class: center, middle # Artificial Intelligence ### Planning --- class: center, middle # The Planning Problem --- # The Planning Problem A planning problem consists of three parts: * A definition of the current state of the world * A definition of a desired state of the world * A definition of the actions the agent can take All of these definitions are done in "some" formal language. -- Logic is a good choice for a formal language! --- # An Example Planning Problem: Blocks World * We have three blocks, A, B, and C * Blocks A and B are on the table, block C is on top of block B * Blocks can be stacked * At the moment, all blocks are on the table next to each other * We want the blocks to be stacked, with C on top of B, and B on top of A --- # An Example Planning Problem: Blocks World <img src="/PF-3341/assets/img/problemstatement.png" width="100%"/> --- # Formalizing the Problem Current state: $$ \text{on}(A, \mathit{table}) \wedge\\\\ \text{on}(B, \mathit{table}) \wedge \\\\ \text{on}(C, B)\\\\ $$ Goal: $$ \text{on}(A, \mathit{table}) \wedge \\\\ \text{on}(B, A) \wedge\\\\ \text{on}(C, B)\\\\ $$ --- # Actions? What can the agent do? * Let's say our agent is a simple robot * It has a gripper * It can pick up a block, and also put it down (on the table, or on top of another block) --- # Formalizing Actions? * First, we need to represent that the robot is holding some block X somehow: `\(\text{holds}(X)\)` * Picking up a block then has a **precondition** and an **effect** * The precondition defines *if* an action can even be taken * The effect defines what the action *changes* in the world --- # Picking Up a Block X Precondition: $$ \forall Y \in \mathit{blocks}: \neg \text{holds}(Y) \wedge \\\\ \forall Y \in \mathit{blocks}: \neg \text{on}(Y, X) $$ Effect: $$ \forall Y \in \mathit{blocks}: \neg \text{on}(X, Y) \wedge \\\\ \neg \text{on}(X, \mathit{table}) \wedge \\\\ \text{holds}(X) $$ --- # Putting a Block X on Top of a Block Y Precondition: $$ \text{holds}(X) \wedge \\\\ \forall Z \in \mathit{blocks}: \neg \text{on}(Z, Y) $$ Effect: $$ \text{on}(X, Y) \wedge \\\\ \neg \text{holds}(X) $$ --- class: small # The Planning Problem Given: * An initial state `\(s_0\)` * Action definitions `\(a = (p,e)\)` consisting of a precondition and an effect * A goal formula `\(g\)` Find a sequence of actions `\(a_0, a_1, \ldots, a_n\)`, such that: * `\(s_i \models p_i\)` * `\(s_{i+1} =\:\text{apply}\:\:e_i\:\:\text{to}\:\:s_i\)` * `\(s_{n+1} \models g\)` Each action can be applied in the state preceding it, and generates a new state using its effect. In the final state the goal condition holds. --- # Planners - A *planner* is an algorithm that solves the planning problem - Two important properties for a planner are: + **Soundness**: Any plan the planner produces is valid, i.e. satisfies the three conditions + **Completeness**: For any planning problem that has a solution, the planner will (eventually) produce one - These properties are "nice to have", but not always necessary! --- class: medium # Actions vs. Action Schemata * Technically "Picking up a block X" (`pickup(X)`) is not an action, because there is no "Block X" in our domain * We used "X" as a free variable to refer to "any block" * `pickup(X)` is an action schema, called **operator**, that will need to be turned into ground actions (without free variables) `pickup(A)`, `pickup(B)`, and `pickup(C)` --- # The Sussman Anomaly Let's look at a slightly different problem: <img src="/PF-3341/assets/img/sussman.png" width="100%"/> --- class: small # The Sussman Anomaly There are two goals: `on(A,B)` and `on(B,C)`. If the planner divides the goal into subgoals: - If it tries to reach a state with A on top of B first, it will put C aside, then put A on B, and then can not fulfill the other goal without undoing the first (by removing A). - If it tries to reach a state with B on top of C first, it will just put B on top of the pile, and then can't satisfy the other goal. A "proper" solution has to interleave solving these two goals: First remove C, then put B on C, then put A on B. This requirement for interleaving is called "the Sussman Anomaly", discovered in the 1970s. Modern Planners do not run into this problem anymore. --- class: center, middle # Planning as Pathfinding --- # Planning As Pathfinding To use a pathfinding algorithm for planning, we need to formulate the planning problem as a graph. Let's start with the "obvious" choice: * Each state is a logical state * Each edge corresponds to an action * In any state, we can take any action whose preconditions are satisfied to generate a new state --- # The State-Space - When we start at the node corresponding to the start state and start expanding actions, we generate the so-called *State-Space* - We can then use a standard A* algorithm to do pathfinding - In fact, many planners do exactly that, with different heuristics --- # State Space Example <img src="/PF-3341/assets/img/planspace.png" width="100%"/> --- # Planning Problem in State Space <img src="/PF-3341/assets/img/problem.png" width="100%"/> --- class: mmedium # Planning Heuristics - In planning, our nodes are (logical) states, and our edges are actions - When we expand a state by applying all actions, we compute a heuristic value for each of the successor states - This heuristic value should estimate how long a plan from that state to a goal state is (at most) - In