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ICFEM2022

 ICFEM2022

Jaime Arias Almeida

October 27, 2022
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  1. Modular Analysis of Tree-Topology Models
    Jaime Arias1, Michal Knapik2, Wojciech Penczek2, and Laure
    Petrucci1
    1LIPN, CNRS UMR 7030, Universit´
    e Sorbonne Paris Nord
    2Institute of Computer Science, Polish Academy of Sciences
    23rd International Conference on Formal Engineering Methods (ICFEM 2022)
    Madrid, Spain, 24-27 October 2022
    Arias, Knapik, Penczek, and Petrucci Modular Analysis of Tree-Topology Models October 2022 1 / 20

    View Slide

  2. Motivation
    Labelled transitions systems (LT S) are a popular formalism for
    modelling complex systems.
    pay
    select
    soda beer
    coin
    τ
    τ
    get soda get beer
    Figure: LT S of a simple beverage vending
    machine [Baier and Katoen, 2008]
    Arias, Knapik, Penczek, and Petrucci Modular Analysis of Tree-Topology Models October 2022 2 / 20

    View Slide

  3. Motivation
    Labelled transitions systems (LT S) are a popular formalism for
    modelling complex systems.
    The behaviour of the entire system is captured by the synchronised
    product of its components.
    pay
    select
    soda beer
    coin
    τ
    τ
    get soda get beer
    Figure: LT S of a simple beverage vending
    machine [Baier and Katoen, 2008]
    Arias, Knapik, Penczek, and Petrucci Modular Analysis of Tree-Topology Models October 2022 2 / 20

    View Slide

  4. Motivation
    l0 l1 l2 lTS
    l¬TS
    ?TF ok
    ?TF
    nok
    ?p
    ok
    ?p nok TS
    !TS ok
    !TS nok
    l0 l2
    l¬TF
    l3 lTF
    ?ST ok
    ?ST
    nok
    !TF nok
    ?GA ok
    ?GA nok
    TF
    !TF ok
    l0
    lp
    l¬p
    p
    !p
    nok
    !p ok
    !p nok
    .
    .
    .
    Figure: Example of an Attack-Defense Tree [Arias et al., 2020]
    Arias, Knapik, Penczek, and Petrucci Modular Analysis of Tree-Topology Models October 2022 3 / 20

    View Slide

  5. Motivation
    Some systems exhibit tree-like synchronisation topologies, e.g.:
    • attack-defense trees (ADTree)
    [Petrucci et al., 2019, Arias et al., 2020, Andr´
    e et al., 2021],
    • hierarchical systems [Aminof et al., 2012],
    • broadcast systems [Belardinelli et al., 2017],
    • multimedia systems [Arias et al., 2016], and
    • workflow models [van der Aalst and van Hee, 2002].
    Arias, Knapik, Penczek, and Petrucci Modular Analysis of Tree-Topology Models October 2022 4 / 20

    View Slide

  6. Motivation
    In practice, the size of the state space of the synchronised product
    grows exponentially with the number of the sub-modules .
    • In the ADTree example: 479 states and 1326 transitions.
    Many efforts are being made to find ways to analyse and represent
    models without building the entire state space.
    Arias, Knapik, Penczek, and Petrucci Modular Analysis of Tree-Topology Models October 2022 5 / 20

    View Slide

  7. Motivation
    In practice, the size of the state space of the synchronised product
    grows exponentially with the number of the sub-modules .
    • In the ADTree example: 479 states and 1326 transitions.
    Many efforts are being made to find ways to analyse and represent
    models without building the entire state space.
    This work
    Compact representation of the state space of the entire product of
    networks of LT Ss that exhibit tree-like synchronisation topologies.
    Arias, Knapik, Penczek, and Petrucci Modular Analysis of Tree-Topology Models October 2022 5 / 20

