Time Constraints in Planning

1
Time Constraints in Planning

• Sudhan Kanitkar
• (sgk205_at_lehigh.edu)

2
References

• Fahiem Bacchus, Michael Ady Planning with
  Resources and Concurrency A Forward Chaining
  Approach
• Ch. 13 Time for Planning
• Ch. 14 Temporal Planning
• http//www.cs.toronto.edu/fbacchus/tlplan-manual.
  html

3
Agenda

• TLPlan Practical Approach
• Functions, Timestamped States, Queues
• Algorithm, Example
• Changes needed in the domain
• A more theoretical but expressive approach
  described in the textbook.

4
TLPlan

• Functions
• Similar to state variable representation
  discussed earlier
• Timestamped States
• Queues

5
Functions

• In traditional planning States are represented as
  databases (sets) of predicate instances and
  operators as making changes to these databases.
• It is needed to add/delete all the predicates
• (drive ?t ?l ?l)
• .
• .
• (forall (?o) (int ?o ?t)
• (and (add (at ?o ?l)) (del (at ?o ?l))))
• )

6
Functions

• Instead of having predicates for all facts we use
  functions.
• Functions seem to analogous to variables in
  programming languages
• They represent values
• Predicate (at ?x ?l) just describes the location
  of object x.
• Instead model the location of the object using a
  function (loc ?x)

7
Functions (Contd)

• (loc ?x) acts just like a variable which
  describes the location of the object x.
• In the drive predicate we make the following
  changes
• (drive ?t ?l ?l)
• .
• .
• (forall (?o) (in ?o ?t)
• (add ( (loc ?o) ?l)))
• )

Recall State-Variable Representation
8
Functions More Examples

• Predicate (refuel ?t) refuels the truck t
• (capacity ?t) is the fuel capacity of the truck
• (fuel ?t) is the current level of fuel
• (fuel-used) is a total fuel used globally
• (refuel ?t)
• (and
• (add ( (fuel-used)
• ( (fuel-used)
• (- (capacity ?t) (fuel ?t)))))
• (add ( (fuel ?t) (capacity ?t)))
• )
• )

9
Forward chaining Planners

• Forward chaining has proved to be useful for
  high-performance planners.
• Domain independent heuristics for search
• Drawback They explore only totally ordered
  sequences of action.
• Hence, modeling concurrent actions with linear
  sequences become problematic
• e.g. Two trucks in two different locations can
  travel simultaneously in parallel.
• Plans generated by GraphPlan

10
Why make time explicit ?

• Model the duration of action
• Model the effects and conditions of an action at
  various points along duration
• Handle goals with relative and absolute temporal
  constraints
• To be able to use events happening in the future
  which are not immediate effects of actions

11
Principle

• In classical planners the effects of an action
  are visible immediately and hence validating the
  preconditions of further action
• This approach suppresses the visibility of
  effects for the duration of action
• Hence the further actions which use these effects
  as preconditions cannot be used.

12
Timestamps

• Associate with each state a timestamp
• Timestamp starts with a fixed start time in the
  initial state
• Denotes the actual time the state will occur
  during the execution of a plan
• Timestamp of a successive state changes only when
  no other action can be applied and it is
  necessary to wait for an action that takes some
  time to finish.
• The effects which are not delayed still become
  available instantaneously

13
Queue

• State also has an event queue
• Queue has updates scheduled to occur at some time
  in the future
• These updates are predicates and time at which
  they become effective
• Each state inherits the pending events of its
  parent state

14
Actions

• s is the current state
• a is an action which is applicable to s only if
  it satisfies all the preconditions of s.
• Applying a to s generates a new successor state
  s
• An action can have two kinds of effects
• Instantaneous effects
• Delayed effects

15
Example

• (def-adl-operator (drive ?t ?l ?l)
• (pre (?t) (truck ?t)
• (?l) (loc ?l)
• (?l) (loc ?l)
• (at ?t ?l)
• )
• (del (at ?t ?l))
• (delayed-effect
• (/ (dist ?l ?l) (speed ?t))
• (arrived-driving ?t ?l ?l)
• (add (at ?t ?l))
• )
• )

Instantaneous Effect
Delayed Effect
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Why two types of effects ??

