plane(P2) & Airport(SFO) & Airport(JFK)',
+                goals='At(C1, JFK) & At(C2, SFO)', 
+                actions=[Action('Load(c, p, a)', 
+                                precond='At(c, a) & At(p, a) & Cargo(c) & Plane(p) & Airport(a)', 
+                                effect='In(c, p) & ~At(c, a)'),
+                         Action('Unload(c, p, a)',
+                                precond='In(c, p) & At(p, a) & Cargo(c) & Plane(p) & Airport(a)',
+                                effect='At(c, a) & ~In(c, p)'),
+                         Action('Fly(p, f, to)',
+                                precond='At(p, f) & Plane(p) & Airport(f) & Airport(to)',
+                                effect='At(p, to) & ~At(p, f)')])
 
 
 def spare_tire():
-    init = [expr('Tire(Flat)'),
-            expr('Tire(Spare)'),
-            expr('At(Flat, Axle)'),
-            expr('At(Spare, Trunk)')]
-
-    def goal_test(kb):
-        required = [expr('At(Spare, Axle)')]
-        return all(kb.ask(q) is not False for q in required)
-
-    # Actions
-
-    # Remove
-    precond_pos = [expr("At(obj, loc)")]
-    precond_neg = []
-    effect_add = [expr("At(obj, Ground)")]
-    effect_rem = [expr("At(obj, loc)")]
-    remove = Action(expr("Remove(obj, loc)"), [precond_pos, precond_neg], [effect_add, effect_rem])
-
-    # PutOn
-    precond_pos = [expr("Tire(t)"), expr("At(t, Ground)")]
-    precond_neg = [expr("At(Flat, Axle)")]
-    effect_add = [expr("At(t, Axle)")]
-    effect_rem = [expr("At(t, Ground)")]
-    put_on = Action(expr("PutOn(t, Axle)"), [precond_pos, precond_neg], [effect_add, effect_rem])
-
-    # LeaveOvernight
-    precond_pos = []
-    precond_neg = []
-    effect_add = []
-    effect_rem = [expr("At(Spare, Ground)"), expr("At(Spare, Axle)"), expr("At(Spare, Trunk)"),
-                  expr("At(Flat, Ground)"), expr("At(Flat, Axle)"), expr("At(Flat, Trunk)")]
-    leave_overnight = Action(expr("LeaveOvernight"), [precond_pos, precond_neg],
-                             [effect_add, effect_rem])
-
-    return PDDL(init, [remove, put_on, leave_overnight], goal_test)
+    """Spare tire problem"""
+
+    return PDDL(init='Tire(Flat) & Tire(Spare) & 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)'),
+                         Action('PutOn(t, Axle)',
+                                precond='Tire(t) & At(t, Ground) & ~At(Flat, Axle)',
+                                effect='At(t, Axle) & ~At(t, Ground)'),
+                         Action('LeaveOvernight',
+                                precond='',
+                                effect='~At(Spare, Ground) & ~At(Spare, Axle) & ~At(Spare, Trunk) & \
+                                        ~At(Flat, Ground) & ~At(Flat, Axle) & ~At(Flat, Trunk)')])
 
 
 def three_block_tower():
-    init = [expr('On(A, Table)'),
-            expr('On(B, Table)'),
-            expr('On(C, A)'),
-            expr('Block(A)'),
-            expr('Block(B)'),
-            expr('Block(C)'),
-            expr('Clear(B)'),
-            expr('Clear(C)')]
+    """Sussman Anomaly problem"""
 