other words, we want to estimate how "difficult" it is to satisfy the goal from each state in our search frontier - Many planners solve a reduced ("relaxed") version of the planning problem to calculate a heuristic value --- # (Some) Planning Heuristics - HSP: Ignore negative literals in effects - FastForward: Do a "Breadth-First Search" with merging of states - FastDownward: Convert Logic into a different representation, eliminate mutual exclusive states, and calculate possible state changes --- class: center, middle # Planning is PSPACE-complete --- # Planning Complexity * You probably know the problem of P vs. NP? * PSPACE is (probably) even worse than NP $$ P \subseteq \mathit{NP} \subseteq \mathit{PSPACE} \subseteq \mathit{EXP} $$ (Any of these subsets may or may not be proper, but at least one of them is) --- # Why is Planning Hard? * We just said that planning is pathfinding on a graph consisting of the states * Pathfinding is solvable in polynomial time * Planning is not? Why? -- * For a pathfinding problem, the input is the graph. For a planning problem the input is the initial state, the goal condition and the actions. The graph is **implicit** (and potentially exponentially large) --- # It gets worse ... <img src="/PF-3335/assets/img/plancomplexity.png" width="80%"/> --- class: medium # The Good News * [All PSPACE-complete planning problems are equal, but some are more equal than others](https://www.aaai.org/ocs/index.php/SOCS/SOCS11/paper/view/4009/4353) * Many interesting planning problems, i.e. the ones that show up in practice, are tractable, even if their generalizations are theoretically hard to solve * Another way to think about it: Since the solutions we usually *want* are not going to be exponential in length, we shouldn't have to generate a graph of exponential size * Also nice: Adding more convenience features doesn't actually make planning harder --- class: center, middle # Applications --- # Some Applications - Logistics (RoboCup Logistics League) - Ride sharing - Video Games - Narrative Generation --- # Goal-Oriented Action Planning * F.E.A.R. (and several other games) used a variant of planning for its enemy AI * Instead of arbitrary goals and states, it greatly restricts what can be represented * This restriction allowed it to be used in a real-time environment * The developers stated that using planning allowed them to easily add new action types --- # Story Generation - We have an initial state of the world - We have actions the different characters can perform - The author defines a goal for the story - For example: In a detective story the goal is that there is a murder and that the detective identifies the murderer --- # Story Example * Sherlock Holmes goes to Irene Adler's house * Sherlock Holmes stabs Irene Adler * Sherlock Holmes identifies himself as the killer * The End. --- # Story Generation as Planning - Planners are usually good at finding efficient solutions - Stories are not efficient! - However, each *character* should act rationally, i.e. efficiently - We could use one planner per character ... --- # Intentional Planning - With one planner per character we may never find a path to the story goal - Instead, we use a "trick": We tell the planner what each character wants to achieve - The planner may then only use actions for a character that helps them achieve their goal - But the overall plan still has to solve the story goal! - Implementations: IPOCL, CPOCL --- # Discourse Planning - Once we have a story (plot), we want to tell it - There are several different media, such as film, text, comics, or combinations thereof - As we discussed during our lecture on Hierarchical Planning, we can use planning to generate this discourse --- # Camera Planning - Suppose we want to make a movie - Each scene is supposed to convey some particular aspect of the story - There are different camera shots, angles, lenses, etc. that we can use - We can frame this as a planning problem by defining the different shots as operators --- # The Battle for the Rune [Link](https://drive.google.com/open?id=1VweROPFFeFyPv4M1rf_FPnZR9NDIuD23) --- class: medium # Resources * [Shakey the Robot](https://www.youtube.com/watch?v=qXdn6ynwpiI) * [STRIPS: A New Approach to the Application of Theorem Proving to Problem Solving](http://ai.stanford.edu/users/nilsson/OnlinePubs-Nils/PublishedPapers/strips.pdf) * [Planning as Heuristic Search](https://www.sciencedirect.com/science/article/pii/S0004370201001084) * [The Computational Complexity of Propositional Strips Planning](https://ai.dmi.unibas.ch/research/reading_group/bylander-aij1994.pdf) * [All PSPACE-complete planning problems are equal, but some are more equal than others](https://www.aaai.org/ocs/index.php/SOCS/SOCS11/paper/view/4009/4353) * [Introduction to STRIPS Planning and Applications in Video-games](https://www.dis.uniroma1.it/~degiacom/didattica/dottorato-stavros-vassos/) --- class: medium # Resources * [ICAPS 2019 Proceedings](https://aaai.org/ojs/index.php/ICAPS) * [Goal-Oriented Action Planning](http://alumni.media.mit.edu/~jorkin/goap.html) * [Narrative Planning: Compilations to Classical Planning](https://arxiv.org/abs/1401.5856) * [Bardic: Generating Multimedia Narrative Reports for Game Logs](https://www.semanticscholar.org/paper/Bardic-%3A-Generating-Multimedia-Narrative-Reports-Barot-Branon/3087daa75239c7990bfc993a5919d14e69373b42)