Dana Nau: Lecture slides for Automated Planning Licensed under the Creative Commons...

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Chapter 14Temporal Planning

Lecture slides forAutomated Planning: Theory and Practice

Dana S. Nau

CMSC 722, AI PlanningUniversity of Maryland, Fall 2004

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Temporal Planning Motivation: want to do planning in situations where actions

have nonzero duration may overlap in time

Need an explicit representation of time Two equivalent approaches:

(1) use logical atoms, and extend the usual planning operators to include temporal conditions on those atoms

» temporal databases (2) use state variables, and specify change and persistence

constraints on the state variables» chronicles

In each case, the chapter gives a planning algorithm that’s like a temporal-planning version of PSP

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Definitions Temporally Qualified Expression (tqe):

expression a@[t1,t2)» a is an atom» t1, t2 are temporal variables (not constants)

• range over the set of real numbers Temporal database: a pair = (F,C)

F is a finite set of tqes 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)

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Example rob1 remains at location loc1, rob2 moves from loc2 to loc3

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Enabling Conditions A set of tqes F supports a tqe e = a@[t1,t2) if F contains

a tqe e' = a'@[t'1,t'2) such that a and a' are unifiable Enabling condition: a set of constraints sufficient to make

a'@[t'1,t'2) a@[t1,t2) Express the unifier as a collection of constraints Add two additional constraints: t'1 ≤ t1 and t2 ≤ t'2

There may be more than one enabling condition F may contain more than one e' that can support e

(e/F) = {enabling conditions for e in F}

If = (F,C), then (e/)= {enabling conditions for e in F that are consistent with C}

For sets of tqes, the obvious generalizations (see next slide)

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Enabling Conditions and Entailment A set of tqes F supports a set of tqes E = {e1, e2, …, en} if there is a

substitution that unifies each ei in E with an e'i = a'i@[t’i1,t’i2) in F Enabling condition:

a set of constraints sufficient to make e'i ei for every i Express the substitution as a collection of constraints For each i, add additional constraints t'i1 ≤ ti1 and ti2 ≤ t'i2

(E/F) = {enabling conditions for E in F}

If = (F,C), then (E/)= {enabling conditions for E in F that are consistent with C}

= (F,C) entails ' = (F',C') if supports F', and there is an enabling condition C'' in (F'/)

such that C entails C'' U C'

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Temporal Planning Operators o = (name(o), precond(o), effects(o), const(o))

name(o): name & variables; last two variables are ts, te precond, effects: sets of tqes const(o): temporal constraints and object constraints

Note: tqes don’t have negations; thus no precond– and effects–

Can get the same effect with domain axioms» will describe later

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Actions Action: partially instantiated operator (like in plan-space planning) An action a is applicable to = (F,C) if

F supports precond(a) there is an enabling condition C' in (a/F) such that

C U const(a) U C' is consistent

The result of applying a to is a set of possible databases e.g., suppose a = pickup(x)@[t1,t2)

» different possible values for x, t1, t2

A temporary definition (we’ll revise this later): 0(,a) = {(F U effects(a), C U const(a) U C' | C' is in (a/F)}

short for (precond(a)/F)

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move(rob1,loc1,loc2)@[ts,te) is enabled Enabling condition is

Example

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Suppose we also have adjacent(loc3,loc2) Then move(rob2,loc3,loc2)@[t's,t'e) is also enabled

Suppose we also have adjacent(loc1,loc3) Then move(rob1,loc1,loc3)@[t''s,t''e) is also enabled

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Domain Axioms Format: set of tqes disjunct of constraints For an axiom r: cond(r) disj(r)

(r/F) is short for (cond(r)/F)

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Negative Effects

Axioms can be used to take the place of negative effects

Axiom: no object can be intwo places at the same time

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Augment with the effects of move(rob1,loc1,loc2)@[ts,te) The axiom is supported for r = r' = rob1, l = loc1, l' = routes Can infer that

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Negative Preconditions

Axioms can also be used to take the place of negative preconditions

Axiom: a robot cannot be in a location atthe same time that the location is free

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Consistency = (F,C) is consistent with an axiom r if

for every enabling condition c in (r/F) there is a condition c' in disj(r) such that C U c U c' is consistent

is consistent with a set of axioms X if is consistent with every axiom in X Consistency condition: a set of constraints that needs to be met to ensure that is

consistent with the supported axioms

There can be more than one consistency condition Each disj(r) may contain more than one constraint c2

Analogy: a consistency condition is like a way to resolve all existing threats

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Satisfaction = (F,C) satisfies a set of axioms X if either (X/) is empty or there is

a consistency condition c that is entailed by C If satisfies X then it will continue to satisfy X when additional

constraints are added If is consistent with X but does not satisfy X then an additional

constraint c in (X/) needs to be added to to make it satisfy X The set of all databases obtained from that satisfy X is

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Plans

sequence.

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Concurrent Interfering

Actions

Here, each action provides some of the conditions needed to enable the other

This could not happen in classical planning, but does happen in temporal planning

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Applying pairs of actions

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Applying sets of actions

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Planning Search space similar to the one for PSP (Chapter 5) CSP-based techniques to handle the constraints

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Algorithm

Similar to a temporal database

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Flaws

Intuitively, open goals are like open goals in PSP, and the ways of resolving them are analogous to how open goals are resolved in PSP.

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Flaws (Continued)

Recall that the primary use of axioms is to entail negative preconditions and effects. Resolving an unsatisfied-axiom flaw consists of asserting those preconditions/effects into the database.

Intuitively, threats are like threats in PSP, and the ways of resolving them are analogous to how threats are resolved in PSP.

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State variables are partially specified functions May never specify the entire function

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Chronicle State-variable equivalent of a temporal database

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