-    def goal_test(kb):
-        required = [expr('On(A, B)'), expr('On(B, C)')]
-        return all(kb.ask(q) is not False for q in required)
-
-    # Actions
-
-    #  Move
-    precond_pos = [expr('On(b, x)'), expr('Clear(b)'), expr('Clear(y)'), expr('Block(b)'),
-                   expr('Block(y)')]
-    precond_neg = []
-    effect_add = [expr('On(b, y)'), expr('Clear(x)')]
-    effect_rem = [expr('On(b, x)'), expr('Clear(y)')]
-    move = Action(expr('Move(b, x, y)'), [precond_pos, precond_neg], [effect_add, effect_rem])
-
-    #  MoveToTable
-    precond_pos = [expr('On(b, x)'), expr('Clear(b)'), expr('Block(b)')]
-    precond_neg = []
-    effect_add = [expr('On(b, Table)'), expr('Clear(x)')]
-    effect_rem = [expr('On(b, x)')]
-    moveToTable = Action(expr('MoveToTable(b, x)'), [precond_pos, precond_neg],
-                         [effect_add, effect_rem])
-
-    return PDDL(init, [move, moveToTable], goal_test)
+    return PDDL(init='On(A, Table) & On(B, Table) & On(C, A) & Block(A) & Block(B) & Block(C) & 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) & Block(b) & Block(y)',
+                                effect='On(b, y) & Clear(x) & ~On(b, x) & ~Clear(y)'),
+                         Action('MoveToTable(b, x)',
+                                precond='On(b, x) & Clear(b) & Block(b)',
+                                effect='On(b, Table) & Clear(x) & ~On(b, x)')])
 
 
 def have_cake_and_eat_cake_too():
-    init = [expr('Have(Cake)')]
+    """Cake problem"""
 
-    def goal_test(kb):
-        required = [expr('Have(Cake)'), expr('Eaten(Cake)')]
-        return all(kb.ask(q) is not False for q in required)
-
-    # Actions
+    return PDDL(init='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)')])
 
-    # Eat cake
-    precond_pos = [expr('Have(Cake)')]
-    precond_neg = []
-    effect_add = [expr('Eaten(Cake)')]
-    effect_rem = [expr('Have(Cake)')]
-    eat_cake = Action(expr('Eat(Cake)'), [precond_pos, precond_neg], [effect_add, effect_rem])
 
-    # Bake Cake
-    precond_pos = []
-    precond_neg = [expr('Have(Cake)')]
-    effect_add = [expr('Have(Cake)')]
-    effect_rem = []
-    bake_cake = Action(expr('Bake(Cake)'), [precond_pos, precond_neg], [effect_add, effect_rem])
+def shopping_problem():
+    """Shopping problem"""
 
-    return PDDL(init, [eat_cake, bake_cake], goal_test)
+    return PDDL(init='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)'),
+                         Action('Go(x, y)',
+                                precond='At(x)',
+                                effect='At(y) & ~At(x)')])
 
 
-class Level():
+class Level:
     """
     Contains the state of the planning problem
     and exhaustive list of actions which use the
     states as pre-condition.
     """
 
-    def __init__(self, poskb, negkb):
-        self.poskb = poskb
-        # Current state
-        self.current_state_pos = poskb.clauses
-        self.current_state_neg = negkb.clauses
-        # Current action to current state link
-        self.current_action_links_pos = {}
-        self.current_action_links_neg = {}
-        # Current state to action link
-        self.current_state_links_pos = {}
-        self.current_state_links_neg = {}
-        # Current action to next state link
+    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_pos = {}
-        self.next_state_links_neg = {}
+        # 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
-        for poseff in self.next_state_links_pos:
-            negeff = poseff
-            if negeff in self.next_state_links_neg:
-                for a in self.next_state_links_pos[poseff]:
-                    for b in self.next_state_links_neg[negeff]:
-                        if {a, b} not in self.mutex:
-                            self.mutex.append({a, b})
-
-        # Interference
-        for posprecond in self.current_state_links_pos:
-            negeff = posprecond
-            if negeff in self.next_state_links_neg:
-                for a in self.current_state_links_pos[posprecond]:
-                    for b in self.next_state_links_neg[negeff]:
-                        if {a, b} not in self.mutex:
-                            self.mutex.append({a, b})
-
-        for negprecond in self.current_state_links_neg:
-            poseff = negprecond
-            if poseff in self.next_state_links_pos:
-                for a in self.next_state_links_pos[poseff]:
-                    for b in self.current_state_links_neg[negprecond]:
-                        if {a, b} not in self.mutex:
-                            self.mutex.append({a, b})
+        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 posprecond in self.current_state_links_pos:
-            negprecond = posprecond
-            if negprecond in self.current_state_links_neg:
-                for a in self.current_state_links_pos[posprecond]:
-                    for b in self.current_state_links_neg[negprecond]:
-                        if {a, b} not in self.mutex:
-                            self.mutex.append({a, b})
+        for posprecond in pos_csl:
+            for negprecond in neg_csl:
+                new_negprecond = Expr(negprecond.op[3:], *negprecond.args)
+                if new_negprecond == posprecond:
+                    for a in self.current_state_links[posprecond]:
+                        for b in self.current_state_links[negprecond]:
+                            if {a, b} not in self.mutex:
+                                self.mutex.append({a, b})
 