    View Slide

  8. Outline
    1 Tree Synchronisation Topology
    2 Sum-of-Squares Product SQu(G)
    3 General Sum-of-Squares GSQ(G)
    4 Experiments
    5 Concluding Remarks
    Arias, Knapik, Penczek, and Petrucci Modular Analysis of Tree-Topology Models October 2022 6 / 20

    View Slide

  9. Tree Synchronisation Topology
    r0
    r1 r2
    r3
    r4
    ?open
    ?chooseL
    ?open
    ?chooseR
    ?chooseL
    beep
    R
    s0
    !open
    M1
    t0 t1 t2
    !chooseL
    τ
    τ
    !chooseR M2
    Figure: Tree Synchronisation Topology (ST )
    Arias, Knapik, Penczek, and Petrucci Modular Analysis of Tree-Topology Models October 2022 7 / 20

    View Slide

  10. Tree Synchronisation Topology
    s0
    s1
    s2
    !chooseL
    τ
    τ
    !chooseR
    Figure: Live-Reset LT S
    Arias, Knapik, Penczek, and Petrucci Modular Analysis of Tree-Topology Models October 2022 8 / 20

    View Slide

  11. Tree Synchronisation Topology
    s0
    s1
    s2
    !chooseL
    τ
    τ
    !chooseR
    Figure: Live-Reset LT S
    s0
    s1
    !a ok
    a
    Figure: Sync-Deadlock
    LT S
    Arias, Knapik, Penczek, and Petrucci Modular Analysis of Tree-Topology Models October 2022 8 / 20

    View Slide

  12. Tree Synchronisation Topology
    s0
    s1
    s2
    !chooseL
    τ
    τ
    !chooseR
    Figure: Live-Reset LT S
    s0
    s1
    !a ok
    a
    Figure: Sync-Deadlock
    LT S
    LT S that are either live-reset or sync-deadlock are said to be
    sync-memoryless.
    Arias, Knapik, Penczek, and Petrucci Modular Analysis of Tree-Topology Models October 2022 8 / 20

    View Slide

  13. Sum-of-Squares Product SQu(G)
    r0
    r1 r2
    r3
    r4
    ?open
    ?chooseL
    ?open
    ?chooseR
    ?chooseL
    beep
    R
    s0
    !open
    M1
    t0 t1 t2
    τ τ
    !chooseL
    !chooseR M2
    Figure: ST Gx
    sI
    sq
    s0r0
    t0r0
    s0r1
    open
    t1r0
    τ
    t2r0
    τ
    t0r1
    open
    t1r1
    τ
    t0r2
    chooseL
    s0r2
    chooseL
    s0r3
    open
    t0r3
    open
    beep
    beep
    t1r3
    τ
    beep
    t2r3
    τ
    beep
    t1r2
    τ
    t2r2
    τ
    t2r1
    τ
    t0r4
    chooseR
    s0r4
    chooseR
    t1r4
    τ
    chooseL
    chooseL
    t2r4
    τ
    Figure: SQu(Gx
    )
    Definition (Sum-of-squares Product)
    Let SQu(G) = Su
    sq
    , sI
    sq
    , Actssq, →sq, Lsq be an LT S s.t. . . .
    Arias, Knapik, Penczek, and Petrucci Modular Analysis of Tree-Topology Models October 2022 9 / 20

    View Slide

  14. Sum-of-Squares Product SQu(G)
    r0
    r1 r2
    r3
    r4
    ?open
    ?chooseL
    ?open
    ?chooseR
    ?chooseL
    beep
    R
    s0
    !open
    M1
    t0 t1 t2
    τ τ
    !chooseL
    !chooseR M2
    Figure: ST Gx
    sI
    sq
    s0r0
    t0r0
    s0r1
    open
    t1r0
    τ
    t2r0
    τ
    t0r1
    open
    t1r1
    τ
    t0r2
    chooseL
    s0r2
    chooseL
    s0r3
    open
    t0r3
    open
    beep
    beep
    t1r3
    τ
    beep
    t2r3
    τ
    beep
    t1r2
    τ
    t2r2
    τ
    t2r1
    τ
    t0r4
    chooseR
    s0r4
    chooseR
    t1r4
    τ
    chooseL
    chooseL
    t2r4
    τ
    Figure: SQu(Gx
    )
    Definition (Sum-of-squares Product)
    Let SQu(G) = Su
    sq
    , sI
    sq
    , Actssq, →sq, Lsq be an LT S s.t. . . .
    Arias, Knapik, Penczek, and Petrucci Modular Analysis of Tree-Topology Models October 2022 9 / 20

