For Friday • Read chapter 12, sections 3-5 • None
Program 2 • Any questions?
Plan-Space Planners • Plan-space planners search through the space of partial plans, which are sets of actions that may not be totally ordered. • Partial-order planners are plan-based and only introduce ordering constraints as necessary (least commitment) in order to avoid unnecessarily searching through the space of possible orderings
Partial Order Plan • Plan which does not specify unnecessary ordering. • Consider the problem of putting on your socks and shoes.
Plans • A plan is a three tuple <A, O, L> – A: A set of actions in the plan, {A 1 ,A 2 ,...A n } – O: A set of ordering constraints on actions {A i <A j , A k <A l ,...A m <A n }. 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
Causal Links and Threats • A causal link, A p Q A c , indicates that action A p has an effect Q that achieves precondition Q for action A c . • A threat, is an action A t that can render a causal link A p Q A c ineffective because: – O {A P < A t < A c } is consistent – A t has ¬Q as an effect
Threat Removal • Threats must be removed to prevent a plan from failing • Demotion adds the constraint A t < A p to prevent clobbering, i.e. push the clobberer before the producer • Promotion adds the constraint A c < A t to prevent clobbering, i.e. push the clobberer after the consumer
Initial (Null) Plan • Initial plan has – A={ A 0 , A } – O={A 0 < A } – L ={} • A 0 (Start) has no preconditions but all facts in the initial state as effects. • A (Finish) has the goal conditions as preconditions and no effects.
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) ) • A 0 : – At(Home) Sells(SM,Banana) Sells(SM,Milk) Sells(HWS,Drill) • A – Have(Drill) Have(Milk) Have(Banana) At(Home)
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,A i > where Q is a precondition of A i 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.
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,A need > 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 {A add Q A need } O ’ = O {A add < A need } if A add is new then A ’ = A {A add } and O ’ =O ’ È {A 0 < A add <A } else A ’ = A
4) Update goal set: Let agenda ’ = agenda - {<Q,A need >} If A add is new then for each conjunct Q i of its precondition, add <Q i , A add > to agenda ’ 5) Causal link protection: For every action A t that threatens a causal link A p Q A c add an ordering constraint by choosing nondeterministically either (a) Demotion: Add A t < A p to O ’ (b) Promotion: Add A c < A t to O ’ If neither constraint is consistent then return failure. 6) Recurse: POP(<A ’ ,O ’ ,L ’ >, agenda ’ )
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) ) • A 0 : – At(Home) Sells(SM,Banana) Sells(SM,Milk) Sells(HWS,Drill) • A – Have(Drill) Have(Milk) Have(Banana) At(Home)
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
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)
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)
GraphPlan • Alternative approach to plan construction • Uses STRIPS operators with some limitations – Conjunctive preconditions – No negated preconditions – No conditional effects – No universal effects
Planning Graph • Creates a graph of constraints on the plan • Then searches for the subgraph that constitutes the plan itself
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).
Planning graph … … …
Constructing the planning graph • Level P 1 : all literals from the initial state • Add an action in level A i if all its preconditions are present in level P i • Add a precondition in level P i if it is the effect of some action in level A i-1 (including no-ops) • Maintain a set of exclusion relations to eliminate incompatible propositions and actions (thus reducing the graph size)
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
Mutual Exclusion relations Inconsistent Interference Effects (prec-effect) Competing Inconsistent Needs Support
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)))
Dinner Date example
Dinner Date example
Observation 1 p p p p A A A ¬q q q q ¬r ¬q ¬q ¬q B B ¬r r r ¬r ¬r Propositions monotonically increase (always carried forward by no-ops)
Observation 2 p p p p A A A ¬q q q q ¬r ¬q ¬q ¬q B B ¬r r r ¬r ¬r Actions monotonically increase
Observation 3 p p p q q q A r r r … … … Proposition mutex relationships monotonically decrease
Observation 4 A A A p p p p q q q q B B B … r r r C C C s s s … … … Action mutex relationships monotonically decrease
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.
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
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
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)
Plan Graph Search If goals are present & non-mutex: Choose action to achieve each goal Add preconditions to next goal set
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.
Recommend
More recommend
