11/21/2006 Massachusetts Institute of Technology Motivation - - PDF document

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11 21 2006
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11/21/2006 Massachusetts Institute of Technology Motivation - - PDF document

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11/21/2006 Massachusetts Institute of Technology Motivation Complex embedded systems DNNF-based Belief State Estimation Paul Elliott Brian Williams 2 Belief State Estimation Approximate Belief State Estimation Behavior Belief

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11/21/2006 1

Massachusetts Institute of Technology

DNNF-based Belief State Estimation

Paul Elliott Brian Williams

2

Motivation

  • Complex embedded systems

3

Belief State Estimation

Belief State Estimation Behavior Model (PCCA) Observations Commands Belief State Ŝ Plant S

4

Approximate Belief State Estimation

Belief State Estimation Behavior Model (PCCA) k-best States Ŝ Observations Commands Plant S

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Compiled Approx Belief State Estimation

Belief State Estimation Behavior Model (PCCA) k-best States Ŝ Compiler (C2D) Observations Commands Plant S Darwiche C2D 2005 v2.20

6

Compiled Approx Belief State Estimation

Belief State Estimation Behavior Model (PCCA) k-best States Ŝ Compiler (C2D) Observations Commands Plant S

Belief State Estimation Behavior Model (PCCA) k-best States Ŝ Compiler Observations Commands Plant S

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7

Prior Work

  • MEXEC (Barrett ’05)

– Trajectories

  • Max-Prod

– sd-DNNF

  • BFBSU (Martin ’05)

– Approximate Belief State

  • Max-Sum-Prod

– Optimal constraint-solver

  • Dynamic Bayesian Networks

– Max-Prod – Sum-Prod – Distribution on variables

Belief State Estimation Behavior Model (PCCA) k-best States Ŝ Compiler Observations Commands Plant S

8

Contributions

  • New encoding of

estimation for compilation

  • New use of sd-DNNF
  • New k-best algorithm for

sd-DNNF

  • New bounded believe

state estimation algorithm

Belief State Estimation Behavior Model (PCCA) k-best States Ŝ Compiler Observations Commands Plant S

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Belief State Estimation Behavior Model (PCCA) k-best States Ŝ Compiler Observations Commands Plant S

Switch Inverter Output Model

Model: Probabilistic Concurrent Constraint Automata (PCCA)

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Belief State Estimation Behavior Model (PCCA) k-best States Ŝ Compiler Observations Commands Plant S

Switch Inverter Output Model Constraints

PCCA: Switch

On Off Failed TurnOff TurnOn !TurnOn !TurnOff 0.001 0.999 0.99 0.01 0.03 0.97 Switch Off On Failed : Sout = Low : Sout = High : Sout ∈ {Low, High}

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Belief State Estimation Behavior Model (PCCA) k-best States Ŝ Compiler Observations Commands Plant S

Switch Inverter Output Model

PCCA: Inverter

Ok Failed 0.99 0.01 Inverter Constraints : Iin ∈ {Low, High} Ok Failed High Low Low High Iout Iin :

12

Belief State Estimation Behavior Model (PCCA) k-best States Ŝ Compiler Observations Commands Plant S

Switch Inverter Output Model

PCCA Model

On Off Ok Failed Failed Low High 0.99 0.01 TurnOff TurnOn !TurnOn !TurnOff 0.001 0.999 0.99 0.01 0.03 0.97 Ok On ^ Ok

^

Off Switch Inverter Constraints

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Belief State Estimation Behavior Model (PCCA) k-best States Ŝ Compiler Observations Commands Plant S

Switch Inverter Output Model

Estimation Function

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Belief State Estimation Behavior Model (PCCA) k-best States Ŝ Compiler Observations Commands Plant S

Switch Inverter Output Model TurnOn Switch Inverter Off Ok On Ok Output Low High Low Switch Inverter Output Ok On Low Ok On

PCCA as a Relation

t t+1

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Belief State Estimation Behavior Model (PCCA) k-best States Ŝ Compiler Observations Commands Plant S

Switch Inverter Output Model TurnOn Switch Inverter Off Ok On Ok Output Low High Low Switch Inverter Output Ok On Low Ok On

PCCA as a Probabilistic Relation

Prev State Next State Next Obs Commands Prev Obs

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Belief State Estimation Behavior Model (PCCA) k-best States Ŝ Compiler Observations Commands Plant S