         # Inconsistent support
         state_mutex = []
@@ -316,32 +285,25 @@ def find_mutex(self):
                 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
+        
+        self.mutex = self.mutex + state_mutex
 
     def build(self, actions, objects):
+        """Populates the lists and dictionaries containing the state action dependencies"""
 
-        # Add persistence actions for positive states
-        for clause in self.current_state_pos:
-            self.current_action_links_pos[Expr('Persistence', clause)] = [clause]
-            self.next_action_links[Expr('Persistence', clause)] = [clause]
-            self.current_state_links_pos[clause] = [Expr('Persistence', clause)]
-            self.next_state_links_pos[clause] = [Expr('Persistence', clause)]
-
-        # Add persistence actions for negative states
-        for clause in self.current_state_neg:
-            not_expr = Expr('not'+clause.op, clause.args)
-            self.current_action_links_neg[Expr('Persistence', not_expr)] = [clause]
-            self.next_action_links[Expr('Persistence', not_expr)] = [clause]
-            self.current_state_links_neg[clause] = [Expr('Persistence', not_expr)]
-            self.next_state_links_neg[clause] = [Expr('Persistence', not_expr)]
+        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.poskb, arg):
+                if a.check_precond(self.kb, arg):
                     for num, symbol in enumerate(a.args):
                         if not symbol.op.islower():
                             arg = list(arg)
@@ -349,47 +311,31 @@ def build(self, actions, objects):
                             arg = tuple(arg)
 
                     new_action = a.substitute(Expr(a.name, *a.args), arg)
-                    self.current_action_links_pos[new_action] = []
-                    self.current_action_links_neg[new_action] = []
+                    self.current_action_links[new_action] = []
 
-                    for clause in a.precond_pos:
+                    for clause in a.precond:
                         new_clause = a.substitute(clause, arg)
-                        self.current_action_links_pos[new_action].append(new_clause)
-                        if new_clause in self.current_state_links_pos:
-                            self.current_state_links_pos[new_clause].append(new_action)
+                        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_pos[new_clause] = [new_action]
-
-                    for clause in a.precond_neg:
-                        new_clause = a.substitute(clause, arg)
-                        self.current_action_links_neg[new_action].append(new_clause)
-                        if new_clause in self.current_state_links_neg:
-                            self.current_state_links_neg[new_clause].append(new_action)
-                        else:
-                            self.current_state_links_neg[new_clause] = [new_action]
-
+                            self.current_state_links[new_clause] = [new_action]
+                   
                     self.next_action_links[new_action] = []
-                    for clause in a.effect_add:
+                    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_pos:
-                            self.next_state_links_pos[new_clause].append(new_action)
-                        else:
-                            self.next_state_links_pos[new_clause] = [new_action]
 
-                    for clause in a.effect_rem:
-                        new_clause = a.substitute(clause, arg)
                         self.next_action_links[new_action].append(new_clause)
-                        if new_clause in self.next_state_links_neg:
-                            self.next_state_links_neg[new_clause].append(new_action)
+                        if new_clause in self.next_state_links:
+                            self.next_state_links[new_clause].append(new_action)
                         else:
-                            self.next_state_links_neg[new_clause] = [new_action]
+                            self.next_state_links[new_clause] = [new_action]
 
     def perform_actions(self):
-        new_kb_pos = FolKB(list(set(self.next_state_links_pos.keys())))
-        new_kb_neg = FolKB(list(set(self.next_state_links_neg.keys())))
+        """Performs the necessary actions and returns a new Level"""
 