• Instantaneous effects make sure that objects in
  question are not reused

• Delayed effects ensure that the timing
  constraints are satisfied

Delayed Effect
(add (at ?t ?l))
17
delayed-action

• Parameters
• delta the time further from the current time
  that the action is time stamped with
• Instantaneous effects change the database of s
  immediately
• Delayed effects are added to the queue of the
  state to be applied later

18
unqueue-event Action

• A mechanism is needed which will remove events
  from the queue when the time is up and update the
  database
• A special action
• Advances the world clock
• Remove all actions scheduled for current time
  from the queue and update the database

19
Planning Algorithm
State Queue pair
Advance to new state
Record Previous State
Non-deterministicOperator or unqueue-event
Record Action
Two types of Updates
New timestamp
Apply all updates with current timestamp from the
queue
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Notes on Algorithm

• The non-deterministic choice operator is realized
  by search.
• The choice of which action to try is made by
  heuristic or domain specific control
• Temporal Control Formula from previous class
• Instead of a plan the final goal state is
  returned
• The sequence of actions leading to the goal can
  be determined using action and prev pointers

21
TLPlan support

• Following actions can be defined for TLPlan
• (delayed-action delta tag formula)
• (wait-for-next-event)
• TLPlan Manual link
• http//www.cs.toronto.edu/fbacchus/tlplan-manual.
  html
• Look for section titled Support for Concurrent
  Planning

22
Thanks Joe Souto http//www.cse.lehigh.edu/munoz
/AIPlanning/classes/Graphplan.ppt
Example
Goal Get cargo at location l0 (at c0 l0)
l1
c0
l0
v0
State (at v0 l0) (at c0 l1)
Queue
State (at c0 l1)
Queue (at v0 l1)
State (at c0 l1) (at v0 l1)
Queue
State (at v0 l1) (in c0 v0)
State (in c0 v0)
Queue (at v0 l0)
State (at v0 l0) (in c0 v0)
State (at v0 l0) (at c0 l0)
Plan move(v0,l0,l1) load(c0,v0,l1) move(v0,l1,l0)
unload(c0,v0,l0)
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Importance of control Formula

• 0 (move v0 l0 l1 f2 f1)
• 20 (event (moving-truck
• v0 l0 l1 f2 f1))
• 20 (load c0 v0 l1 s1 s0)
• 20 (move v0 l1 l0 f1 f0)
• 40 (event (moving-truck
• v0 l1 l0 f1 f0))
• 40 (unload c0 v0 l0 s0 s1)

• 0 (move v0 l0 l1 f1 f0)
• 0 (move v1 l1 l0 f2 f1)
• 20 (event ...
• 20 (move v0 l1 l0 f1 f0)
• 20 (load c0 v1 l0 s2 s1)
• 20 (load c1 v1 l0 s1 s0)
• 20 (unload c0 v1 l0 s0 s1)
• 20 (donate l2 l0 f2 f1 f0 f0 f1)
• 20 (load c0 v1 l0 s1 s0)

Note Redundant actions
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Changes in Domain File

• (define (domain mprime-strips)
• (types space vehicle cargo)
• (predicates
• (at ?v ?l)
• (conn ?l1 ?l2)
• (has-fuel ?l ?f)
• (fuel-neighbor ?f1 ?f2)
• (in ?c ?v)
• (has-space ?v ?s)
• (space-neighbor ?s1 ?s2)
• (not-equal ?l1 ?l2)
• )
• ..
• ..
• ..

(declare-described-symbol (predicate cargo-at
2) (predicate vehicle-at 2) (predicate conn
2) (predicate has-fuel 2) (predicate
fuel-neighbor 2) (predicate in 2) (predicate
has-space 2) (predicate space-neighbor 2)
(predicate not-equal 2) ) .. .. ..
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Changes in Domain File

• (action move
• parameters (
• ?v - vehicle
• ?l1 ?l2 - location
• ?f1 ?f2 fuel)
• precondition
• (and
• (at ?v ?l1)
• ..
• (fuel-neighbor ?f2 ?f1))
• effect
• (and
• (not (at ?v ?l1))
• ..
• (has-fuel ?l1 ?f2)))

(def-adl-operator (move ?v ?l1 ?l2 ?f1 ?f2)
(pre (?v ?l1) (vehicle-at ?v ?l1) (?l2)
(conn ?l1 ?l2) (?f1) (has-fuel ?l1 ?f1)
(?f2) (fuel-neighbor ?f2 ?f1)) (del
(vehicle-at ?v ?l1) (has-fuel ?l1 ?f1))
(delayed-action 20 (moving-truck ?v ?l1 ?l2
?f1 ?f2) (add (vehicle-at ?v
?l2) (has-fuel ?l1 ?f2) )))
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Changes in Domain File

• Add operator to unqueue events
• (def-adl-operator (event)
• (wait-for-next-event)
• )
• Add to the top of the domain file
• (enable concurrent-planning))

