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A Structural Characterization of Temporal Dynamic Controllability. Paul Morris NASA Ames Research Center Moffett Field, CA 94035, U.S.A. [10,20]. A. B. Simple Temporal Networks (Dechter, Meiri, Pearl 1991). Special type of CSP Variables: Event times

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a structural characterization of temporal dynamic controllability

A Structural Characterization of Temporal Dynamic Controllability

Paul Morris

NASA Ames Research Center

Moffett Field, CA 94035, U.S.A.

simple temporal networks dechter meiri pearl 1991

[10,20]

A

B

Simple Temporal Networks(Dechter, Meiri, Pearl 1991)
  • Special type of CSP
  • Variables: Event times
  • Binary Constraints: Event separations
  • Issue: Determine consistency

The time of B is somewhere

between 10 and 20 after A.

temporal constraints in planning requirements versus contingencies

[10,20]

A

B

Want B to occur at some time

between 10 and 20 after A.

Temporal Constraints in Planning:Requirements versus Contingencies

Requirement

Agent must somehow ensure

bounds are satisfied

[10,20]

Contingency

Nature ensures bounds are satisfied

Nature chooses B

A

B

Know B will occur at a time

between 10 and 20 after A.

(precise time is uncertain)

contingencies are observable
Contingencies are Observable
  • Before a contingent link finishes, we can observe that it has not finished yet.
  • After a contingent link finishes, we can observe how long it took.
stn with uncertainty stnu vidal ghallab 1996 etc

[4,4]

[5,5]

B

C

B

C

[2,3]

[1,2]

[2,3]

[1,2]

A

A

D

D

STN with Uncertainty (STNU)(Vidal & Ghallab 1996, etc.)

[4,5]

B

C

STN with contingent link

[2,3]

[1,2]

A

D

Projection2

Projection1

max

min

Courtesy Morris, Muscettola, & Vidal IJCAI-2001

stn stnu solutions
STN/STNU Solutions
  • STN Problem

Solution = Schedule

  • STNU Dynamic Controllability Problem

Solution = Dynamic Strategy

  • mapping from projections to schedules
  • depends only on past
  • must not depend on future

Courtesy Morris, Muscettola, & Vidal IJCAI-2001

example

A

C

B

D

Example

[2,4]

[0,1]

[-2,+2]

[2,4]

Strategy: execute A and C simultaneously

Courtesy Morris, Muscettola, & Vidal IJCAI-2001

example1

A

C

C

B

D

D

Example

[2,4]

[0,1]

[-2,+2]

[2,4]

Strategy: execute A and C simultaneously

[2,4]

A

B

tighten

[0,0]

[-2,+2]

[2,4]

Courtesy Morris, Muscettola, & Vidal IJCAI-2001

example wait tightening
ExampleWait Tightening

[0,4]

A

B

[-2,+2]

[0,4]

C

A

B

[-2,+2]

tighten

Wait(B;2)

C

C must wait for B or until 2 units after A

Wait-for-event with timeout

stn stnu solutions1
STN/STNU Solutions
  • STN Problem

Solution = Schedule

  • STNU Dynamic Controllability Problem

Solution = Tightened Network

Courtesy Morris, Muscettola, & Vidal IJCAI-2001

previous new results
Previous & New Results

STNU Dynamic Controllability Problem

  • Morris, Muscettola & Vidal (IJCAI01)
    • Tractable; pseudo-polynomial (bounded values)
  • Morris & Muscettola (AAAI05)
    • Strong-polynomial O(N^5)
  • This paper:
    • Strong-polynomial O(N^4)

Courtesy Morris, Muscettola, & Vidal IJCAI-2001

approach element2 mimic stn ideas
Distance graph

Paths of edges

Consistent  no neg cycle

Neg cycle  O(N) neg cycle

Bellman-Ford O(N) cutoff

Labeled distance graph

paths of label-edges??

DC  no ?? cycle

?? cycle  simpler ??