Switch Inverter Output Model TurnOn Switch Inverter Off Ok On Ok Output Low High Low Switch Inverter Output Ok On Low Ok On

PCCA as a Probabilistic Relation

Bt+1(S) Bt(S)/#M(S) #M(S,O)/#M(S) 0.960 0.989 Prev State Next State Next Obs Commands Prev Obs

∏τ

1 1

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Belief State Estimation Behavior Model (PCCA) k-best States Ŝ Compiler Observations Commands Plant S

Switch Inverter Output Model TurnOn Switch Inverter Off Ok On Ok Output Low High Low Switch Inverter Output Ok On Low Ok On

sd-DNNF: Relation Encoding

Bt+1(S) Bt(S)/#M(S) #M(S,O)/#M(S) 0.960 0.989 Prev State Next State Next Obs Commands Prev Obs

∏τ

1 1

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Belief State Estimation Behavior Model (PCCA) k-best States Ŝ Compiler Observations Commands Plant S

Switch Inverter Output Model TurnOn Switch Inverter Off Ok On Ok Output Low High Low Switch Inverter Output Ok On Low Ok On

sd-DNNF: Model Counting

#M(S,O)/#M(S) 0.960 0.989 Prev State Next State Next Obs Commands Prev Obs

∏τ

1 1 Bt+1(S) Bt(S)/#M(S)

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Belief State Estimation Behavior Model (PCCA) k-best States Ŝ Compiler Observations Commands Plant S

Switch Inverter Output Model TurnOn Switch Inverter Off Ok On Ok Output Low High Low Switch Inverter Output Ok On Low Ok On

sd-DNNF: Probabilities

#M(S,O)/#M(S) 0.960 0.989 Prev State Next State Next Obs Commands Prev Obs

∏τ

1 1 Bt+1(S) Bt(S)/#M(S)

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Belief State Estimation Behavior Model (PCCA) k-best States Ŝ Compiler Observations Commands Plant S

Switch Inverter Output Model TurnOn Switch Inverter Off Ok On Ok Output Low High Low Switch Inverter Output Ok On Low Ok On

sd-DNNF: States

#M(S,O)/#M(S) 0.960 0.989 Prev State Next State Next Obs Commands Prev Obs

∏τ

1 1 Bt+1(S) Bt(S)/#M(S)

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Belief State Estimation Behavior Model (PCCA) k-best States Ŝ Compiler Observations Commands Plant S

Switch Inverter Output Model

Estimation

0.960 0.989 TurnOn Switch Inverter Off Ok On Ok Output Low High Low Switch Inverter Output Ok On Low Ok On

Bt+1(S)

Ok On

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Belief State Estimation Behavior Model (PCCA) k-best States Ŝ Compiler Observations Commands Plant S

Switch Inverter Output Model

Estimation: k-best Approx

0.960 0.989 TurnOn Switch Inverter Off Ok On Ok Output Low High Low Switch Inverter Output Ok On Low Ok On

argmax Bt+1(S)

Ok On #M(S,O)/#M(S) ∏τ Bt(S)/#M(S)

k 23

Belief State Estimation Behavior Model (PCCA) k-best States Ŝ Compiler Observations Commands Plant S

Switch Inverter Output Model

Estimation: k-best Approx

0.960 0.989 TurnOn Switch Inverter Off Ok On Ok Output Low High Low Switch Inverter Output Ok On Low Ok On

argmax Bt+1(S)

Ok On #M(S,O)/#M(S) ∏τ Bt(S)/#M(S)

Max-Sum-Product-Sum

k 24

Belief State Estimation Behavior Model (PCCA) k-best States Ŝ Compiler Observations Commands Plant S

Switch Inverter Output Model

Estimation: k-best Approx (Division)

0.960 0.989 TurnOn Switch Inverter Off Ok On Ok Output Low High Low Switch Inverter Output Ok On Low Ok On

argmax Bt+1(S)

Ok On #M(S,O)/#M(S) ∏τ Bt(S)/#M(S)

Max-Sum-Product-Sum

k

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Belief State Estimation Behavior Model (PCCA) k-best States Ŝ Compiler Observations Commands Plant S

Switch Inverter Output Model

Estimation: k-best Approx (Division)