-        return Level(new_kb_pos, new_kb_neg)
+        new_kb = FolKB(list(set(self.next_state_links.keys())))
+        return Level(new_kb)
 
 
 class Graph:
@@ -398,20 +344,25 @@ class Graph:
     Used in graph planning algorithm to extract a solution
     """
 
-    def __init__(self, pddl, negkb):
+    def __init__(self, pddl):
         self.pddl = pddl
-        self.levels = [Level(pddl.kb, negkb)]
-        self.objects = set(arg for clause in pddl.kb.clauses + negkb.clauses for arg in clause.args)
+        self.kb = FolKB(pddl.init)
+        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.pddl.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:
@@ -426,69 +377,63 @@ class GraphPlan:
     Returns solution for the planning problem
     """
 
-    def __init__(self, pddl, negkb):
-        self.graph = Graph(pddl, negkb)
+    def __init__(self, pddl):
+        self.graph = Graph(pddl)
         self.nogoods = []
         self.solution = []
 
     def check_leveloff(self):
-        first_check = (set(self.graph.levels[-1].current_state_pos) ==
-                       set(self.graph.levels[-2].current_state_pos))
-        second_check = (set(self.graph.levels[-1].current_state_neg) ==
-                        set(self.graph.levels[-2].current_state_neg))
+        """Checks if the graph has levelled off"""
 
-        if first_check and second_check:
+        check = (set(self.graph.levels[-1].current_state) == set(self.graph.levels[-2].current_state))
+
+        if check:
             return True
 
-    def extract_solution(self, goals_pos, goals_neg, index):
-        level = self.graph.levels[index]
-        if not self.graph.non_mutex_goals(goals_pos+goals_neg, index):
-            self.nogoods.append((level, goals_pos, goals_neg))
+    def extract_solution(self, goals, index):
+        """Extracts the solution"""
+
+        level = self.graph.levels[index]    
+        if not self.graph.non_mutex_goals(goals, index):
+            self.nogoods.append((level, goals))
             return
 
-        level = self.graph.levels[index-1]
+        level = self.graph.levels[index - 1]    
 
-        # Create all combinations of actions that satisfy the goal
+        # Create all combinations of actions that satisfy the goal    
         actions = []
-        for goal in goals_pos:
-            actions.append(level.next_state_links_pos[goal])
+        for goal in goals:
+            actions.append(level.next_state_links[goal])    
 
-        for goal in goals_neg:
-            actions.append(level.next_state_links_neg[goal])
+        all_actions = list(itertools.product(*actions))    
 
-        all_actions = list(itertools.product(*actions))
-
-        # Filter out the action combinations which contain mutexes
-        non_mutex_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:
+            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:
+        for action_list in non_mutex_actions:        
             if [action_list, index] not in self.solution:
                 self.solution.append([action_list, index])
 
-                new_goals_pos = []
-                new_goals_neg = []
-                for act in set(action_list):
-                    if act in level.current_action_links_pos:
-                        new_goals_pos = new_goals_pos + level.current_action_links_pos[act]
-
-                for act in set(action_list):
-                    if act in level.current_action_links_neg:
-                        new_goals_neg = new_goals_neg + level.current_action_links_neg[act]
+                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):
+                if abs(index) + 1 == len(self.graph.levels):
                     return
-                elif (level, new_goals_pos, new_goals_neg) in self.nogoods:
+                elif (level, new_goals) in self.nogoods:
                     return
                 else:
-                    self.extract_solution(new_goals_pos, new_goals_neg, index-1)
+                    self.extract_solution(new_goals, index - 1)
 
