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Planning Burr H. Settles CS-540, UW-Madison Summer 2003 Announcements (7/7) Lots of homework business! HW#3 out (due Monday, 7/14) HW#2 due today HW#1 almost graded, back tomorrow
Planning Burr H. Settles CS-540, UW-Madison Summer 2003 Announcements (7/7) Lots of homework business! HW#3 out (due Monday, 7/14) HW#2 due today HW#1 almost graded, back tomorrow Reminders: Midterm review on Wednesday (7/9) Midterm on Thursday, in class (7/10) No class on Friday (7/11) 1 2 Announcements (7/8) Planning Homework #1 is not quite graded! Check my mailbox in CS 5 th floor after 5pm today Otherwise, collect them at the review tomorrow There is a solution for HW#1 on the webpage Homework #3 due date extended (Wed. 7/16) About the exam: Closed book, but you may bring a 1-sided handwritten 8½ 11 sheet of notes and a calculator I threw together a midterm study guide available on the course webpage under exam section 3 Problem: Mechanically and efficiently find a sequence of actions that, when executed, achieve a goal Given: Initial state, goal state, and actions Find: A plan: a sequence of actions that when applied, beginning with the initial state, transforms the world into a goal state 4 Assumptions with Planning Assumptions with Planning Goal is a conjunction of sub-goals: To achieve a goal, you must achieve a set of sub-goals Actions are atomic Are not divisible into sub-actions Actions are sequential No two actions can be executed concurrently Actions are deterministic: No uncertainty in performing an action 5 The agent is the sole cause of change in the environment World is accessible (i.e. the agent knows all it need to know about the environment) Closed World Assumption: State description lists all that is true Anything else is assumed false The planning task is very difficult, even with such a simplified framework! 6 1 Classic Planning Problems Planning As Search Dressing Initial state: socks, shoes, and pants off Goal state: socks on, under shoes (on correct feet), under pants Actions: PutOnPants, PutOnSock(f), PutOnShoe(f) Blocks World Initial state: some configuration of blocks on a table Goal State: another configuration (stacked?) Actions: Pickup(x), Putdown(x), Stack(x,y), Unstack(x,y) Shopping Initial state: at home, with no items Goal state: at home, having a list of items Actions: Go(store), Buy(item), etc 7 State-space search: State representation Operators/actions state Goal test Note: This is how we approached the water jugs problem back in lecture 3 8 Planning As Search Planning Using Logic Doesn t allow for real reasoning about states and actions Operators are just used to generate next state Can t reason about operator definition or ordering Causes exploration of dead-end successor states (possibly even illegal ones!) Goal test is just used to determine if goal reached Can t reason about goal definition No knowledge of determining how to best achieve the goal Note: heuristic is simply the distance from goal Weak representation Weak ability to reason about the world 9 By using knowledge-based agents, we can capture reasonable information things about the agent s actions and their effect on the world If I move forward, I m in the next room If I pick up a gold brick, then I am holding it If I am holding something, my hand is not empty The problem here is dealing with time: If I move forward again, I m in a different room The results of each action are now relative to the sequence of actions before 10 Situation calculus extends FOL to deal with such time-sensitive dilemmas for planning (Sec. 10.3) Situations are states that are generated from applying an action to another situation Result(a, s) is the function that returns the situation when applying action a to situation s Fluents are predicates/functions that vary from one situation to the next, such as the location of the agent, or what it may be holding Atemporals/eternals are predicates/functions that do not depend on a time stamp e.g. Dog(Lassie) or LeftLegOf(John) There are two types of axioms (or rules) in situation