0.960 0.989 TurnOn Switch Inverter Off Ok On Ok Output Low High Low Switch Inverter Output Ok On Low Ok On

argmax Bt+1(S)

Ok On #M(S,O)/#M(S) ∏τ Bt(S)/#M(S)

Max-Sum-Product-Sum

k 26

Belief State Estimation Behavior Model (PCCA) k-best States Ŝ Compiler Observations Commands Plant S

Off On Ok Low High Low Ok On Failed Failed High Bt 0.03 1 0.97 0.999 0.001 Switch Inverter Output Switch Inverter Output

C2D 2005 v2.20: Compilation

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Belief State Estimation Behavior Model (PCCA) k-best States Ŝ Compiler Observations Commands Plant S

Switch Inverter Output Switch Inverter Output Off On Ok Low High Low Ok On Failed Failed High Bt 0.03 1 0.97 0.999 0.001

Labeled sd-DNNF

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Estimation Algorithm

  • 1. Assign known values and

probabilities

  • 2. First Pass: Compute all

summations

  • 3. Merge summation results
  • 4. Second Pass: Extract the

k-best solutions

  • 1. Upward part: Compute the

probabilities and paths of the k-best estimates

  • 2. Downward part: Follow the

paths to extract the k-best estimates

Belief State Estimation Behavior Model (PCCA) k-best States Ŝ Compiler Observations Commands Plant S

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Belief State Estimation Behavior Model (PCCA) k-best States Ŝ Compiler Observations Commands Plant S

Switch Inverter Output Switch Inverter Output Off On Ok Low High Low Ok On Failed Failed High Bt 0.03 1 0.97 0.999 0.001 Low Given: On Bt( ) = 1

Algorithm: Initial Assignment

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Belief State Estimation Behavior Model (PCCA) k-best States Ŝ Compiler Observations Commands Plant S

Switch Inverter Output Switch Inverter Output Off On Ok Low High Low Ok On Failed Failed High Bt 0.03 1 0.97 0.999 0.001

Algorithm: Initial Assignment

Off On Ok Low Failed Low Given: On Bt( ) = 1 1

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Belief State Estimation Behavior Model (PCCA) k-best States Ŝ Compiler Observations Commands Plant S

Switch Inverter Output Switch Inverter Output Off On Ok Low High Low Ok On Failed Failed High Bt 0.03 1 0.97 0.999 0.001

Algorithm: Summation

Off Failed Low Given: On Bt( ) = 1 0.999 0.001 1/1 1/2 Ok Low On 1

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Algorithm: Summation (Simplified)

Switch Inverter Output Switch Inverter Output Low Given: On Bt( ) = 1

Belief State Estimation Behavior Model (PCCA) k-best States Ŝ Compiler Observations Commands Plant S

On Ok Low Low Ok On Failed High Bt 0.999 0.001 Ok Low On 0.999 0.001 1/1 1/2 1

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Algorithm: Summation (Collect)

Switch Inverter Output Switch Inverter Output Low Given: On Bt( ) = 1 0.999 0.001 1/1 1/2

Belief State Estimation Behavior Model (PCCA) k-best States Ŝ Compiler Observations Commands Plant S

On Ok Low Low Ok On Failed High Bt 0.999 0.001 Ok Low On 1

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Algorithm: Merge Results

Switch Inverter Output Switch Inverter Output Low Given: On Bt( ) = 1 0.999 0.001 1/1 1/2

Belief State Estimation Behavior Model (PCCA) k-best States Ŝ Compiler Observations Commands Plant S

Ok On Failed

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Algorithm: Merge Results

Switch Inverter Output Switch Inverter Output Low Given: On Bt( ) = 1 0.999 0.0005

Belief State Estimation Behavior Model (PCCA) k-best States Ŝ Compiler Observations Commands Plant S

Ok On Failed

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Switch Inverter Output Switch Inverter Output Low Given: On Bt( ) = 1

Belief State Estimation Behavior Model (PCCA) k-best States Ŝ Compiler Observations Commands Plant S

Ok On Failed

Algorithm: k-Best Probabilities

0.999 0.0005

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Switch Inverter Output Switch Inverter Output Low Given: On Bt( ) = 1

Belief State Estimation Behavior Model (PCCA) k-best States Ŝ Compiler Observations Commands Plant S

Ok On Failed

Algorithm: k-Best Probabilities

0.999 0.0005 [1] [1] [1]