    View Slide

  15. Sum-of-Squares Product SQu(G)
    r0
    r1 r2
    r3
    r4
    ?open
    ?chooseL
    ?open
    ?chooseR
    ?chooseL
    beep
    R
    s0
    !open
    M1
    t0 t1 t2
    τ τ
    !chooseL
    !chooseR M2
    Figure: ST Gx
    sI
    sq
    s0r0
    t0r0
    s0r1
    open
    t1r0
    τ
    t2r0
    τ
    t0r1
    open
    t1r1
    τ
    t0r2
    chooseL
    s0r2
    chooseL
    s0r3
    open
    t0r3
    open
    beep
    beep
    t1r3
    τ
    beep
    t2r3
    τ
    beep
    t1r2
    τ
    t2r2
    τ
    t2r1
    τ
    t0r4
    chooseR
    s0r4
    chooseR
    t1r4
    τ
    chooseL
    chooseL
    t2r4
    τ
    Figure: SQu(Gx
    )
    Definition (Sum-of-squares Product)
    Let SQu(G) = Su
    sq
    , sI
    sq
    , Actssq, →sq, Lsq be an LT S s.t. . . .
    Arias, Knapik, Penczek, and Petrucci Modular Analysis of Tree-Topology Models October 2022 9 / 20

    View Slide

  16. Sum-of-Squares Product SQu(G)
    r0
    r1 r2
    r3
    r4
    ?open
    ?chooseL
    ?open
    ?chooseR
    ?chooseL
    beep
    R
    s0
    !open
    M1
    t0 t1 t2
    τ τ
    !chooseL
    !chooseR M2
    Figure: ST Gx
    sI
    sq
    s0r0
    t0r0
    s0r1
    open
    t1r0
    τ
    t2r0
    τ
    t0r1
    open
    t1r1
    τ
    t0r2
    chooseL
    s0r2
    chooseL
    s0r3
    open
    t0r3
    open
    beep
    beep
    t1r3
    τ
    beep
    t2r3
    τ
    beep
    t1r2
    τ
    t2r2
    τ
    t2r1
    τ
    t0r4
    chooseR
    s0r4
    chooseR
    t1r4
    τ
    chooseL
    chooseL
    t2r4
    τ
    Figure: SQu(Gx
    )
    Definition (Sum-of-squares Product)
    Let SQu(G) = Su
    sq
    , sI
    sq
    , Actssq, →sq, Lsq be an LT S s.t. . . .
    Arias, Knapik, Penczek, and Petrucci Modular Analysis of Tree-Topology Models October 2022 9 / 20

    View Slide

  17. Sum-of-Squares Product SQu(G)
    r0
    r1 r2
    r3
    r4
    ?open
    ?chooseL
    ?open
    ?chooseR
    ?chooseL
    beep
    R
    s0
    !open
    M1
    t0 t1 t2
    τ τ
    !chooseL
    !chooseR M2
    Figure: ST Gx
    sI
    sq
    s0r0
    t0r0
    s0r1
    open
    t1r0
    τ
    t2r0
    τ
    t0r1
    open
    t1r1
    τ
    t0r2
    chooseL
    s0r2
    chooseL
    s0r3
    open
    t0r3
    open
    beep
    beep
    t1r3
    τ
    beep
    t2r3
    τ
    beep
    t1r2
    τ
    t2r2
    τ
    t2r1
    τ
    t0r4
    chooseR
    s0r4
    chooseR
    t1r4
    τ
    chooseL
    chooseL
    t2r4
    τ
    Figure: SQu(Gx
    )
    Definition (Sum-of-squares Product)
    Let SQu(G) = Su
    sq
    , sI
    sq
    , Actssq, →sq, Lsq be an LT S s.t. . . .
    Arias, Knapik, Penczek, and Petrucci Modular Analysis of Tree-Topology Models October 2022 9 / 20