calculus: Possibility axioms: say when it is possible to perform a certain action At(Agent, x, s) Adjacent(x, y) Poss(Go(x, y), s) Gold(g) At(g, x, s) At(Agent, x, s) Poss(Pickup(g), s) Holding(g, s) Poss(Putdown(g), s) Effect axioms: defines what happens in the environment when a possible action is executed Poss(Go(x, y), s) At(Agent, y, Result(Go(x,y), s)) Poss(Pickup(g), s) Holding(g, Result(Pickup(g), s)) Poss(Putdown(g), s) Holding(g, Result(Putdown(g), s)) 13 Fortunately, situation calculus allows us to express what actions are reasonable as well as what will change when an action is taken Unfortunately, it doesn t say anything about what stays the same! Frame axioms specify what does not change when a certain action is applied e.g. If I go into a room that had gold in it during the last situation, then the gold is still there Many axioms are required (for each action even!) 14 Situation calculus with frame axioms is a strong representation However, the approach is not very modular each new predicate requires axioms to be added for each of the possible actions Inference procedures are very weak the representation is too fine-grained 15 Planning Solution Combine the two approaches: Simplify the representation language Allow reasoning about how to achieve the goal Inference procedure is faster than resolution Open up the representation of states, operators, and goal test Rather than blindly applying operators, try to reason about which ones are most important Reduces the number of nodes that are considered 16 STRIPS Representation Representation for Planning STRIPS (STandard Research Institute Problem Solver): Facts: ground literals with variables Situations: conjunction of facts Goal: conjunction of positive literals Variables allowed, assume all variables are existential Operators/Actions: Action name Preconditions: conjunction of positive literals that defines if action is legal/applicable Effects: conjunction of positive literals (called the add list) and negative literals (called the delete list) Assumption: everything stays the same unless explicitly on the Operator Examples: Action name: Buy(x) Preconditions: At(s), Sells(s,x) Effects: Have(x) Action name: Pickup(x) Preconditions: OnTable(x), Clear(x), HandEmpty Effects: Holding(x), OnTable(x), Clear(x), HandEmpty delete list (avoids frame problem) Planning as Seach More on the Nature of Plans Situation-space search: Search space: all possible situations (i.e. states) Node: situation (i.e. world state) Edges: actions node: initial situation Goal node: situation where all of the sub-goals solved Plan: sequence of actions in path from start to goal Plan-space search: Search space: all possible plans More later A plan is complete if and only if every precondition is achieved A precondition is achieved if and only if it is the effect of an earlier step (and no intervening steps undo it) Situation-Space Planners Example Progression: Forward Chaining Like state-space search except for representation Inefficient due to large situation space to explore Regression: Backward Chaining (e.g. Prolog) from the goal state and solve its sub-goals (preconditions) More efficient and goal-directed than progression (fewer applicable operators) Putting on pants, socks, and shoes : PantsOff, SockOff(L), SockOff(R), ShoeOff(L), ShoeOff(R) Goal: PantsOn, SockOn(L), SockOn(R), ShoeOn(L),ShoeOn(R) Operators: PutOnPants: Pre: PantsOff, ShoeOff(L), ShoeOff(R) Eff: PantsOn, PantsOff PutOnSock(x): Pre: ShoeOff(x), SockOff(x), Eff: SockOn(x), SockOff(x) PutOnShoe(x): Pre: ShoeOff(x), SockOn(x) Eff: ShoeOn(x), ShoeOff(x) Goal-Stack Regression Planner Key Assumption in STRIPS Goal stack: what to do next Current situation: facts that are true Pick order of achieving (sub-)goals Find operator that achieves the (sub-)goal Push the operator onto stack Push its preconditions (in some order) onto stack When eventually get back to original goal, check that all of the preconditions that were needed to be satisfied are still satisfied 23 Sub-goals are independent of each other Divide and conquer the problem