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Switch Inverter Output Switch Inverter Output Low Given: On Bt( ) = 1

Belief State Estimation Behavior Model (PCCA) k-best States Ŝ Compiler Observations Commands Plant S

Ok On Failed

Algorithm: k-Best Probabilities

[0.999] [0.0005] [1] [1] [1]

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Switch Inverter Output Switch Inverter Output Low Given: On Bt( ) = 1

Belief State Estimation Behavior Model (PCCA) k-best States Ŝ Compiler Observations Commands Plant S

Ok On Failed

Algorithm: k-Best Probabilities

[0.999] [0.0005] [1] [1] [1] [0.999, 0.0005]

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Switch Inverter Output Switch Inverter Output Low Given: On Bt( ) = 1

Belief State Estimation Behavior Model (PCCA) k-best States Ŝ Compiler Observations Commands Plant S

Ok On Failed

Algorithm: k-Best Probabilities

[0.999] [0.0005] [1] [1] [1] [0.999, 0.0005] [0.999, 0.0005]

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Switch Inverter Output Switch Inverter Output Low Given: On Bt( ) = 1 P(1)=0.999 P(2)=0.0005 [0.999] [0.0005] [1] [1] [1] [0.999, 0.0005] [0.999, 0.0005]

Belief State Estimation Behavior Model (PCCA) k-best States Ŝ Compiler Observations Commands Plant S

Ok On Failed

Algorithm: k-Best Solutions

1 2

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Switch Inverter Output Switch Inverter Output Low Given: On Bt( ) = 1 P(1)=0.999 P(2)=0.0005 [0.999] [0.0005] [1] [1] [1] [0.999, 0.0005] [0.999, 0.0005]

Belief State Estimation Behavior Model (PCCA) k-best States Ŝ Compiler Observations Commands Plant S

Ok On Failed

Algorithm: k-Best Solutions

1 2 1 2 1, 2

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Switch Inverter Output Switch Inverter Output Low Given: On Bt( ) = 1 P(1)=0.999 P(2)=0.0005 [0.999] [0.0005] [1] [1] [1] [0.999, 0.0005] [0.999, 0.0005]

Belief State Estimation Behavior Model (PCCA) k-best States Ŝ Compiler Observations Commands Plant S

Ok On Failed

Algorithm: k-Best Solutions

1 2 1, 2 1 2 1 2

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Switch Inverter Output Switch Inverter Output Low Given: On Bt( ) = 1 P(1)=0.999 P(2)=0.0005 [0.999] [0.0005] [1] [1] [1] [0.999, 0.0005] [0.999, 0.0005]

Belief State Estimation Behavior Model (PCCA) k-best States Ŝ Compiler Observations Commands Plant S

Ok On Failed

Algorithm: k-Best Solutions

1 2 1 2 1, 2 1 2 1 2

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Switch Inverter Output Switch Inverter Output Low Given: On Bt( ) = 1 P(1)=0.999 P(2)=0.0005 [0.999] [0.0005] [1] [1] [1] [0.999, 0.0005] [0.999, 0.0005]

Belief State Estimation Behavior Model (PCCA) k-best States Ŝ Compiler Observations Commands Plant S

Ok On Failed

Algorithm: k-Best Solutions

On Ok Failed Ok 1 2 1 2 1, 2 1 2 1 2

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Results

  • EDL Model

– 42 Variables – 10 State Variables – 4.4 Values/Variable

  • Model Size (k > 3)

– (71 ± 2.5) × 103 × k nodes – (244 ± 4.7) × 103 × k edges

  • Model Size (k < 3)

– (4.4 ± 0.37) × 103 × k nodes – (14.8 ± 0.34) × 103 × k edges

  • Algorithm Complexity

– O(k2 en) space – O(k3 en) time

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Conclusion

  • New Estimation Algorithm

– Linear in the size of the sd-DNNF, cubic time with k – Estimates k belief states

  • New Algorithm for computing Max-Sum-Product on

an sd-DNNF

– Operates in two phases

  • 1. Computes Sum-Product
  • 2. Computes Max-Product
  • An encoding of PCCA estimation as a probabilistic

relation

48

Questions?

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TurnOn sd-DNNF

Off On Ok Low High Low Ok On Failed Failed High Bt 0.03 1 0.97 0.999 0.001 Failed Failed 0.99 0.01 1