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For Friday

For Friday. Read chapter 12, sections 1-3 Program 2 due. Program 2. Any questions?. Uncertainty questions?. State-Space Planners. State­space (situation space) planning algorithms search through the space of possible states of the world searching for a path that solves the problem.

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For Friday

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  1. For Friday • Read chapter 12, sections 1-3 • Program 2 due

  2. Program 2 • Any questions?

  3. Uncertainty questions?

  4. State-Space Planners • State­space (situation space) planning algorithms search through the space of possible states of the world searching for a path that solves the problem. • They can be based on progression: a forward search from the initial state looking for the goal state. • Or they can be based on regression: a backward search from the goals towards the initial state • STRIPS is an incomplete regression­based algorithm.

  5. Plan-Space Planners • Plan­space planners search through the space of partial plans, which are sets of actions that may not be totally ordered. • Partial­order planners are plan­based and only introduce ordering constraints as necessary (least commitment) in order to avoid unnecessarily searching through the space of possible orderings

  6. Partial Order Plan • Plan which does not specify unnecessary ordering. • Consider the problem of putting on your socks and shoes.

  7. Plans • A plan is a three tuple <A, O, L> • A: A set of actions in the plan, {A1 ,A2 ,...An} • O: A set of ordering constraints on actions {Ai <Aj , Ak <Al ,...Am <An}. These must be consistent, i.e. there must be at least one total ordering of actions in A that satisfy all the constraints. • L: a set of causal links showing how actions support each other

  8. Causal Links and Threats • A causal link, Ap®QAc, indicates that action Ap has an effect Q that achieves precondition Q for action Ac. • A threat, is an action A t that can render a causal link Ap®QAc ineffective because: • O È {AP < At < Ac} is consistent • At has ¬Q as an effect

  9. Threat Removal • Threats must be removed to prevent a plan from failing • Demotion adds the constraint At < Ap to prevent clobbering, i.e. push the clobberer before the producer • Promotion adds the constraint Ac < At to prevent clobbering, i.e. push the clobberer after the consumer

  10. Initial (Null) Plan • Initial plan has • A={ A0, A¥} • O={A0 < A¥} • L ={} • A0 (Start) has no preconditions but all facts in the initial state as effects. • A¥ (Finish) has the goal conditions as preconditions and no effects.

  11. Example Op( Action: Go(there); Precond: At(here); Effects: At(there), ¬At(here) ) Op( Action: Buy(x), Precond: At(store), Sells(store,x); Effects: Have(x) ) • A0: • At(Home) Sells(SM,Banana) Sells(SM,Milk) Sells(HWS,Drill) • A¥ • Have(Drill) Have(Milk) Have(Banana) At(Home)

  12. POP Algorithm • Stated as a nondeterministic algorithm where choices must be made. Various search methods can be used to explore the space of possible choices. • Maintains an agenda of goals that need to be supported by links, where an agenda element is a pair <Q,Ai> where Q is a precondition of Ai that needs supporting. • Initialize plan to null plan and agenda to conjunction of goals (preconditions of Finish). • Done when all preconditions of every action in plan are supported by causal links which are not threatened.

  13. POP(<A,O,L>, agenda) 1) Termination: If agenda is empty, return <A,O,L>. Use topological sort to determine a totally ordered plan. 2) Goal Selection: Let <Q,Aneed> be a pair on the agenda 3) Action Selection: Let A add be a nondeterministically chosen action that adds Q. It can be an existing action in A or a new action. If there is no such action return failure. L’ = L È {Aadd®QAneed} O’ = O È {Aadd < Aneed} if Aadd is new then A’ = A È {Aadd} and O’=O’ È {A0 < Aadd <A¥} else A’ = A

  14. 4) Update goal set: Let agenda’= agenda - {<Q,Aneed>} If Aadd is new then for each conjunct Qi of its precondition, add <Qi , Aadd> to agenda’ 5) Causal link protection: For every action At that threatens a causal link Ap®QAc add an ordering constraint by choosing nondeterministically either (a) Demotion: Add At < Ap to O’ (b) Promotion: Add Ac < At to O’ If neither constraint is consistent then return failure. 6) Recurse: POP(<A’,O’,L’>, agenda’)

  15. Example Op( Action: Go(there); Precond: At(here); Effects: At(there), ¬At(here) ) Op( Action: Buy(x), Precond: At(store), Sells(store,x); Effects: Have(x) ) • A0: • At(Home) Sells(SM,Banana) Sells(SM,Milk) Sells(HWS,Drill) • A¥ • Have(Drill) Have(Milk) Have(Banana) At(Home)

  16. Example Steps • Add three buy actions to achieve the goals • Use initial state to achieve the Sells preconditions • Then add Go actions to achieve new pre-conditions