27
Changes in Problem File
(define (state0) (not-equal l0 l1) (not-equal
l0 l2) (not-equal l1 l0) .. ) (define
goal0 (cargo-at c0 l0) (cargo-at c1 l2) )

• define
• (problem strips-mprime-. .-c4)
• (domain mprime-strips)
• (objects f0 f1 f2 - fuel
• ..
• c0 c1 - cargo)
• (init
• (not-equal l0 l1)
• (not-equal l0 l2)
• .
• .
• )
• (goal
• (and
• (at c0 l0)
• ..
• (at c1 l2)
• ))

28
Break

• After the break we will look at the one
  theoretical approach

29
Formal Representation

• Formal representation of a temporal planning
  domain has following objects
• Symbols
• Relations
• Rigid Relations
• Flexible Relations
• Constraints
• Temporal Constraints
• Binding Constraints

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Symbols

• Constant Symbols
• Objects which remain constant over time or state
  changes
• Objects of classes such as robot, crane
• Variable Symbols
• Objects whose value changes over time or state
  changes
• e.g. temporal variables ranging over R

31
Relations

• Rigid Relations
• Relations which do not change over time or state
  transitions
• e.g. adjacent(loc1,loc2)
• Flexible Relations
• Also called Fluents
• Relations which invalidate/validate over a period
  of time
• e.g. at(robot1,loc1)

32
Constraints

• Binding constraints
• Temporal constraints
• If t1 and t2 are two temporal variables and r is
  a constraint defined on them
• r 2P
• P lt,gt,
• 2PF,lt,,gt,lt,,gt,,gt,lt,P

33
Temporally Qualified Expression

• A temporally qualified expression (tqe) is an
  expression of the form
• p(?1,, ?k)_at_ts,te
• p is a flexible relation
• ?1,, ?k are constants or object variables
• ts,te are temporal variables such that tsltte
• A tqe asserts that for the time range tstltte the
  relation p(?1,, ?k)holds

34
Temporal Database

• A temporal database is a pair
• F (F,C)
• F is a finite set of tqes
• C is a finite set of temporal and object
  constraints

35
Textbook. Pg 312
36
Enabling Conditions

• In the temporal database shown previously there
  are two instances of tqe free(l)_at_t,t).
• This tqe holds w.r.t to database only if one of
  the following holds
• lloc3, t0 t, t t5
• lloc2, t6 t, t t7
• These two sets of constraints are called enabling
  conditions for the tqe to be supported by F
• One of them has to be consistent with C for the
  database to support the tqe.

37
Definitions

• A set F supports a tqe e p(?1,,?k)_at_t1,t2 iff
  there is in F a tqe p(?1,,?k)_at_t1,t2 and a
  substitution s such that s(p(?1,,?k))
  s(p(?1,,?k)) and
• An enabling condition for e in F is conjunction
  of the temporal constraints t1 t1 and t2 t2
  with binding constraints of s.
• ?(e/F) is set of all the possible enabling
  conditions for e in F.
• ?(e/F) is set of all the possible enabling
  conditions for a set of tqes e in F. In this case
  F is said to support e.
• A temporal database F(F,C) supports a set of
  tqes e if all the enabling conditions c ? ?(e/F)
  are consistent with C.
• F(F,C) supports another database (F,C) when F
  supports F and there is an enabling condition c
  ? ?(F/F) such that CU c is consistent with C.

38
Temporal Planning Operators

• Its a tuple
• o (name(o), precond(o), effects(o), const(o))
• name is an expression of form o(x1,xk, ts, te)
  such that o is an operator, x1,xk are object
  variables, ts, te are temporal variables
• precond(o) and effects(o) are tqes
• const(o) is a conjunction of constraints

39
Temporal Planning Operator
Textbook. Pg 315

• Action is a partially instantiated operator
• If preconditions and constraints of an action
  hold then action will run from ts to te.
• effects describe the new tqes that result from an
  action

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Applicability of an Action

• An action a is applicable to a temporal database
  (F,C) if and only if precond(a) is supported by F
  and there is an enabling condition c in ?(a/F)for
  the a such that C U const(a) U c is consistent
  with the set of constraints
• G(F,a) (F U effects(a),
• C U const(a) U c c ? ?(a/F)
• Note that actions are applied to database and the
  result is a set databases since action can be
  applied differently at different times.