         # Level-Order multiple solutions
         solution = []
@@ -507,28 +452,125 @@ def extract_solution(self, goals_pos, goals_neg, index):
 
 
 def spare_tire_graphplan():
+    """Solves the spare tire problem using GraphPlan"""
+
     pddl = spare_tire()
-    negkb = FolKB([expr('At(Flat, Trunk)')])
-    graphplan = GraphPlan(pddl, negkb)
+    graphplan = GraphPlan(pddl)
+
+    def goal_test(kb, goals):
+        return all(kb.ask(q) is not False for q in goals)
+
+    goals = expr('At(Spare, Axle), At(Flat, Ground)')
+
+    while True:
+        graphplan.graph.expand_graph()
+        if (goal_test(graphplan.graph.levels[-1].kb, goals) and graphplan.graph.non_mutex_goals(goals, -1)):
+            solution = graphplan.extract_solution(goals, -1)
+            if solution:
+                return solution
+        
+        if len(graphplan.graph.levels) >= 2 and graphplan.check_leveloff():
+            return None
+
+
+def have_cake_and_eat_cake_too_graphplan():
+    """Solves the cake problem using GraphPlan"""
+
+    pddl = have_cake_and_eat_cake_too()
+    graphplan = GraphPlan(pddl)
+
+    def goal_test(kb, goals):
+        return all(kb.ask(q) is not False for q in goals)
+
+    goals = expr('Have(Cake), Eaten(Cake)')
+
+    while True:
+        graphplan.graph.expand_graph()
+        if (goal_test(graphplan.graph.levels[-1].kb, goals) and graphplan.graph.non_mutex_goals(goals, -1)):
+            solution = graphplan.extract_solution(goals, -1)
+            if solution:
+                return [solution[1]]
+
+        if len(graphplan.graph.levels) >= 2 and graphplan.check_leveloff():
+            return None
+
+
+def three_block_tower_graphplan():
+    """Solves the Sussman Anomaly problem using GraphPlan"""
+
+    pddl = three_block_tower()
+    graphplan = GraphPlan(pddl)
 
     def goal_test(kb, goals):
         return all(kb.ask(q) is not False for q in goals)
 
-    # Not sure
-    goals_pos = [expr('At(Spare, Axle)'), expr('At(Flat, Ground)')]
-    goals_neg = []
+    goals = expr('On(A, B), On(B, C)')
 
     while True:
-        if (goal_test(graphplan.graph.levels[-1].poskb, goals_pos) and
-                graphplan.graph.non_mutex_goals(goals_pos+goals_neg, -1)):
-            solution = graphplan.extract_solution(goals_pos, goals_neg, -1)
+        if (goal_test(graphplan.graph.levels[-1].kb, goals) and graphplan.graph.non_mutex_goals(goals, -1)):
+            solution = graphplan.extract_solution(goals, -1)
             if solution:
                 return solution
+
         graphplan.graph.expand_graph()
-        if len(graphplan.graph.levels) >=2 and graphplan.check_leveloff():
+        if len(graphplan.graph.levels) >= 2 and graphplan.check_leveloff():
             return None
 
 
+def air_cargo_graphplan():
+    """Solves the air cargo problem using GraphPlan"""
+
+    pddl = air_cargo()
+    graphplan = GraphPlan(pddl)
+
+    def goal_test(kb, goals):
+        return all(kb.ask(q) is not False for q in goals)
+
+    goals = expr('At(C1, JFK), At(C2, SFO)')
+
+    while True:
+        if (goal_test(graphplan.graph.levels[-1].kb, goals) and graphplan.graph.non_mutex_goals(goals, -1)):
+            solution = graphplan.extract_solution(goals, -1)
+            if solution:
+                return solution
+
+        graphplan.graph.expand_graph()
+        if len(graphplan.graph.levels) >= 2 and graphplan.check_leveloff():
+            return None
+
+
+def shopping_graphplan():
+    pddl = shopping_problem()
+    graphplan = GraphPlan(pddl)
+
+    def goal_test(kb, goals):
+        return all(kb.ask(q) is not False for q in goals)
+
+    goals = expr('Have(Milk), Have(Banana), Have(Drill)')
+
+    while True:
+        if (goal_test(graphplan.graph.levels[-1].kb, goals) and graphplan.graph.non_mutex_goals(goals, -1)):
+            solution = graphplan.extract_solution(goals, -1)
+            if solution:
+                return solution
+
+        graphplan.graph.expand_graph()
+        if len(graphplan.graph.levels) >= 2 and graphplan.check_leveloff():
+            return None
+
+
+def linearize(solution):
+    """Converts a level-ordered solution into a linear solution"""
+
+    linear_solution = []
+    for section in solution[0]:
+        for operation in section:
+            if not (operation.op[0] == 'P' and operation.op[1].isupper()):
+                linear_solution.append(operation)
+
+    return linear_solution
+
+
 def double_tennis_problem():
     init = [expr('At(A, LeftBaseLine)'),
             expr('At(B, RightNet)'),
@@ -770,21 +812,6 @@ def job_shop_problem():
     with resource and ordering constraints.
 
