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A Distributed Logical Filter for Connected Row Convex Constraints

Hong Xu
November 06, 2017

A Distributed Logical Filter for Connected Row Convex Constraints

The presentation slides of the paper "T. K. Satish Kumar, Hong Xu, Zheng Tang, Anoop Kumar, Craig Milo Rogers, and Craig A. Knoblock. A distributed logical filter for connected row convex constraints. In Proceedings of the 29th IEEE International Conference on Tools with Artificial Intelligence (ICTAI), 96–101. 2017. doi:10.1109/ICTAI.2017.00026".

More details: http://www.hong.me/papers/kumar2017.html
Link to the published paper: https://doi.org/10.1109/ICTAI.2017.00026

Hong Xu

November 06, 2017
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  1. A Distributed Logical Filter for
    Connected Row Convex Constraints
    T. K. Satish Kumar Hong Xu Zheng Tang Anoop Kumar Craig Milo Rogers
    Craig A. Knoblock
    [email protected], [email protected], {zhengtan, anoopk, rogers, knoblock}@isi.edu
    November 6, 2017
    Information Sciences Institute, University of Southern California
    The 29th IEEE International Conference on Tools with Artificial Intelligence (ICTAI 2017)
    Boston, Massachusetts, the United States of America

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  2. Executive Summary
    The Kalman filter and its distributed variants are successful methods in
    state estimation in stochastic models. We develop the analogues in
    domains described using constraints.
    Kumar et al. (Information Sciences Institute, USC) A Distributed Logical Filter for Connected Row Convex Constraints 1 / 19

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  3. Agenda
    What is Filtering and What is Its Motivation
    The Connected Row Convex (CRC) Filter
    Distributed Connected Row Convex (CRC) Filter
    Kumar et al. (Information Sciences Institute, USC) A Distributed Logical Filter for Connected Row Convex Constraints 2 / 19

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  4. Agenda
    What is Filtering and What is Its Motivation
    The Connected Row Convex (CRC) Filter
    Distributed Connected Row Convex (CRC) Filter

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  5. Motivation
    Navigation system (Zarchan et al. 2015)
    Image by Hervé Cozanet (CC BY-SA 3.0). Retreived from:
    https://commons.wikimedia.org/wiki/File:
    Navigation_system_on_a_merchant_ship.jpg
    Econometrics (Schneider 1988)
    Image retrieved from:
    https://i.ytimg.com/vi/vEP4RIOKuE4/hqdefault.jpg
    Kumar et al. (Information Sciences Institute, USC) A Distributed Logical Filter for Connected Row Convex Constraints 3 / 19

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  6. Filtering
    In a partially observable or uncertain environment, an agent often needs
    to maintain its belief state (a representation of its knowledge about the
    world) based on
    • What are the beliefs at previous time steps?
    • What does the agent observe at the current time step?
    Filtering denotes any method whereby an agent updates its belief state.
    Kumar et al. (Information Sciences Institute, USC) A Distributed Logical Filter for Connected Row Convex Constraints 4 / 19

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  7. Example: The Kalman Filter
    Prediction step
    Based on e.g.
    physical model
    Prior knowledge
    of state
    Update step
    Compare prediction
    to measurements
    Measurements
    Next timestep
    Output estimate
    of state
    Kalman filter (Kalman 1960)
    by Petteri Aimonen (CC0). Retreived from: https://commons.wikimedia.org/wiki/File:Basic_concept_of_Kalman_filtering.svg
    Kumar et al. (Information Sciences Institute, USC) A Distributed Logical Filter for Connected Row Convex Constraints 5 / 19

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  8. Agenda
    What is Filtering and What is Its Motivation
    The Connected Row Convex (CRC) Filter
    Distributed Connected Row Convex (CRC) Filter

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  9. Logical Filter
    A logical filter applies to domains that are described using logical
    formulae or constraints.
    Here, we are interested in the connected row convex (CRC) filter (Kumar
    et al. 2006), a logical filter that uses CRC constraints.
    Kumar et al. (Information Sciences Institute, USC) A Distributed Logical Filter for Connected Row Convex Constraints 6 / 19

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  10. Motivation: Multi-Robot Localization
    by James McLurkin. Retreived from: https://people.csail.mit.edu/jamesm/project-MultiRobotSystemsEngineering.php
    Kumar et al. (Information Sciences Institute, USC) A Distributed Logical Filter for Connected Row Convex Constraints 7 / 19

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  11. Constraint Satisfaction Problems (CSPs)
    • N variables X = {X1
    , X2
    , . . . , XN
    }.
    • Each variable Xi
    has a discrete-valued domain D(Xi
    ).
    • M constraints C = {C1
    , C2
    , . . . , CM
    }.
    • Each constraint Ci
    specifies allowed and disallowed assignments of
    values to a subset S(Ci
    ) of the variables.
    • Find an assignment a of values to these variables such that all
    constraints allow it.
    • Known to be NP-hard.
    Kumar et al. (Information Sciences Institute, USC) A Distributed Logical Filter for Connected Row Convex Constraints 8 / 19

