Causal-link Planning II

1
Causal-link Planning II

• José Luis Ambite

2
Planning as Search
State Space Plan Space
Algorithm Progression, Regression POP
Nodes World States Partial Plans
Edges/ Transitions Actions For example, in BW move-A-from-B-to-C move-B-from-A-to-Table move-C-from-B-to-A Plan refinements Step addition Step reuse Demotion Promotion
3
POP algorithm

• POP((A, O, L), agenda, actions)
• If agenda () then return (A, O, L)
• Pick (Q, aneed) from agenda
• aadd choose(actions) s.t. Q ?effects(aadd)
• If no such action aadd exists, fail.
• L L ? (aadd, Q, aneed) O O ? (aadd
  lt aneed)
• agenda agenda - (Q, aneed)
• If aadd is new, then A A ? aadd and
• ?P ?preconditions(aadd), add (P, aadd) to
  agenda
• For every action at that threatens any causal
  link (ap, Q, ac) in L
• choose to add at lt ap or ac lt at to O.
• If neither choice is consistent, fail.
• POP((A, O, L), agenda, actions)

Termination
Goal Selection
Action Selection
Update goals

• Protect causal links
• Demotion at lt ap
• Promotion ac lt at

4
More expressive action representation

• Actions with variables
• Conditional effects
• Disjunctive preconditions
• Universal quantification
• -gt UCPOP Penberthy Weld 92

5
Propositional STRIPS

• Move-C-from-A-to-Table
• precondition (and (on C A) (clear C))
• effects (and (on C Table)
• (not (on C A))
• (clear A))
• With n blocks gt O(n3) actions!
• Many actions not relevant for goal

6
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 to handle actions with
variables (1)

• Plan (A, O, L, B), where
• A set of actions in the plan
• O temporal orderings between actions (a lt b)
• L causal links linking actions via a literal
• B binding constraints (co-designation and non
  co-designation)
• Unification (instead of matching)
• To satisfy (Q, aneed) we can use an action aadd
  such that P ? effects(aadd) and MGU(P, Q, B) ? ?
• For example, use a new action move(?b ?x ?y) to
  satisfy on(A C), since MGU(on(?b ?y), on(A C),
  B) ((?b A) (?y C)) ? ?

8
Modifications to POP to handle actions with
variables (2)

• Distinct variables for new action instances
• Move(?b1 ?x1 ?y1), Move(?b2 ?x2 ?y2),
• Add codesignation constrains in the preconditions
  of a new action to the bindings B
• After adding move(?b ?x ?y) to satisfy on(A C),
• B B ? ((?b ? ?x) (?x ? ?y) (?b A) (?y
  C))
• (?b ? ?y) and (?y ? Table) are already
  satisfied since ?b and ?y are bound to constants

9
Modifications to POP to handle actions with
variables (3)

• Threat resolution
• Delay threat checks until ground values are known
  for variables, or
• Separation add (inequality) binding that
  ensures condition does not unify.
• Ensure all actions eventually grounded
• Require ground initial state (no variables)
• Safe operators vars(effects) ? vars (precs)

10
POP with variables
((?b1 B) (?y1 C) (?b1 ? ?x1) (?x1 ? ?y1)
(?y1 ? Table))
11
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))))

12
Conditional Effects

• (when P Q)
• means if P holds in the state before the action
  is applied, then Q will hold in the state
  resulting from the application of the action
• In the situation calculus
• P(x, s) ? Q(y, do(a,s))

13
Modifications to POP to handle conditional effects

• Allow conditional effects to be used for causal
  links
• Add the antecedent of conditional effect to the
  agenda.
• Threat resolution by confrontation
• Add the negation of the antecedent of the
  conditional effect to the agenda.
• Handle negated goals
• Same as positive goals
• Closed world assumption for initial state

14
Disjunctive preconditions

• Ex (and (on ?x ?y)
• (or (clear ?x) (big-and-flat
  ?x)))
• Modifications to POP
• Put (or Q1 Q2) on the agenda when the action is
  selected.
• When (or Q1 Q2) is picked from the agenda, choose
  either Q1 or Q2 to work on.
• Note No disjunctive effects

15
Universal quantification in preconditions and
effects

• Move-briefcase(?b ?loc1 ?loc2)
• preconds (and (briefcase ?b) (at ?b ?loc1)
• (forall ((padlock ?p)) (not (locked ?b
  ?p)))
• (? ?loc1 ?loc2))
• effects (and (at ?b ?loc2)
• (not (at ?b ?loc1))
• (forall ((object ?x))
• (when (in ?x ?b) (and (at ?x ?loc2)
• 
  (not (at ?x ?loc1))))

16
Universal quantification restricted to finite,
static types

• Assume a finite, static set of typed objects
• No object creation
• No object destruction
• Example
• Extension(briefcase ?b) B1
• Extension(padlock ?p) P1, P2
• Extension(object ?x) B1, P1, P2, O1, O2, O3,
  

17
Approach Replace quantified expressions with
ground literals

• Since quantification is over a finite set (type),
  replace forall goals with the conjunction of
  all the ground formulas obtained from the
  instances of the type.
• Universal base
• Y(?t1x ?(x)) Y(?(c1)) ? ? Y(?(cn))
• where the instances of t1 are (c1, .., cn)
• Example if Extension(padlock ?x) p1, p2,
  then
• (forall ((padlock ?x)) (not (locked ?x ?b)))
• ? (and (not (locked p1 ?b)) (not (locked p2
  ?b)))

18
Modifications to POP to handlequantification

• Replace a universally quantified goal with its
  universal base.
• Use ground literals from the universal base of a
  quantified effect as needed for causal links.
• Consider threats when their bindings refer to
  universally quantified variables.

19
Briefcase example
A0
(briefcase B)
(at B home)
(in P B)
(at P home)
(at B office)
(at P home)
Ainf
20
Briefcase exampleNo threats yet
A0
(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 office)
(not (at B ?l))
(at ?o1 office)
(not (at ?o1 ?l))
(at B office)
(at P home)
Ainf
21
Briefcase example(?l home) gt threat to (at P
home)
A0
(briefcase B)
(at B home)
(in P B)
(at P home)
(briefcase B)
(at B home)
(in ?o1 B)
move B home office
(not (at B home))
(at B office)
(at ?o1 office)
(not (at ?o1 home))
(at B off)
(at P home)
Ainf
22
Briefcase exampleSolve threat by confrontation
A0
(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 home office
(not (at B home))
(at B office)
(at ?o1 office)
(not (at ?o1 home))
(at B off)
(at P home)
Ainf
23
Briefcase exampleFinal Plan
A0
(briefcase B)
(at B home)
(in P B)
(at P home)
(in P B)
take-out P B
(not (in P B))
(briefcase B)
(at B home)
(in ?o1 B)
(not (in P B))
move B home office
(not (at B home))
(at B office)
(at ?o1 office)
(not (at ?o1 home))
(at B off)
(at P home)
Ainf
24
Quantified goal example
25
Quantified goal example
26
Quantified goal example
27
(No Transcript)