     Example:
-    >>> from planning import *
-    >>> p = job_shop_problem()
-    >>> p.goal_test()
-    False
-    >>> p.act(p.jobs[1][0])
-    >>> p.act(p.jobs[1][1])
-    >>> p.act(p.jobs[1][2])
-    >>> p.act(p.jobs[0][0])
-    >>> p.act(p.jobs[0][1])
-    >>> p.goal_test()
-    False
-    >>> p.act(p.jobs[0][2])
-    >>> p.goal_test()
-    True
-    >>>
     """
     init = [expr('Car(C1)'),
             expr('Car(C2)'),
diff --git a/tests/test_planning.py b/tests/test_planning.py
index c10c0e9ba..375c4e26a 100644
--- a/tests/test_planning.py
+++ b/tests/test_planning.py
@@ -1,20 +1,20 @@
 from planning import *
 from utils import expr
-from logic import FolKB
+from logic import FolKB, conjuncts
 
 
 def test_action():
-    precond = [[expr("P(x)"), expr("Q(y, z)")], [expr("Q(x)")]]
-    effect = [[expr("Q(x)")], [expr("P(x)")]]
-    a=Action(expr("A(x,y,z)"), precond, effect)
-    args = [expr("A"), expr("B"), expr("C")]
-    assert a.substitute(expr("P(x, z, y)"), args) == expr("P(A, C, B)")
-    test_kb = FolKB([expr("P(A)"), expr("Q(B, C)"), expr("R(D)")])
+    precond = 'At(c, a) & At(p, a) & Cargo(c) & Plane(p) & Airport(a)'
+    effect = 'In(c, p) & ~At(c, a)'
+    a = Action('Load(c, p, a)', precond, effect)
+    args = [expr("C1"), expr("P1"), expr("SFO")]
+    assert a.substitute(expr("Load(c, p, a)"), args) == expr("Load(C1, P1, SFO)")
+    test_kb = FolKB(conjuncts(expr('At(C1, SFO) & At(C2, JFK) & At(P1, SFO) & At(P2, JFK) & Cargo(C1) & Cargo(C2) & Plane(P1) & Plane(P2) & Airport(SFO) & Airport(JFK)')))
     assert a.check_precond(test_kb, args)
     a.act(test_kb, args)
-    assert test_kb.ask(expr("P(A)")) is False
-    assert test_kb.ask(expr("Q(A)")) is not False
-    assert test_kb.ask(expr("Q(B, C)")) is not False
+    assert test_kb.ask(expr("In(C1, P2)")) is False
+    assert test_kb.ask(expr("In(C1, P1)")) is not False
+    assert test_kb.ask(expr("Plane(P2)")) is not False
     assert not a.check_precond(test_kb, args)
 
 
@@ -62,18 +62,19 @@ def test_spare_tire():
 
     assert p.goal_test()
 
-def test_double_tennis():
-    p = double_tennis_problem()
-    assert p.goal_test() is False
 