without worrying about other parts of the problem e.g. With putting on socks: the order doesn t matter; putting on left sock first doesn t preclude putting on the right Whole plan is sum of all sub-plans Sussman anomaly Sub-goals interfere with each other e.g. Blocks world tower, can t fix with reordering Thus, STRIPS is incomplete: (i.e. can t always find a plan even if one exists) 24 4 The Sussman Anomaly Interleaving in Planning B C A State A B C Goal State Stacking A on top of B precludes us from stacking B on top of C We cannot pick it up because it is no longer clear! Imagine stacking 100 blocks 25 Non-interleaving planners All of the steps for a sub-goal must occur atomically Given two sub-goals G 1 and G 2, either all the steps for achieving G 1 occur before G 2, or vice-versa STRIPS is non-interleaving because it uses a stack mechanism (solves one sub-goal at a time) Interleaving planners Can intermix the order of sub-goal steps This solves the Sussman anomaly 26 Partial-Order Plans (Sec. 11.3) Principle of Least Commitment Total-order planner (linear): Maintains a partial solution as a totally ordered list of steps found so far e.g. STRIPS e.g. Situation-space progression/regression planners Partial-order planner (non-linear): Only maintains partial order Constraints on the ordering of steps in the plan 27 Principle of Least Commitment: don t make an ordering choice unless required to do so Property of partial-order planners (POP) Not a property of situation-space planners: they commit to an ordering when an operator is applied Keep the ordering choice as general as possible Reduces the amount of backtracking needed Don t waste time undoing steps 28 Planning as Search: Revisited POP Example Situation-space search: Search space: all possible situations (i.e. states) etc Plan-space search: Search space: all possible partial-order plans Node: a partially-order plan Edges: add/delete/modify steps of previous node s plan or add temporal and causal constraint between existing steps node: initial partial-order plan, start finish where start: pre = none, eff = positive literals defining start state and finish: pre = goal of conjunctive literals, eff = none Goal node: a complete plan that solves all sub-goals Types of Links Solving Open Preconditions Ordering constraints: S 1 S 2 : S 1 before S 2 S 1 must occur before S 2 but not necessarily immediately before it Thin links Causal constraints: S 1 c S 2 : S 1 achieves c for S 2 S 1 has a literal c in its effect list that is needed to satisfy part of the precondition for S 2 Records the purpose of a step in the plan Thick links S 1 S 2 S 1 c S 2 31 A open (i.e. unsatisfied) precondition is one that does not have a causal link to it How is an open precondition p for step S solved? Step addition: add new plan step R that contains p in its Effects list Simple establishment: find an existing plan step R prior to S that has p in its Effects list Then add a causal and ordering links from R to S To keep the search focused, the planner only adds steps that achieve an open precondition 32 Example: Shopping Problem Example: Shopping Problem, Sells(GS, Cookies), Sells(GS, Milk), Sells(HWS, Drill), Sells(GS, Cookies), Sells(GS, Milk), Sells(HWS, Drill) At(store), Sells(store, Cookies) Plan Step Addition Buy(x): Pre: At(store) Sells(store,x) Eff: Have(x) Simple Establishment Example: Shopping Problem ing the Algorithm, Sells(GS, Cookies), Sells(GS, Milk), Sells(HWS, Drill) The algorithm is finished when every precondition in every step has a causal link 35 The algorithm fails if a precondition cannot be satisfied or an ordering constraint cannot be met e.g. S 1 S 2 and S 2 S A Flawed Shopping Plan A Flawed Shopping Plan At(here) Go(GS) Plan Step Addition Go(there): Pre: At(here) Eff: At(there), At(here) Simple Establishment Go(GS) Go(HWS) Threat Removal (Declobbering) Threat Removal Threat: step that deletes (clobbers) a needed effect S 2 requires an effect of S 1 (i.e. there is a causal link between S 1 and S 2 ), but the effect of S 3 is to undo the needed effect Thus S 3 can t occur