An Introduction to Artificial Intelligence CE 40417

1. An Introduction to Artificial IntelligenceCE 40417 Chapter 11 – Planning Ramin Halavati (halavati@ce.sharif.edu) In which we see how an agent can take advantage of a problem to construct complex plans of actions.

2. What is planning • We have some Operators. • We have a Current state. • We have a Goal State. • We want to know: How to arrange the operators to reach the Goal State from Current State.

3. Air Cargo Transfer Example • What’s in Domain: • We have a set of airports as SFO,JFK, ... • We have a set of cargos as C1, C2, … • We have some airplanes as P1, P2, … • State: • Plans and Cargos are at specific airports and we want to change the positions. • Actions: • Load (Cargo, Plane, Airport) • Fly (Plane, Airport1, Airport2) • Unload (Cargo, Plane, Airport)

4. A B C C A B Blocks World Example • Domain Objects: • A set of blocks and a table. • States: • Blocks are stacked on each other and the table and we want to change their positions. • Actions: • PickUp( Block ), • PutDown( Block) • Unstack( Block, Block) • Stack( Block, Block )

5. Domain Definition Example 1 AIR CARGO TRANPORT DOMAIN: • Objects: • SFO , JFK, C1 , C2 , P1 , P2. • Predicates: • At( C1, SFO ) • In( C1, P2 ) • Plane( P1) • Cargo(C1) • Actions: • …

6. Domain Definition Ex.1(cont.) • Actions: • Name, • Parameters, • Preconditions, • Effects. • LOAD( c, p , a ) • Prec.: At(c,a), At(p,a), Cargo(c),Plane(p),Airport(a). • Effects: ~At(c,a), In(c,p)

7. Domain Definition Ex.1 (cont.) • Actions: • UNLOAD( c, p , a ) • Prec.: In(c,p), At(p,a), Cargo(c),Plane(p),Airport(a). • Effects: At(c,a), ~In(c,p) • FLY( p , a1, a2 ) • Prec.: At(p,a1) Plane(p),Airport(a1) ,Airport(a2). • Effects: ~At(p,a1), At(p,a2)

8. Domain Definition Example 2 BLOCKS WORLD DOMAIN • Objects: • A, B, C, … (the blocks) & the ROBOT. • Predicates: • On( x , y ). • OnTable( x ). • Holding( x ). • HandEmpty. • Clear( x ). • OnTable( x ). • Block( x ).

9. X Y Domain Definition Ex.2(cont.) • Actions: • UnStack( x , y ): • Prec: On(x,y), HandEmpty, Clear( x ). • Effects: ~On(x,y), Holding(x), ~Clear(x), Clear(y). • Stack( x,y ): • Prec: Holding(x), Clear(y) • Effects: On(x,y), HandEmpty, ~Holding(x), Clear( x ), ~Clear(y). NOTE: Nothing about what is block and what’s not.

10. Domain Definition Ex.2(cont.) • Actions: • PickUp( x ): • Prec: HandEmpty, Clear( x ), OnTable( x ). • Effects: Holding(x), ~Clear(x), ~OnTable( x ). • PutDown( x ): • Prec: Holding(x). • Effects: OnTable( x ), HandEmpty, Clear( x ).

11. A B C C A B Propblem Definition Ex.2 • PROBLEM DEFINITION: • Initial State: On(C,A), Clear(C), ~Clear(A), OnTable(A), Clear(B), OnTable(B), HandEmpty. • Goal State: HandEmpty, Clear(A), On(A,B), On(B,C), OnTable(C).

12. Simplest Approach • It’s all about SEARCH. • States: As described before. • Next State Generator: Which actions are applicable, apply every one of em. • Path Cost: One for each action. • Goal Test: Has goal state reached?

13. C A B C A B C Initialstate B A B A C A B C B B C A B A B C C A C A A A Goal B C B C B C A A C B

14. C C B A B A C C C A A A B B B C B C A B A B C A Simplest Approach • Progression (Forward Search): • Start from Initial State, move forward till you reach goal state. . . . NOTE: Backtracking is mandatory.

15. Simplest Approach • Regression (Backward Search): • Put Goal State’s predicates in Agenda. • Recursively fetch an item from agenda • Find something to satisfy it. • Remove all its effects from agenda. • Add all its preconditions to agenda.

16. Regression Example On(A,B), On(B,C), OnTable(C) 1. Pick Goal: On(A,B) 2. Choose Action: Stack(A,B) 3. Add actions preconditions to agenda and remove its effects from it. On(B,C), OnTable(C), Holding(A), Clear(B). 1. Pick Goal: Holding(A) 2. Choose Action: PickUp(A). 3. ... On(B,C), OnTable(C), Clear(B), HandEmpty, OnTable(A) ...

17. Pure Search Approaches • Heuristics: • Using Relaxed Domain Definition: • Assume actions have no precondition. • Assume actions have no negative effects. • … • (They are all admissible). • Sub-Goal Independence Assumption • Assuming each goal can be achieved with a sub-plan, regardless of other necessities. • (Not necessarily admissible, depends on the domain).

18. Simplest Approach • What’s wrong with Search? • Branching Factor may be too big. • The search space is reversible, resulting in infinite loops and repeated states. • Simple Search is the least that we can do.

19. Partial Order Planning PARTIAL ORDER PLANNING

20. Partial Order Planning • We do not need to start from the beginning of plan and march to end. • Some steps, facts, etc. are more important, we can decide on them ahead. • We can impose least possible commitments during the task.