-    solution = [expr("Go(A, RightBaseLine, LeftBaseLine)"),
-                expr("Hit(A, Ball, RightBaseLine)"),
-                expr("Go(A, LeftNet, RightBaseLine)")]
+def test_spare_tire_2():
+    p = spare_tire()
+    assert p.goal_test() is False
+    solution_2 = [expr('Remove(Spare, Trunk)'),
+                  expr('Remove(Flat, Axle)'),
+                  expr('PutOn(Spare, Axle)')]
 
-    for action in solution:
+    for action in solution_2:
         p.act(action)
 
     assert p.goal_test()
+
     
 def test_three_block_tower():
     p = three_block_tower()
@@ -100,10 +101,24 @@ def test_have_cake_and_eat_cake_too():
     assert p.goal_test()
 
 
+def test_shopping_problem():
+    p = shopping_problem()
+    assert p.goal_test() is False
+    solution = [expr('Go(Home, SM)'), 
+                expr('Buy(Banana, SM)'), 
+                expr('Buy(Milk, SM)'), 
+                expr('Go(SM, HW)'), 
+                expr('Buy(Drill, HW)')]
+
+    for action in solution:
+        p.act(action)
+
+    assert p.goal_test()
+
+
 def test_graph_call():
     pddl = spare_tire()
-    negkb = FolKB([expr('At(Flat, Trunk)')])
-    graph = Graph(pddl, negkb)
+    graph = Graph(pddl)
 
     levels_size = len(graph.levels)
     graph()
@@ -111,49 +126,100 @@ def test_graph_call():
     assert levels_size == len(graph.levels) - 1
 
 
-def test_job_shop_problem():
-    p = job_shop_problem()
-    assert p.goal_test() is False
+def test_graphplan():
+    spare_tire_solution = spare_tire_graphplan()
+    spare_tire_solution = linearize(spare_tire_solution)
+    assert expr('Remove(Flat, Axle)') in spare_tire_solution
+    assert expr('Remove(Spare, Trunk)') in spare_tire_solution
+    assert expr('PutOn(Spare, Axle)') in spare_tire_solution
 
-    solution = [p.jobs[1][0],
-                p.jobs[0][0],
-                p.jobs[0][1],
-                p.jobs[0][2],
-                p.jobs[1][1],
-                p.jobs[1][2]]
+    cake_solution = have_cake_and_eat_cake_too_graphplan()
+    cake_solution = linearize(cake_solution)
+    assert expr('Eat(Cake)') in cake_solution
+    assert expr('Bake(Cake)') in cake_solution
 
-    for action in solution:
-        p.act(action)
+    air_cargo_solution = air_cargo_graphplan()
+    air_cargo_solution = linearize(air_cargo_solution)
+    assert expr('Load(C1, P1, SFO)') in air_cargo_solution
+    assert expr('Load(C2, P2, JFK)') in air_cargo_solution
+    assert expr('Fly(P1, SFO, JFK)') in air_cargo_solution
+    assert expr('Fly(P2, JFK, SFO)') in air_cargo_solution
+    assert expr('Unload(C1, P1, JFK)') in air_cargo_solution
+    assert expr('Unload(C2, P2, SFO)') in air_cargo_solution
+
+    sussman_anomaly_solution = three_block_tower_graphplan()
+    sussman_anomaly_solution = linearize(sussman_anomaly_solution)
+    assert expr('MoveToTable(C, A)') in sussman_anomaly_solution
+    assert expr('Move(B, Table, C)') in sussman_anomaly_solution
+    assert expr('Move(A, Table, B)') in sussman_anomaly_solution
+
+    shopping_problem_solution = shopping_graphplan()
+    shopping_problem_solution = linearize(shopping_problem_solution)
+    assert expr('Go(Home, HW)') in shopping_problem_solution
+    assert expr('Go(Home, SM)') in shopping_problem_solution
+    assert expr('Buy(Drill, HW)') in shopping_problem_solution
+    assert expr('Buy(Banana, SM)') in shopping_problem_solution
+    assert expr('Buy(Milk, SM)') in shopping_problem_solution
+
+
+# def test_double_tennis():
+#     p = double_tennis_problem()
+#     assert p.goal_test() is False
+
+#     solution = [expr("Go(A, RightBaseLine, LeftBaseLine)"),
+#                 expr("Hit(A, Ball, RightBaseLine)"),
+#                 expr("Go(A, LeftNet, RightBaseLine)")]
+
+#     for action in solution:
+#         p.act(action)
+
+#     assert p.goal_test()
+
+
+# def test_job_shop_problem():
+#     p = job_shop_problem()
+#     assert p.goal_test() is False
+
+#     solution = [p.jobs[1][0],
+#                 p.jobs[0][0],
+#                 p.jobs[0][1],
+#                 p.jobs[0][2],
+#                 p.jobs[1][1],
+#                 p.jobs[1][2]]
+
+#     for action in solution:
+#         p.act(action)
+
+#     assert p.goal_test()
 