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  12. A Filter Based on Constraints: Framework
    X
    1
    0
    X
    2
    0
    X
    N
    0
    X
    1
    t
    X
    2
    t
    X
    N
    t
    X
    1
    t+1
    X
    2
    t+1
    X
    N
    t+1
    Time = 0 Time = t Time = t+1
    observations at 0 observations at t observations at t+1
    • Observations at t are modeled as
    constraints on the variables at t.
    • Transitions from t to t + 1 are
    modeled as the constraints between
    variables at t and t + 1.
    • At each time step t + 1, the belief
    state is defined by all allowed
    assignments of values to variables
    at t + 1 that satisfy observation
    constraints at t + 1, and have a
    consistent extension to variables at
    t (with Markovian assumption).
    Kumar et al. (Information Sciences Institute, USC) A Distributed Logical Filter for Connected Row Convex Constraints 9 / 19

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  13. A Filter Based on Constraints: Framework
    X
    1
    0
    X
    2
    0
    X
    N
    0
    X
    1
    t
    X
    2
    t
    X
    N
    t
    X
    1
    t+1
    X
    2
    t+1
    X
    N
    t+1
    Time = 0 Time = t Time = t+1
    observations at 0 observations at t observations at t+1
    But…
    in general, determining the existence of
    a consistent extension to the previous
    time step requires looking further back.
    We’d like to have compact information.
    Solution:
    Connected Row Convex (CRC) Constraints
    Kumar et al. (Information Sciences Institute, USC) A Distributed Logical Filter for Connected Row Convex Constraints 10 / 19

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  14. The Connected Row Convex (CRC) Constraint
    0
    1
    0
    0
    0
    1
    1
    1
    0
    0
    0
    0
    1
    1
    1
    0
    0
    0
    0
    0
    0
    0
    1
    0
    0
    d
    j1
    d
    j2
    d
    j3
    d
    j4
    d
    j5
    d
    i1
    d
    i2
    d
    i3
    d
    i4
    d
    i5
    X
    i
    X
    j
    (a)
    0
    1
    0
    0
    0
    1
    0
    0
    0
    0
    0
    0
    1
    1
    1
    0
    0
    1
    0
    0
    0
    0
    1
    0
    0
    d
    j1
    d
    j2
    d
    j3
    d
    j4
    d
    j5
    d
    i1
    d
    i2
    d
    i3
    d
    i4
    d
    i5
    X
    i
    X
    j
    (b)
    ‘1’: Allowed assignment ‘0’: Disallowed assignment
    Row convex constraint: All ‘1’s in each row are consecutive
    CRC constraint: Row convex + The ‘1’s in any two successive rows/columns
    intersect or are consecutive after removing empty rows/columns
    Kumar et al. (Information Sciences Institute, USC) A Distributed Logical Filter for Connected Row Convex Constraints 11 / 19

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  15. The Connected Row Convex (CRC) Constraint
    • Path consistency: For any consistent assignment of values to any two
    variables Xi
    and Xj
    , there exists a consistent extension to any other
    variable Xk
    .
    • After enforcing path consistency on constraint networks with only CRC
    constraints, all constraints are still CRC. This is not true for row
    convex constraints.
    Kumar et al. (Information Sciences Institute, USC) A Distributed Logical Filter for Connected Row Convex Constraints 12 / 19

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  16. The Connected Row Convex (CRC) Constraint
    (X
    1
    = d
    11
    )
    (X
    1
    = d
    11
    , X
    2
    = d
    22
    )
    (X
    1
    = d
    11
    , X
    2
    = d
    22
    , X
    3
    = d
    31
    )
    (X
    1
    = d
    11
    , X
    2
    = d
    22
    , X
    3
    = d
    31
    , X
    4
    = d
    43
    )
    (X
    1
    = d
    11
    , X
    2
    = d
    22
    , X
    3
    = d
    31
    , X
    4
    = d
    43
    , X
    5
    = ?)
    ()
    CSP search tree
    Kumar et al. (Information Sciences Institute, USC) A Distributed Logical Filter for Connected Row Convex Constraints 13 / 19

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  17. The Connected Row Convex (CRC) Constraint
    0
    1
    0
    0
    1
    1
    0
    0
    1
    1
    1
    0
    1
    0
    1
    1
    0
    0
    1
    1
    d
    51
    d
    52
    d
    53
    d
    54
    d
    55
    X
    1
    = d
    11
    X
    5
    X
    2
    = d
    22
    X
    3
    = d
    31
    X
    4
    = d
    43
    d
    51
    d
    52
    d
    53
    d
    54
    d
    55
    X
    1
    = d
    11
    X
    5
    X
    2
    = d
    22
    X
    3
    = d
    31
    X
    4
    = d
    43
    Row convexity implies global consistency in path consistent constraint networks
    Kumar et al. (Information Sciences Institute, USC) A Distributed Logical Filter for Connected Row Convex Constraints 14 / 19