  17. Handling Threat • Cannot resolve threat to At(Home) preconditions of both Go(HWS) and Go(SM). • Must backtrack to supporting At(x) precondition of Go(SM) from initial state At(Home) and support it instead from the At(HWS) effect of Go(HWS). • Since Go(SM) still threatens At(HWS) of Buy(Drill) must promote Go(SM) to come after Buy(Drill). Demotion is not possible due to causal link supporting At(HWS) precondition of Go(SM)

  18. Example Continued • Add Go(Home) action to achieve At(Home) • Use At(SM) to achieve its precondition • Order it after Buy(Milk) and Buy(Banana) to resolve threats to At(SM)

  19. Graph Form • Directed, leveled graph • 2 types of nodes: • Proposition: P • Action: A • 3 types of edges (between levels) • Precondition: P -> A • Add: A -> P • Delete: A -> P • Proposition and action levels alternate • Action level includes actions whose preconditions are satisfied in previous level plus no-op actions (to solve frame problem).

  20. … … … Planning graph

  21. Constructing the planning graph • Level P1: all literals from the initial state • Add an action in level Ai if all its preconditions are present in level Pi • Add a precondition in level Pi if it is the effect of some action in level Ai-1 (including no-ops) • Maintain a set of exclusion relations to eliminate incompatible propositions and actions (thus reducing the graph size)

  22. Mutual Exclusion relations • Two actions (or literals) are mutually exclusive (mutex) at some stage if no valid plan could contain both. • Two actions are mutex if: • Interference: one clobbers others’ effect or precondition • Competing needs: mutex preconditions • Two propositions are mutex if: • All ways of achieving them are mutex

  23. Mutual Exclusion relations Inconsistent Effects Interference (prec-effect) Competing Needs Inconsistent Support

  24. Dinner Date example • Initial Conditions: (and (garbage) (cleanHands) (quiet)) • Goal: (and (dinner) (present) (not (garbage)) • Actions: • Cook :precondition (cleanHands) :effect (dinner) • Wrap :precondition (quiet) :effect (present) • Carry :precondition :effect (and (not (garbage)) (not (cleanHands)) • Dolly :precondition :effect (and (not (garbage)) (not (quiet)))

  25. Dinner Date example

  26. Dinner Date example

  27. Observation 1 p ¬q ¬r p q ¬q ¬r p q ¬q r ¬r p q ¬q r ¬r A A A B B Propositions monotonically increase (always carried forward by no-ops)

  28. Observation 2 p ¬q ¬r p q ¬q ¬r p q ¬q r ¬r p q ¬q r ¬r A A A B B Actions monotonically increase

  29. Observation 3 p q r … p q r … p q r … A Proposition mutex relationships monotonically decrease

  30. Observation 4 A A A p q r s … p q r s … p q r s … p q … B B B C C C Action mutex relationships monotonically decrease

  31. Observation 5 Planning Graph ‘levels off’. • After some time k all levels are identical • Because it’s a finite space, the set of literals never decreases and mutexes don’t reappear.

  32. Valid plan A valid plan is a planning graph where: • Actions at the same level don’t interfere • Each action’s preconditions are made true by the plan • Goals are satisfied

  33. GraphPlan algorithm • Grow the planning graph (PG) until all goals are reachable and not mutex. (If PG levels off first, fail) • Search the PG for a valid plan • If none is found, add a level to the PG and try again

  34. Searching for a solution plan • Backward chain on the planning graph • Achieve goals level by level • At level k, pick a subset of non-mutex actions to achieve current goals. Their preconditions become the goals for k-1 level. • Build goal subset by picking each goal and choosing an action to add. Use one already selected if possible. Do forward checking on remaining goals (backtrack if can’t pick non-mutex action)

  35. Plan Graph Search • If goals are present & non-mutex: • Choose action to achieve each goal • Add preconditions to next goal set

  36. Termination for unsolvable problems • Graphplan records (memoizes) sets of unsolvable goals: • U(i,t) = unsolvable goals at level i after stage t. • More efficient: early backtracking • Also provides necessary and sufficient conditions for termination: • Assume plan graph levels off at level n, stage t > n • If U(n, t-1) = U(n, t) then we know we’re in a loop and can terminate safely.

  37. Dinner Date example • Initial Conditions: (and (garbage) (cleanHands) (quiet)) • Goal: (and (dinner) (present) (not (garbage)) • Actions: • Cook :precondition (cleanHands) :effect (dinner) • Wrap :precondition (quiet) :effect (present) • Carry :precondition :effect (and (not (garbage)) (not (cleanHands)) • Dolly :precondition :effect (and (not (garbage)) (not (quiet)))

  38. Dinner Date example

  39. Dinner Date example

  40. Dinner Date example

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