-    assert p.goal_test()
 
-def test_refinements() :
-    init = [expr('At(Home)')]
-    def goal_test(kb):
-        return kb.ask(expr('At(SFO)'))
+# def test_refinements():
+#     init = [expr('At(Home)')]
+#     def goal_test(kb):
+#         return kb.ask(expr('At(SFO)'))
         
-    library = {"HLA": ["Go(Home,SFO)","Taxi(Home, SFO)"],
-    "steps": [["Taxi(Home, SFO)"],[]],
-    "precond_pos": [["At(Home)"],["At(Home)"]],
-    "precond_neg": [[],[]],
-    "effect_pos": [["At(SFO)"],["At(SFO)"]],
-    "effect_neg": [["At(Home)"],["At(Home)"],]}
-    # Go SFO
-    precond_pos = [expr("At(Home)")]
-    precond_neg = []
-    effect_add = [expr("At(SFO)")]
-    effect_rem = [expr("At(Home)")]
-    go_SFO = HLA(expr("Go(Home,SFO)"),
-                      [precond_pos, precond_neg], [effect_add, effect_rem])
-    # Taxi SFO
-    precond_pos = [expr("At(Home)")]
-    precond_neg = []
-    effect_add = [expr("At(SFO)")]
-    effect_rem = [expr("At(Home)")]
-    taxi_SFO = HLA(expr("Go(Home,SFO)"),
-                      [precond_pos, precond_neg], [effect_add, effect_rem])
-    prob = Problem(init, [go_SFO, taxi_SFO], goal_test)
-    result = [i for i in Problem.refinements(go_SFO, prob, library)]
-    assert(len(result) == 1)
-    assert(result[0].name == "Taxi")
-    assert(result[0].args == (expr("Home"), expr("SFO")))
+#     library = {"HLA": ["Go(Home,SFO)","Taxi(Home, SFO)"],
+#     "steps": [["Taxi(Home, SFO)"],[]],
+#     "precond_pos": [["At(Home)"],["At(Home)"]],
+#     "precond_neg": [[],[]],
+#     "effect_pos": [["At(SFO)"],["At(SFO)"]],
+#     "effect_neg": [["At(Home)"],["At(Home)"],]}
+#     # Go SFO
+#     precond_pos = [expr("At(Home)")]
+#     precond_neg = []
+#     effect_add = [expr("At(SFO)")]
+#     effect_rem = [expr("At(Home)")]
+#     go_SFO = HLA(expr("Go(Home,SFO)"),
+#                       [precond_pos, precond_neg], [effect_add, effect_rem])
+#     # Taxi SFO
+#     precond_pos = [expr("At(Home)")]
+#     precond_neg = []
+#     effect_add = [expr("At(SFO)")]
+#     effect_rem = [expr("At(Home)")]
+#     taxi_SFO = HLA(expr("Go(Home,SFO)"),
+#                       [precond_pos, precond_neg], [effect_add, effect_rem])
+#     prob = Problem(init, [go_SFO, taxi_SFO], goal_test)
+#     result = [i for i in Problem.refinements(go_SFO, prob, library)]
+#     assert(len(result) == 1)
+#     assert(result[0].name == "Taxi")
+#     assert(result[0].args == (expr("Home"), expr("SFO")))