Chapter 14 Temporal Planning

Transcription

1 Lecture slides for Automated Planning: Theory and Practice Chapter 14 Temporal Planning Dana S. Nau CMSC 722, AI Planning University of Maryland, Spring

2 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? 2

3 Temporal Planning The book presents two equivalent approaches: 1. Use logical atoms, and extend the usual planning operators to include temporal conditions on those atoms» Chapter 14 calls this the state-oriented view 2. Use state variables, and specify change and persistence constraints on the state variables» Chapter 14 calls this the time-oriented view In each case, the chapter gives a planning algorithm that s like a temporal-planning version of PSP 3

4 The Time-Oriented View We ll concentrate on the time-oriented view : Sections 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 4

5 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 We ll be sloppy and call it a state variable rather than a collection of state variables 5

6 Example DWR domain, with robot r1 container c1 ship Uranus locations loc1, loc2 cranes crane2, crane4 r1 is in loc1 at time t 1, leaves loc1 at time t 2, enters loc2 at time t 3, leaves loc2 at time t 4, enters l at time t 5 c1 is in pile1 until time t 6, held by crane2 from t 6 to t 7, sits on r1 until t 8, held by crane4 until t 9, and sits on p until t 10 or later Uranus stays at dock5 from t 11 to t 12 6

7 Temporal assertion: Temporal Assertions Event: an expression of the form : (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 are they? 7

8 Temporal assertion: Temporal Assertions Event: an expression of the form : (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 are they? To prevent potential confusion about x s value at the endpoints 8

9 Chronicle: a pair Φ = (F,C) Chronicles F is a finite set of temporal assertions C is a finite set of constraints» temporal constraints and object constraints C must be consistent (i.e., there must exist variable assignments that satisfy it) Timeline: a chronicle for a single state variable The book writes F and C in a calligraphic font Sometimes I will, more often I ll just use italics 9

10 Example Timeline for rloc(r1): 10

11 A timeline (F,C) is consistent if C is consistent Consistency 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 consistent iff the timelines for all of its state variables are consistent This is stronger than the usual notion of a consistent set of constraints Here, the separation constraints must actually be entailed by C It s sort of like saying that (F,C) contains no threats 11

12 Example Let (F,C) include the timelines given earlier, plus some additional constraints: t 1 t 6, t 7 < t 2, t 3 t 8, t 9 < t 4, attached(p, loc2) Above, I ve drawn the entire set of time constraints All pairs of temporal assertions are either disjoint or refer to the same value at the same point, so (F,C) is consistent 12

13 Support and Enablers Let α be the assertion or the assertion Intuitively, a chronicle Φ = (F,C) supports α when Φ contains an assertion β that we can use to establish x = v at some time s <t,» β is called the support for α it is consistent to have v to persist over [s,t), it is consistent to have α be true Formally, Φ = (F,C) supports α if there is an assertion β in F of the form β = x@s:(w',w) or β = x@[s',s)@w and a set of separation constraints C' that makes the following chronicle consistent:» (F U {α, x@[s,t):v}, C U C' U {w=v, s < t}) The pair δ = ({α, x@[s,t):v}, C' U {w=v, s < t}) is an enabler for α Analogous to a causal link in PSP Can either be absent from Φ or already in Φ Just like there could be more than one possible causal link in PSP, there can be more than one possible enabler 13

14 Example We can support α = rloc(r1)@t:(routes, loc3) in two ways: β 1 = rloc(r1)@t 2 :(loc1, routes) β 2 = rloc(r1)@t 4 :(loc2, routes) An enabler for β 2 is δ = ({rloc(r1)@[t 4, t):routes, rloc(r1)@t: (routes, loc3)}, {t 4 < t < t 5 } 14