21. PickUp(B) STACK(A,B) STACK(B,C) OnTable(B) H.E. Holding(A) Clear(B) Holding(B) Clear(C) On(B,C) H.E. Clear(B) Holding(B) ~Clear(B) ~OnTable(B) On(A,B) H.E. Clear(A) A B C C A B START: On(C,A) OnTable(A) OnTable(B) Clear(C) Clear(A) END : On(A,B) On(B,C) OnTable(C) Note: Not all results of each action is mentioned.

22. Ordering

23. Partial Order Planning • Assume an action called START. • No precondition. • All ‘Initial State’ as its effects. • Assume an action called END. • All ‘Goal State’ as its precondition. • No Effect.

24. Partial Order Planning • Partial Plan is a (A,O,L,Agenda) where: • A: set of actions in the plan • Initially {Start, End} • O: temporal orderings between actions • Initially {Start<End} • Agenda: open preconditions that need to be satisfied along with the action which needs them. • Initially all preconditions of End such as {(BeHome,End),(HaveMoney,End)}.

25. Clear(B) Unstack(C,B) Putdown(A,B) Q A1 A2 Partial Order Planning • L: The set of Causal Links • Initially Empty. • Causal Link: Action A2 has precondition Q that is established in the plan by action A1.

26. STACK(B,C) Holding(B) Clear(C) On(B,C) H.E. Clear(B) Partial Order Planning • Example: • A ={Start, Stack(B,C), End} • O ={Start<End, Stack(B,C)<End} • L ={(Stack(B,C),On(B,C),End)} • Agenda={(On(A,B),End), (OnTable(C),End), (Holding(B),Stack(B,C), (Clear(C),Stack(B,C)}. START: On(C,A) OnTable(A) OnTable(B) Clear(C) Clear(A) END: On(A,B) On(B,C) OnTable(C)

27. Partial Order Planning • A causal link (A1, Q, A2) represents the assertion that the role of A1 is to establish proposition Q for A2. This tells future search steps to “protect” Q in the interval between A1 and A2. • Action B threatens causal link (A1, Q, A2) if: 1. B has Q as a delete effect, and 2. B could come between A1 and A2, i.e. O  (A1 < B < A2) is consistent. • For example: PutDown(C,B) is a threat for: Clear(B) Unstack(C,B) PutDown(A,B)

28. Finally, POP’s Code. POP(<A,O,L>, agenda) 1. Termination: If agenda is empty return <A,O,L> 2. Goal selection: Let <Q,Aneed> be a pair on the agenda 3. Action selection: Let Aadd = choose an action that adds Qif no such action exists, then return failure LetL’= L  {Aadd→ Aneed}, and letO’= O  {Aadd< Aneed}. IfAadd is newly instantiated, thenA’ = A{Aadd} and O’= O  {A0< Aadd< A} (otherwise, let A’ = A) Q 4. Updating of goal set:Letagenda’ = agenda -{<Q,Aneed>}. IfAadd is newly instantiated, then for each conjunction, Qi, of its precondition, add <Qi,Aadd> to agenda’ • 5. Causal link protection: For every action At that might threaten a causal link Ap→ Ac, add a consistentordering constraint, either • Demotion: Add At< Ap to O’ • Promotion: Add Ac< At to O’ • Inequality constraints • If neither constraint is consistent, then return failure p 6. Recursive invocation: POP((<A’,O’,L’>, agenda’)

29. Last POP Notes. • Using Variables: • You need not add UnStack(A,B) when you need Clear(B). Just add Unstack(x,B) and add binding as a next step. • Heuristics: • What to do in ChooseGoal and ChooseAction?

30. … … … Planning Graph • Main Idea: • To construct a graph of possible outcomes.

31. Dinner Date Domain • 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)))

32. Dinner Date Graph

33. Mutual Exclusion Classes Interference (Prec-Effect) Inconsistent Effects Inconsistent Support Competing Needs

34. 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)

35. 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

36. Observation 3 p q r … p q r … p q r … A Proposition Mutex relationships monotonically decrease

37. 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

38. Observation 5 (Sum Up) 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.

39. Graph Plan Algorithm Graph Plan 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 non found, add a level to the PG and try again

40. Search for a solution plan

41. Search 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)

42. Just Another Planning Approach Planning By Logic (SAT-Plan): • Convert the planning problem into a logic problem. • Solve the logic problem.

43. SAT Plan Example INITIAL STATE: At(P1,SFO)0 AT(C1,JFK)0  Plane(P1)  Cargo(C1)  Airport(SFO)  Airport(JFK). RULES: At(x,y)t FLY(x,y,z)t  Airplane(x)  Airport(y)  Airport(z) At(x,z)t+1  At(x,y)t+1 … GOAL STATE: AT(C1,SFO)x

44. Sum Up • POP: Most human-like. • Graph Plan: Winner of planning contests. • SAT Plan: Widely used in real problem as: • Hardcode logic solvers. • Mathematics and Optimization. • Note: Combinations are also used.

45. EXERCISES & Projects Implement either POP or Graph-Plan. As Exercise: On a hard-coded domain without variable-instantiating. – Send To: sharifian@ce.sharif.edu – Subject: AIEX-C11 As Project: Read the domain as PDDL, have variable instantiation, and all.