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Chapter 14

Temporal Planning

Lecture slides for

Automated Planning: Theory and Practice

Temporal Planning

• Motivation: want to do planning in situations

where actions

– have nonzero duration

– may overlap in time

• Need an explicit representation of time

• In Chapter 10 we studied a “temporal” logic

– Its notion of time is too simple: a sequence of discrete

events

– Many real-world applications require continuous time

– How to get this?

The Time-Oriented View

• We’ll concentrate on the “time-oriented view”: Sections 14.3.1–

• It produces a simpler representation
• State variables seem better suited for the task
• States not defined explicitly
• Instead, can compute a state for any time point, from the values

of the state variables at that time

State Variables

• A state variable is a partially specified function telling what is true at some time t - cpos(c1) : time → containers U cranes U robots - Tells what c1 is on at time t - rloc(r1) : time → locations - Tells where r1 is at time t
• Might not ever specify the entire function
• cpos( c ) refers to a collection of state variables
  • But we’ll be sloppy and just call it a state variable

Temporal Assertions

• Temporal assertion:
  • Event : an expression of the form x @ t : ( v 1 , v 2 )
    • At time t , x changes from v 1 to v 2 ≠ v 1
  • Persistence condition : x @[ t 1 , t 2 ) : v
    • x = v throughout the interval [ t 1 , t 2 )
  • where
    • t, t 1 , t 2 are constants or temporal variables
    • v , v 1 , v 2 are constants or object variables
• Note that the time intervals are semi-open
  • Why?

Temporal Assertions

• Temporal assertion:
  • Event : an expression of the form x @ t : ( v 1 , v 2 )
    • At time t , x changes from v 1 to v 2 ≠ v 1
  • Persistence condition : x @[ t 1 , t 2 ) : v
    • x = v throughout the interval [ t 1 , t 2 )
  • where
    • t, t 1 , t 2 are constants or temporal variables
    • v , v 1 , v 2 are constants or object variables
• Note that the time intervals are semi-open
  • Why?
  • To prevent potential confusion about x ’s value at the endpoints

Example

• Timeline for rloc(r1):

Inconsistency in the book between Figure 14. and Example 14.

C-consistency

• A timeline ( F,C ) is c-consistent (chronicle-consistent) if
  • C is consistent, and
  • Every pair of assertions in F are either disjoint or they refer to the same value and/or time points: - If F contains both x @[ t 1 ,t 2 ): v 1 and x @[ t 3 ,t 4 ): v 2 , then C must entail { t 2 ≤ t 3 }, { t 4 ≤ t 1 }, or { v 1 = v 2 } - If F contains both x @ t :( v 1 , v 2 ) and x @[ t 1 ,t 2 ): v , then C must entail { t < t 1 }, { t 2 < t }, { v = v 2 , t 1 = t }, or { t 2 = t , v = v 1 } - If F contains both x @ t :( v 1 , v 2 ) and x @ t' :( v' 1 , v' 2 ), then C must entail { t ≠ t' } or { v 1 = v' 1 , v 2 = v' 2 }
• ( F,C ) is c-consistent iff every timeline in (F,C) is c-consistent
• The book calls this consistency, not c-consistency
  • But it’s a stronger requirement than ordinary mathematical consistency
• Mathematical consistency: C doesn’t contradict the separation constraints
• c-consistency: C must actually entail the separation constraints
  • It’s sort of like saying that ( F,C ) contains no threats

Support and Enablers

• Let α be either x @ t :( v,v' ) or x @[ t,t' ): v
  • Note that α requires x = v either at t or just before t
• Intuitively, a chronicle Φ = ( F,C ) supports α if
  • F contains an assertion β that we can use to establish x = v at some time s < t ,
    • β is called the support for α
  • and if it’s consistent with Φ for v to persist over [ s,t ) and for α be true
• Formally, Φ = ( F,C ) supports α if
  • F contains an assertion β of the form β = x @ s :( w',w ) or β = x @[ s',s ): w , and
  • ∃ separation constraints C' such that the following chronicle is c-consistent:
    • ( F ∪ { x@ [ s,t ): v , α}, C ∪ C' ∪ { w=v, s < t })
  • C' can either be absent from Φ or already in Φ
• The chronicle δ = ({ x@ [ s,t ): v , α}, C' ∪ { w=v, s < t }) is an enabler for α
  • Analogous to a causal link in PSP
• Just as there could be more than one possible causal link in PSP, there can be more than one possible enabler

Example

• Let Φ be as shown
• Then Φ supports α 1 = rloc(r1)@ t :(routes, loc3) in two different ways: - β 1 establishes rloc(r1) = routes at time t 2 - this can support α 1 if we constrain t 2 < t < t 3 - enabler is δ 1 = ({rloc(r1)@[ t 2 , t ):routes, α 1 }, { t 2 < t < t 3 } - β 2 establishes rloc(r1) = routes at time t 4 - this can support α 1 if we constrain t 4 < t < t 5 - enabler is δ 2 = ({rloc(r1)@[ t 4 , t ):routes, α 1 }, { t 4 < t < t 5 }

β 1 = rloc(r1)@ t 2 :(loc1, routes)

β 2 = rloc(r1)@ t 4 :(loc2, routes)

Example

• Let Φ be as shown
• Let α 1 be the same as before: α 1 = rloc(r1)@ t :(routes, loc3)
• Let α 2 = rloc(r1)@[ t',t'' ):loc
• Then Φ supports{α 1 , α 2 } in four different ways: - As before, for α 1 we can

use either β 1 and δ 1

or β 2 and δ 2

• We can support α 2 with β 3 = rloc(r1)@ t 5 :(routes, l )
  • Enabler is δ 3 = ({rloc(r1)@[ t 5 , t' ):loc3, α 2 }, { l = loc3, t 5 < t' })
• Or we can support α 2 with α 1
  • If we supported α 1 with β 1 and enabled it with δ 1 , the enabler for α 2 is δ 4 = ({rloc(r1)@[ t,t' ):loc3, α 2 }, { t < t' < t 3 })
  • If we supported α 1 with β 1 and enabled it with δ 2 , then replace t 3 with t 5 in δ 4

One Chronicle Supporting Another

• Let Φ ' = ( F',C' ) be a chronicle, and suppose Φ = ( F,C ) supports F'.
• Let δ 1 , …, δk be all the possible enablers of Φ '
  • For each δi , let δ'i = δ 1 ∪ C'
• If there is a δ'i such that Φ ∪ δ'i is c-consistent,
  • Then Φ supports Φ ' , and δ'i is an enabler for Φ '
  • If δ'i ⊆ Φ, then Φ entails Φ '
• The set of all enablers for Φ ' is θ(Φ/Φ ' ) = { δ'i : Φ ∪ δ'i is c-consistent}

Example: Operator for Moving a

Robot

move( t (^) s , t (^) e , t 1 , t 2 , r , l , l' ) =

Applying a Set of

Actions

• Just like several temporal assertions can support each other, several actions can also support each other - Let π = { a 1 , …, a (^) k } be a set of actions - Let Φπ = ∪ i ( F ( a (^) i ), C ( ai )) - If Φ supports Φπ then π is applicable to Φ - Result is a set of chronicles

• Example:
  • Suppose Φ asserts that at time t 0 , robots r1 and r2 are at adjacent locations loc1 and loc
  • Let a 1 and a 2 be as shown
  • Then Φ supports { a 1 , a 2 } with l 1 = loc1, l 2 = loc2, l' 1 = loc2, l' 2 = loc1, t 0 < t (^) s < t 1 < t' 2 , t 0 < t' (^) s < t' 1 < t 2

a 1

a 2