??? cutoff

ApproachElement2: mimic STN ideas

STN Methods

STNU Methods

labeled distance graph aaai05

[0,y]

A

B

[x,y]

B

A

Labeled Distance Graph (aaai05)

y

B

A

-x

b:0

B

A

B:-y

Wait(B;z)

B:-z

C

C

A

A

labeled path distance
Labeled-Path Distance

Add up edge weights, ignoring labels

x

c:y

E:z

B

C

D

A

Path Distance = x+y+z

NOT TRUE!!

Theorem???: DC iff no labeled negative cycle??

Theorem: DC iff no allowed labeled negative cycle 

allowed paths moat edges
Allowed Paths: Moat Edges

first D with BD negative

b:0

A

B

C

D

moat edge

for AB

b:0

B:-u

A

B

C

A

Unusable moat edge

allowed paths moat edges1
Allowed Paths: Moat Edges

Derived edge

same path distance lower-case free

b:0

A

B

C

D

Usable moat edge

dynamic controllability characterization
Dynamic Controllability Characterization

Theorem: DC if and only if no allowed negative cycle

A pathis allowed if every lower-case edge has a usable moat edge in the path

(These paths are semi-reducible: they can be transformed into paths without lower-case edges)

nesting lemma
Nesting Lemma
  • <lower-edge…moat edge> subpaths cannot partially overlap.
  • The subpaths are either nested or disjoint.
  • Thus they lead to a parenthesization of a path.

b:0 1 D:-3 d:0 3 B:-2 b:0 -2 ABDCDBABE

Depth of nesting = 2

negative cycle simplification

x

y

Negative Cycle Simplification

If x+y < 0 then x < 0 or y < 0

Can eliminate repetitions

in STN negative cycles

BUT

moat edge

x

y

Cannot eliminateall repetitions

in STNUallowed negative cycles

lc edge

negative cycle simplification1
Negative Cycle Simplification

HOWEVER:

nested repetition

A

B

A

B

C

D

E

F

Nested repetitions CAN be eliminated!

 neg cycle with depth of nesting at most O(K).

(K = no. of contingent links)

outline of dc test algorithm
Outline of DC Test Algorithm
  • Match up <lc-edge,moat-edge> usable pairs from the inside out; add derived edge.
  • Each iteration eliminates one level of nesting.
  • Detects allowed neg. cyc. in O(K) iterations.
  • Each iteration propagates forward from each contingent link.
  • Overall O(N^4) complexity.
dc execution needs
DC Execution Needs
  • Test if network is DC (previous slides)
  • Regress waits as far as they will go
  • Bypass certain lower-case edges
    • lower-case edge followed by negative ordinary path
  • Ensures contingent edges will not be squeezed during execution.
regression paths tower edges
Regression Paths: Tower Edges

nearest D with DA non-negative

B:-x

A

B

C

D

tower edge

for BA

D:-3 2 b:0 B:-1 2 CDBA B E

Precursor path of derived wait can be parenthesized

no nested repetitions in wait derivations
No Nested Repetitionsin Wait Derivations

upper-case nested repetition

A

B

A

B

C

D

E

F

negative

Upper-case nested repetition would have to be negative

Would be allowed negative cycle

Cannot occur if verified DC

Regression of waits can be done in O(N^4) time

dc execution needs1
DC Execution Needs
  • Test if network is DC (previous slides)
    • O(N^4) complexity 
  • Regress waits as far as they will go
    • O(N^4) complexity 
  • Bypass certain lower-case edges
    • lower-case edge followed by negative ordinary path
    • also O(N^4) complexity
conclusions
Conclusions
  • Simplification: contingent links with zero lower bound
  • Dynamic Controllability Structural Characterization: DC  no negative cycle of specific type
  • Parenthesization based on moat edges
  • Can eliminate nested repetitions, limit depth
  • Strong polynomial O(N^4) algorithm for DC
    • Compare: STN consistency testing is O(N^3)