1. Causal-link planning 2
Jim Blythe

2. Useful bells and whistles
Last class: state-space and plan-space planning, the POP algorithm, principle of least commitment.
This class: improving the action representation.
Variablised actions
Disjunctive preconditions
Conditional effects
Universal quantifiers

3. Reminder: blocks world (Sussman anomaly)
Initial:
Goal: A C B A B C
State I: (on-table A) (on C A) (on-table B) (clear B) (clear C)
Goal: (on A B) (on B C)

4. STRIPS representation for actions:
Move-C-from-A-to-Table:
precondition: (and (on C A) (clear C))
effects:
(and (on C Table)
(not (on C A))
(clear A))
The explicit effects are the only changes to the state.

5. POP algorithm POP((A, O, L), agenda, PossibleActions):
If agenda is empty, return (A, O, L)
Pick (Q, An) from agenda
Ad = choose an action that adds Q.
If no such action exists, fail.
Add the link Ad Ac to L and the ordering Ad < Ac to O
If Ad is new, add it to A.
Remove (Q, An) from agenda. If Ad is new, for each of its preconditions P add (P, Ad) to agenda.
For every action At that threatens any link
Choose to add At < Ap or Ac < At to O.
If neither choice is consistent, fail.
POP((A, O, L), agenda, PossibleActions) Q Ap Ac Q

6. 1. ‘Operators’ --- action schemata
Move ?b from ?x to ?y
parameters: ?b, ?x, ?y
preconds: (and (on ?b ?x) (clear ?b) (clear ?y)
(≠ ?b ?x) (≠ ?b ?y) (≠ ?x ?y)
(≠ ?y Table))
effects:
(and (on ?b ?y)
(not (on ?b ?x))
(clear ?x)
(not (clear ?y)))

7. Modifications to POP
Keep a list of bindings with a partial plan (A, O, L, B)
Choose action Ad adding E, with MGU(Q, E, B) defined.
When adding an action, add bindings preconditions to B.
Delay threat checks until ground values are known for variables.
(why do we know they eventually will be?)

8. Work on open precondition (on B C)
Spot the threat! A0
(on C A)
(on-table A)
(on-table B)
(clear C)
(clear B)
(clear ?b2)
(clear ?y2)
(on ?b2 Table)
A2: move ?b2 from Table to ?y2
(clear ?b1)
(clear ?y1)
(on ?b1 Table)
-(on ?b2 Table)
-(clear ?y2)
(on ?b2 ?y2)
A1: move ?b1 from Table to ?y1
[[?b2 A] [?y2 B]]
-(on ?b1 Table)
-(clear ?y1)
(on ?b1 ?y1)
[[?b1 B] [?y1 C]]
(on A B)
(on B C)
Ainf

9. 2. Disjunctive preconditions
Preconditions: (and (on ?x ?y)
(or (clear ?x) (big-and-flat ?x)))
Put (or Q1 Q2) on the agenda when the action is selected.
When it’s picked from the agenda, choose either Q1 or Q2 to work on. (Must try all choices for completeness).

10. 3. Conditional effects Move ?b from ?x to ?y parameters: ?b, ?x, ?y
preconds: (and (on ?b ?x) (clear ?b) (clear ?y)
(≠ ?b ?x) (≠ ?b ?y) (≠ ?x ?y)
effects:
(and (on ?b ?y)
(not (on ?b ?x))
(clear ?x)
(when (≠ ?y Table) (not (clear ?y))))

11. Modifications to planning algorithm
Allow conditional effects to be used for causal links.
Add the condition to the agenda.
Allow threat resolution by “confrontation”: if the threat is from a conditional effect, add its negation to the agenda.

12. 4. Universal quantification
Move-briefcase:
preconds: (and (briefcase ?b) (at ?b ?loc)
(forall ((padlock ?p)) (not (locked ?b ?p)))
(≠ ?dest ?loc))
effects: (and (at ?b ?dest)
(not (at ?b ?loc))
(forall ((object ?x))
(when (in ?x ?b)
(and (at ?x ?dest) (not (at ?x ?loc))))
Assume a finite, static set of typed objects.

13. Approach: replace with ground literals
Replace “forall” goals with the conjunction of all the implied ground goals.
“Universal base”
where the instances of t1 are (C1, .., Cn) and
so (forall ((padlock ?x)) (not (locked ?x ?b)))
 (and (not (locked p1 ?b)) (not (locked p2 ?b)) …)

14. Modifications to planner
Replace a universally quantified goal with its universal base.
Only use ground literals from the universal base of a quantified effect as they are needed for causal links.
Consider threats when their bindings refer to universally quantified variables.

15. Briefcase example A0 move B ?l office Ainf (briefcase B) (at B home)
(in P B)
(at P home)
(briefcase B)
(at B ?l)
(in ?o1 B)
move B ?l office
(at B off)
(not (at B ?l))
(at ?o1 off)
(not (at ?o1 ?l))
(at B off)
(at P home)
Ainf

16. Briefcase 2 A0 move B ?l office Ainf (briefcase B) (at B home)
(in P B)
(at P home)
(briefcase B)
(at B home)
(in ?o1 B)
move B ?l office
(at B off)
(not (at B home))
(at ?o1 off)
(not (at ?o1 home))
(at B off)
(at P home)
Ainf

17. Briefcase 3 A0 move B ?l office Ainf (briefcase B) (at B home)
(in P B)
(at P home)
(briefcase B)
(at B home)
(in ?o1 B)
(not (in P B))
move B ?l office
(at B off)
(not (at B home))
(at ?o1 off)
(not (at ?o1 home))
(at B off)
(at P home)
Ainf

18. Briefcase 4 A0 take-out P B move B ?l office Ainf (briefcase B)
(at B home)
(in P B)
(at P home)
take-out P B
(briefcase B)
(at B home)
(in ?o1 B)
(not (in P B))
move B ?l office
(at B off)
(not (at B home))
(at ?o1 off)
(not (at ?o1 home))
(at B off)
(at P home)
Ainf