Synthesizing Robust Plans under
Incomplete Domain Models
Tuan Nguyen and Subbarao Kambhampati
Department of CSE, Arizona State University
Minh Do
Palo Alto Research Center
Acknowledgement: Funding from ONR grants N00014-09-1-0017,
N00014-07-1-1049, NFS grant IIS-0905672, and by DARPA and the
U.S. Army Research Laboratory under contract W911NF-11-C-0037.
Planning – The Traditional View
Problem instance
a1
a2
PLANNER
a3
Domain model
“Valid” plan
Deterministic
actions
Stochastic
non-deterministic
actions
add “p”
delete “q”
“Probabilistic”
plan/policy
0.4 : add “p”
0.6 : delete “q”
COMPLETE/FULL MODEL
Laborious and error-prone!!!
Model-lite Planning as Generalized Planning
Model-lite Planning
A Domain Incompleteness View
Deterministic
actions
Stochastic,
non-deterministic
actions
I/O types
Task dependency
(e.g. workflows management,
web service composition)
Missing some
preconditions/effects of actions
(e.g. Garland & Lesh, AAAI-05)
There are known
knowns; there are
things we know that we
know. There are known
unknowns; that is to
say, there are things
that we now know we
don’t know. But there
are also unknown
unknowns; there are
things we do not know
we don’t know.
Problem Formulation

Incomplete domain ̃D = ⟨ F , A⟩


Proposition setF = { p1 , p 2 ,... , p n }
Actiona∈ A
Add (a)
q
p
Pre(a)
a
̃Pre(a) p '
w
pre
a
( p')
-
-
Del (a)
r
q'
̃ Add (a)
w del
a (q ' )
r'
del
a
w (r ' )
̃Del (a)
Different from stochastic
effects!
Problem Formulation

Transition function
π
I
̃D
?
̃D
Completion set
⟨⟨ ̃D ⟩⟩
D1
D2
D2
K
π
I
Di
π
γ (π, I , ̃D )=
γ (π, I , Di )
U
D ∈⟨⟨ ̃D ⟩⟩
i
Dj
Assumption: “Inapplicable” action
causes state unchanged
Challenges

Language for domain incompleteness

A robustness measure for plans

Generating robust plans
Incompleteness Annotation: Modeling
Issues

Incompleteness annotations can be at

Schema level

Grounded level

Or in between
Incompleteness Annotation: Modeling
A tourist planning to
Issues
have food in a small
Restriction on variable values
q( x1 )
p( x 3)
a( x 1 , x 2 , x 3 )
-
r ( x2)
town is not sure if she
needs to have cash.
Her action
have_food(M: Meals,
C: Town) has possible
precondition
need_cash(M: Meals,
C:Town): when
(C=the town)
q ' ( x1 ,C 3 )
p ' (C 1 , x 2 )
-
Possible add:
q ' ( x1 , x3 ): when( x 3 = C 3 )
Possible precondition:
p ' ( x1 , x 2 ): when ( x 1= C 1 )
The domain writer knows that p ' ( x1 , x 2 )
is NOT a precondition of a( x 1 , x 2 , x 3 )
when x1≠ C 1 , and may be in other
cases
r ' ( x 1 ,C 2 )
Possible delete:
r ' ( x 1 , x 2): when( x 2= C 2 )
Incompleteness Annotation: Modeling
Issues
Restriction on variables:

Possible preconditions/effects depending on
values of some variables, but such values are
have_food(M: Meals,
q( x1 )
unknown!
C: Town) has possible
p( x 3)
p ' ( x 1 , x 2)
a( x 1 , x 2 , x 3 )
Possible precondition:
p ' ( x1 , x 2 ): depends x1
p ' ( x1writer
, x 2 ):knows
depends
The domain
that xp1' ( x1 , x 2 )
is a possible precondition of a( x 1 , x 2 , x 3 )
when x1 has some specific value, but
unknown.
-
r ( x2)
q ' ( x1 , x 3 )
-
precondition
need_cash(M: Meals,
C:Town): depends C
Possible add:
q ' ( x1 , x3 ): depends x 3
r ' ( x 1 , x 2)
Possible delete:
r ' ( x 1 , x 2): depends x 2
Incompleteness Annotation: Modeling
Issues