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  18. The Connected Row Convex (CRC) Filter
    X
    1
    t
    X
    2
    t
    X
    N
    t
    X
    1
    t+1
    X
    2
    t+1
    X
    N
    t+1
    Time = t Time = t+1
    observations at t observations at t+1
    X
    1
    t-1
    X
    2
    t-1
    X
    N
    t-1
    Time = t-1
    observations at t-1
    If all constraints are CRC, enforcing
    path consistency between every two
    consecutive time steps t and t + 1
    leads to new CRC constraints
    between variables at t + 1. These CRC
    constraints contain all information of
    consistent assignments.
    Kumar et al. (Information Sciences Institute, USC) A Distributed Logical Filter for Connected Row Convex Constraints 15 / 19

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  19. Example: Multi-Robot Localization (Kumar et al. 2006)
    (X
    i
    t, Y
    i
    t)
    (X
    i
    t+1, Y
    i
    t+1)
    (a) Each robot estimate its own
    movement.
    H
    H
    K
    a
    1
    a
    2
    (b) Each robot estimate its
    distances from other robots.
    X
    Y
    0
    Y – X = U
    Y – X = L
    (c) The constraints are CRC.
    (Kumar et al. 2006, Fig. 18)
    Kumar et al. (Information Sciences Institute, USC) A Distributed Logical Filter for Connected Row Convex Constraints 16 / 19

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  20. Agenda
    What is Filtering and What is Its Motivation
    The Connected Row Convex (CRC) Filter
    Distributed Connected Row Convex (CRC) Filter

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  21. The Distributed Kalman Filter
    The distributed version of the Kalman filter has been successful in state
    estimation in wireless sensor networks (Rao et al. 1993), including large
    scale systems (Olfati-Saber 2007).
    What about a distributed CRC filter?
    Kumar et al. (Information Sciences Institute, USC) A Distributed Logical Filter for Connected Row Convex Constraints 17 / 19

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  22. Distributed Connected Row Convex (CRC) Filter
    n
    1
    S
    1
    = {X
    1
    , X
    2
    }
    n
    2
    S
    2
    = {X
    2
    , X
    4
    }
    S
    3
    = {X
    1
    , X
    3
    , X
    4
    }
    n
    3
    n
    4
    n
    5
    S
    4
    = {X
    4
    , X
    6
    }
    S
    5
    = {X
    3
    , X
    4
    , X
    5
    }
    X
    3
    t
    X
    4
    t
    X
    5
    t
    X
    3
    t+1
    X
    4
    t+1
    X
    5
    t+1
    X
    3
    0
    X
    4
    0
    X
    5
    0
    X
    4
    0
    X
    6
    0
    X
    4
    t
    X
    6
    t
    X
    4
    t+1
    X
    6
    t+1
    • Each agent is in charge of a
    subset of variables, and all
    constraints involving those
    variables.
    • The system evolves using
    distributed path consistency.
    • Improving distributed path
    consistency algorithms is key to
    the success of the CRC filter.
    Kumar et al. (Information Sciences Institute, USC) A Distributed Logical Filter for Connected Row Convex Constraints 18 / 19

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  23. Conclusion
    • Filtering denotes any method whereby an agent updates its belief
    state.
    • The Kalman filter is well known in stochastic models.
    • A logical filter is a filter that uses logical formulae or constraints.
    • The CRC filter is the long-pursued logical analogue of the Kalman filter.
    • The distributed CRC filter is a logical analogue of the distributed
    Kalman filter and requires distributed path consistency.
    Kumar et al. (Information Sciences Institute, USC) A Distributed Logical Filter for Connected Row Convex Constraints 19 / 19

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  24. References I
    R. E. Kalman. “A New Approach to Linear Filtering and Prediction Problems”. In: Journal of Basic
    Engineering. D 82 (1960), pp. 35–45. doi: 10.1115/1.3662552.
    T. K. S. Kumar and S. Russell. “On Some Tractable Cases of Logical Filtering”. In: Proceedings of the 16th
    International Conference on Automated Planning and Scheduling. 2006, pp. 83–92.
    R. Olfati-Saber. “Distributed Kalman Filtering for Sensor Networks”. In: Proceedings of the 46th IEEE
    Conference on Decision and Control. 2007, pp. 5492–5498.
    B. S. Y. Rao, H. F. Durrant-Whyte, and J. A. Sheen. “A Fully Decentralized Multi-Sensor System for Tracking
    and Surveillance”. In: International Journal of Robotics Research 12.1 (1993), pp. 20–44.
    W. Schneider. “Analytical uses of Kalman filtering in econometrics—A survey”. In: Statistical Papers 29.1
    (1988), pp. 3–33. issn: 1613-9798. doi: 10.1007/BF02924508.
    P. Zarchan and H. Musoff. “Fundamentals of Kalman Filtering: A Practical Approach”. In: (2015). doi:
    10.2514/4.102776.

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