between S 1 and S 2 It must occur either before S 1 (demotion) Add link S 3 S 1 Or after S 2 (promotion) Add link S 2 S 3 Go(GS) There is no way to remove the threat that each Go action poses to another so try a new plan At(GS) Go(HWS) Threat Removal Completing the Plan Demotion of Threat At(GS) At(GS) S 1 :Go(GS) S 3 :Go(HWS) Go(GS) Go(HWS) At(here) At(HWS) Go(Home) S 2 : Historical AI Planning State-space search (STRIPS) can be directed using logic, but is still incomplete Partially-ordered planners are complete, but are practically limited in the number of steps they can accurately plan Planning was sort of a dead AI research area for a while Modern AI Planning Since 1992, there have been several new approaches to the planning task discovered (e.g. Graph-Plan and SAT-Plan) that can find plans up to thousands of steps long CS-731 goes into these approaches in detail D. Weld, Recent advances in AI planning, AI Magazine,1999 Excellent coverage of these new approaches Graph-Plan (Sec. 11.4) Graph-Plan A. Blum and M. Furst, Fast Planning Through Planning Graph Analysis, Artificial Intelligence, 1997 Propositionalize actions and situations Construct a planning graph Levels (e.g. time steps) with potential action nodes Include persistence actions (inactions) to deal with frame prob. Link actions to situation nodes between each level Indicate which situation descriptions are mutually // basic graph-plan algorithm (p.399) GRAPH = initial state graph GOALS = problem goals loop forever { if GOALS non-mutex in last level of graph then { SOL = extract_sol(graph, GOALS, len(graph)) if SOL failure then return SOL else if no_sol(graph) then return failure } GRAPH = expand_graph(graph, problem definition) } // See textbook or paper for more details on computing // mutex, algorithmically finding solutions, etc... exclusive with mutex links Graph-Plan SAT-Plan (Sec. 11.5) : Have(Cake) Eaten(Cake) Goal: Have(Cake) Eaten(Cake) Action: Eat(Cake) Precond = Have(Cake) / Effect = Have(Cake) Eaten(Cake) Action: Bake(Cake) Precond = Have(Cake) / Effect = Have(Cake) Have(Cake) Eaten(Cake) S 0 A 0 Eat(Cake) S 1 A 1 Have(Cake) Have(Cake) Eaten(Cake) Eaten(Cake) Bake(Cake) Eat(Cake) S 2 Have(Cake) Have(Cake) Eaten(Cake) Eaten(Cake) 47 H.A. Kautz and B. Selman, Planning as Satisfiability, Proceedings of the Tenth European Conference on Artificial Intelligence (ECAI), 1992 Recall that a planning environment can be expressed in situation calculus Axioms of the form α β (rather α β) Recall that plans are considered to be a conjunction of sub-goals: state axioms goals 48 8 SAT-Plan SAT-Plan The basic idea with SAT-Plan: Describe the environment in situation calculus Propositionalize all the axioms (disjunctions), enumerated for each of an arbitrary number of steps Conjoin all instantiated rules with the initial state and goal descriptions This provides us with a PL formula in CNF, which we can try to solve using HC, SA, Tabu, GAs, etc. 49 Problem anomaly reverse medium hanoi # of variables # of clauses 933 2,533 3,025 3,798 SAT-Plan 0.1 sec 4 sec 1.2 sec 13 hours SAT-PLAN isn t necessarily complete Using local search, can get stuck in local optima Using exhaustive heuristic search (e.g. DPLL), it is complete but can take a long time 50 Summary Summary Planning agents search to find a sequence of actions to achieve a goal using a flexible representation of states, operators, goals, plans STRIPS language describes actions in terms of their preconditions and effects Not feasible to search through the entire space as was done with search agents Regression planning focuses the search STRIPS assumes sub-goals are independent POP uses principle least commitment, declobbering 51 Partial-Order Planning (POP) is a sound and complete planning algorithm, but can be limited by plan length Recent advances in AI planning reduce the planning environment to other problems (Graphs, SAT formulas) that can be solved using other methods 52 Next Lecture After the Midterm: Machine Learning! 53 9
Planning Class 7,8

Planning Class 7,8

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