Correlated incompleteness
p
Pre(a)
a
̃Pre(a) p '
w
pre
a
( p')
-
-
q
Add (a)
r
Del (a)
q'
̃ Add (a)
w del
a (q ' )
r'
del
a
w (r ' )
If p' is realized as a precondition
of a, then more likely that r' will be
delete effect of the action.
̃Del (a)
Challenges

Language for domain incompleteness

A robustness measure for plans

Generating robust plans
A Robustness Measure for Plans

A plan in ̃D may fail or succeed in reaching
̃D
goal states
Completion set
⟨⟨ ̃D ⟩⟩
Plan execution
reaches goal
state?
D1
D2
D2
yes
no
no
Plan robustness: Cummulative probability
mass of complete models under which the plan
succeeds.
K
A Spectrum of Robust Planning
Problems

Robustness assessment

Maximally robust plan generation

Generating plans with desired level of robustness

Cost sensitive robust plan generation

Incremental robustification
Challenges

Language for domain incompleteness

A robustness measure for plans

Generating robust plans
to Conformant Probabilistic Planning Problem
Conformant Probabilistic Planning
Problem
Domain D' = ⟨ F ' , A' ⟩

Proposition set F '

Action a ' ∈ A'
Mutually
exclusive and
exhaustive
Preconditions Pre(a ' )⊆ F '
Conditional effects e= (cons(e) , O(e)= {( Pr (ϵ) , add (ϵ) , del (ϵ))})
0.7
0.3
a'
0.2
0.8
Problem P ' = ⟨ D ' ,b I ,G ' ,ρ⟩
Compilation Approach: An Example
Compiled “pick-up”
Compilation Example
Correctness of the compilation
Experimental Results
Logistics

Two cities C 1 and C 2 each with a downtown and airport.

Heavy packages at the downtown areas

Robots Ri ,1 , ... , Ri , m at the airport of the city C i
Source of incompleteness: robots R1, j , R 2, j were made
from the same manufacturer, having possible
precondition that packages should not be heavy to pick.

Goals: move packages from C 1 to C 2 and vice versa.
Plans can be made more robust by using robots from different manufacturers after
moving them into the downtown area, with the cost of increasing the plan length.
Experimental Results
Satellite

Two satellites S 1 and S 2 orbiting the planet Earth

Imagers Li ,1 ,... , Li , m installed on S i
Source of incompleteness: lense of L1, j , L 2, j
were
made from the type of material M j and can produce
possible effect that images taken are mangled.

Goals: images taken in some mode at some direction.
Plans can be made more robust by using additional instruments, which might be in
different satellites, but should be of different types of material and can also take an
image of the interested mode at some direction.
Experimental Results
Logistics
Satellite
Number of
m
models: 2
Observations
Fixed the number of models:

Plan tends to be longer with increasing robustness threshold
Fixed the robustness threshold:

The maximal robustness value of plans that can be returned
increases with higher number of manufacturers.
Related work



K-faults plans (Jensen et al 2004)
Plan evaluation with incomplete models
(Garland & Lesh, 2002)
Planning and Acting in Incomplete Domains
(Bryce & Weber, 2011)

Robust temporal planning (Fox 2006)

Handling incompleteness at the atomic level:

MDP with uncertain transition probabilities
(Satia & Lave 1973; Delago & Sanner 2009)

Bounded parameter MDP (Givan, Leach,
Dean 2000)
Conclusion & Future work

Introduce planning with incomplete models

Incompleteness annotations

Robustness measure for plans

A spectrum of robust planning problems


Finding a plan with at least a robustness
value: compilation approach
Future work:

Heuristic approach utilizing annotations

Plan robustification, and other problems in
the spectrum.
Thank you!
Q&A
Backup
Uniform distribution: 6/8 robustness
0.7
0.3
Belief state b
a'
0.2
0.8
Resulting belief
state b'
R(π, ̃P )=
∑
D i ∈ ⟨⟨ ̃D ⟩⟩ , γ (π, I , G)∣%equal G
Pr ( Di )
Problem Formulation
q
Pre(a) p
a
p'
-
r
q'
r'
Problem Formulation
q
Add (a)
r
Del (a)
q'
̃ Add (a)
p
Pre(a)
a
̃Pre(a) p '
pre
wa ( p ' )
-
-
del
wa ( p ' )
r'
del
a
w ( p' )
̃Del (a)