41
Domain Axioms

• The operators described till now do not express
  the negative effects of the actions
• The action thus keeps on increasing the size of
  the database where we might have conflicting
  statements appearing.
• Domain axioms is the mechanism used to overcome
  this shortcoming.
• Domain axiom is a conditional expression of the
  form
• p cond(p) ? disj(p)
• cond(p) is a set of tqes
• disj(p) is a disjunction of temporal and object
  constraints

42
Domain Axiom (Contd)

• Consider a scenario which has two robots r and r
  an two locations l and l
• - at(r,l)_at_ts,te),at(r,l)_at_ts,te) ?
• (r ? r) v (l l) v (te ts) v (te ts)
• - at(r,l)_at_t1,t1),free(l)_at_t2,t2) ?
• (l ? l) v (t1 t2) v (t2 t1)

43
Domain Axiom Support

• Let p be an axiom and F(F,C) be a temporal
  database such that cond(p) is supported by F and
  ?(p/F) is set of enabling conditions for cond(p)
  in F.
• F is consistent with p iff for each enabling
  condition c1 in ?(p/F) there is atleast one
  disjunct c2 in disj(p) such that C U c1 U c2 is
  consistent set of constraints.
• This means that for every for every tqe to be
  supported by F, there is needs to be atleast one
  disjunct in disj(p) which is consistent with F or
  C.
• A consistency condition for F w.r.t a set of
  axioms X is

• A set of all such conditions is denoted by ?(X/F)

44
Domain Axioms- Actions

• So for a set of axioms to be applicable the
  consistency condition needs to satisfied
• As result we get a new set of databases as

• Earlier it was mentioned that effect of applying
  an action a to F is a set of databases.
• Many of these databases may not be consistent
  with X
• So we now restrict that definition to only those
  databases which are consistent with X as follows

45
Temporal Planning Domain

• A temporal Planning domain is the triple
• D (?F , O, X)
• - ?F is set of all temporal databases that can
  be defined
• - O is a set of temporal planning operators
• - X is a set of domain axioms

46
Temporal Planning Problem

• Is the triple P (D, F0, Fg)
• D is the planning domain
• F0 (F,C) is the initial state of the domain
• Fg (G,Cg) is the goal state of the domain
• The statement of the problem is given by
• P (O, X, F0, Fg)

47
TPS Procedure
Note the similarity with Plan-space Planning
approach
48
TPS Procedure

• It maintains the data structure O.
• O F, G, K, p
• F F, C is the current temporal database
• G is a set of tqes corresponding to current open
  goals
• K C1,,C2 is the set of pending enabling
  conditions of actions, consistency conditions of
  axioms
• p is a set of actions corresponding to current
  plan

49
Flaws Open Goals

• A tqe in F can support a tqe e ? G if there is an
  enabling condition ?(e/F). Updates are
• K ? K U ?(e/F)
• G ? G e
• Updates owing to action a for this goal
• p ? p U a
• F ? F U effects(a)
• C ? C U const(a)
• G ? (G e) U precond(a)
• K ? K U ?(a/F)

50
Flaws - Axioms and Threat

• Unsatisfied Axioms
• These flaws are possible inconsistencies of
  instances of F w.r.t to the axioms of X.
• A resolver is a set of consistency conditions
  ?(X/F)
• K ? K U ?(X/F)
• Threats
• Over the period of time we have kept on adding
  new constraints which are required to be solved
  to K.
• For every Ci in K, the resolver is a constraint c
  such that
• C ? C U c
• K ? K - Ci

51
Thank You

• sgk205_at_lehigh.edu

52
Thank You
53
Example

• move(r,l,l)_at_ts,te
• precond at(r,l)_at_t1,ts)
• free(l)_at_t2,te)
• effects at(r,routes)_at_ts,te)
• at(r,l)_at_te,t3)
• free(l)_at_t4,t5)
• const ts lt t4 lt t2
• adjacent(l,l)

Temporal Constraints
Binding Constraints
54
Example
Goal Get cargo at location l0
l1
c0
l0
v0
State (at v0 l0) (at c0 l1)
Queue
State (at c0 l1)
Queue (at v0 l1)
State (at c0 l1) (at v0 l1)
Queue
State (at v0 l1) (in c0 v0)
State (in c0 v0)
Queue (at v0 l0)
State (at v0 l0) (in c0 v0)
State (at v0 l0) (at c0 l0)
Plan move(v0,l0,l1) load(c0,v0,l1) move(v0,l1,l0)
unload(c0,v0,l0)
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Example
Goal Get cargo at location l0
l1
c0
l0
v0
move(v0,l0,l1) load(c0,v0,l1) move(v0,l1,l0) unloa
d(c0,v0,l0)