15 Φ = (F,C) supports a set of assertions E = {α 1,, α k } if there is a set of enablers φ = {δ 1,, δ k } having the following properties: the chronicle is (F,C) U δ 1 U U δ k is consistent each α i is supported by everything other than α i» i.e., α i is supported by (F U E {α i }, C} Note that some of the assertions in E may support each other! φ = {δ 1,, δ k } is an enabler for E 15

16 Example Four ways to support the pair of assertions α 1 = rloc(r1)@t:(routes, loc3) α 2 = rloc(r1)@[t',t''):loc3) To support α 1, use either β 1 = rloc(r1)@t 2 :(loc1, routes) or β 2 = rloc(r1)@t 4 :(loc2, routes) To support α 2, use either α 1, or β =rloc(r1)@t 5 :(routes,l) with the binding l=loc3 16

17 Enabling and Supporting Another Chronicle 17

18 Chronicles as Planning Operators Chronicle planning operator: a pair o = (name(o), (F(o), C(o))) such that name(o) is an expression of the form o(t s, t e,, v 1, v 2, )» o is an operator symbol» t s, t e,, v 1, v 2, are all the temporal and object variables in o (F(o), C(o)) is a chronicle Action: a (partially) instantiated operator a If Φ supports (F(a), C(a)), then a is applicable to Φ a may be applicable in several ways, so the result is a set of chronicles γ(φ,a) = {Φ φ φ θ(a/φ)} 18

19 Example: moving a robot 19

20 Applying a Set of Actions If we want to execute a set of actions, they may support each other! Couldn t happen in classical planning, because the actions that support a i must finish before a i starts In temporal planning, the actions can overlap Let π = {a 1,, a k } be the set of actions Let Φ π = i (F(a i ),C(a i )) If Φ supports Φ π then π is applicable to Φ Result is a set of chronicles γ(φ,π) = {Φ φ φ θ(φ π /Φ)} 20

21 Domains and Problems Temporal planning domain: a pair D = (Λ Φ,O) O = {all chronicle planning operators in the domain} Λ Φ = {all chronicles allowed in the domain} Temporal planning problem on D: a triple P = (D,Φ 0,Φ g ) D is the domain Φ 0 and Φ g are initial chronicle and goal chronicle O is the set of chronicle planning operators Statement of the problem P: a triple P = (O, Φ 0, Φ g ) O is the set of chronicle planning operators Φ 0 and Φ g are initial chronicle and goal chronicle Solution plan: a set of actions π = {a 1,, a n } such that at least one chronicle in γ(φ 0,π) entails Φ g 21

22 set of open goals set of sets of enablers As in plan-space planning, there are two kinds of flaws: Open goal: a tqe that isn t yet enabled Threat: an enabler that hasn t yet been incorporated into Φ {θ(α/φ)} 22

23 Let α be an open goal Case 1: Φ supports α Resolving Open Goals Resolver: any enabler for α that s consistent with Φ Refinement:» G G {α}» K K θ(α/φ) Case 2: Φ doesn t support α Resolver: an action a = (F(a),C(a)) that supports α» We don t yet require a to be supported by Φ Refinement:» π π {a}» Φ Φ (F(a), C(a))» G G F(a) put all of a s tqes into G» K K θ(a/φ) put a s set of enablers into K 23

24 Resolving Threats Threat: each enabler in K that isn t yet entailed by Φ is threatened For each C in K, we need only one of the enablers in C» They re alternative ways to achieve the same thing Threat means something different here than in PSP, because we won t try to entail all of the enablers» Just the one we select Resolver: any enabler φ in C that is consistent with Φ Refinement:» K K C» Φ Φ φ 24

25 Example Φ 0 is as shown: Φ g = Φ 0 U (E,{}), where E = {α 1 = rloc(r1)@t:(routes, loc3) α 2 = rloc(r1)@[t',t''):loc3)} As before, let β 1 = rloc(r1)@t 2 :(loc1, routes) β 2 = rloc(r1)@t 4 :(loc2, routes) Also, let β = rloc(r1)@t 5 :(routes, l) 25