    View Slide

  18. Sum-of-Squares Product SQu(G)
    r0
    r1 r2
    r3
    r4
    ?open
    ?chooseL
    ?open
    ?chooseR
    ?chooseL
    beep
    R
    s0
    !open
    M1
    t0 t1 t2
    τ τ
    !chooseL
    !chooseR M2
    Figure: ST Gx
    sI
    sq
    s0r0
    t0r0
    s0r1
    open
    t1r0
    τ
    t2r0
    τ
    t0r1
    open
    t1r1
    τ
    t0r2
    chooseL
    s0r2
    chooseL
    s0r3
    open
    t0r3
    open
    beep
    beep
    t1r3
    τ
    beep
    t2r3
    τ
    beep
    t1r2
    τ
    t2r2
    τ
    t2r1
    τ
    t0r4
    chooseR
    s0r4
    chooseR
    t1r4
    τ
    chooseL
    chooseL
    t2r4
    τ
    Figure: SQu(Gx
    )
    Definition (Sum-of-squares Product)
    Let SQu(G) = Su
    sq
    , sI
    sq
    , Actssq, →sq, Lsq be an LT S s.t. . . .
    Arias, Knapik, Penczek, and Petrucci Modular Analysis of Tree-Topology Models October 2022 9 / 20

    View Slide

  19. Sum-of-Squares Product SQu(G)
    r0
    r1 r2
    r3
    r4
    ?open
    ?chooseL
    ?open
    ?chooseR
    ?chooseL
    beep
    R
    s0
    !open
    M1
    t0 t1 t2
    τ τ
    !chooseL
    !chooseR M2
    Figure: ST Gx
    sI
    sq
    s0r0
    t0r0
    s0r1
    open
    t1r0
    τ
    t2r0
    τ
    t0r1
    open
    t1r1
    τ
    t0r2
    chooseL
    s0r2
    chooseL
    s0r3
    open
    t0r3
    open
    beep
    beep
    t1r3
    τ
    beep
    t2r3
    τ
    beep
    t1r2
    τ
    t2r2
    τ
    t2r1
    τ
    t0r4
    chooseR
    s0r4
    chooseR
    t1r4
    τ
    chooseL
    chooseL
    t2r4
    τ
    Figure: SQu(Gx
    )
    Definition (Sum-of-squares Product)
    Let SQu(G) = Su
    sq
    , sI
    sq
    , Actssq, →sq, Lsq be an LT S s.t. . . .
    Arias, Knapik, Penczek, and Petrucci Modular Analysis of Tree-Topology Models October 2022 9 / 20

    View Slide

  20. Sum-of-Squares Product SQu(G)
    r0
    r1 r2
    r3
    r4
    ?open
    ?chooseL
    ?open
    ?chooseR
    ?chooseL
    beep
    R
    s0
    !open
    M1
    t0 t1 t2
    τ τ
    !chooseL
    !chooseR M2
    Figure: ST Gx
    sI
    sq
    s0r0
    t0r0
    s0r1
    open
    t1r0
    τ
    t2r0
    τ
    t0r1
    open
    t1r1
    τ
    t0r2
    chooseL
    s0r2
    chooseL
    s0r3
    open
    t0r3
    open
    beep
    beep
    t1r3
    τ
    beep
    t2r3
    τ
    beep
    t1r2
    τ
    t2r2
    τ
    t2r1
    τ
    t0r4
    chooseR
    s0r4
    chooseR
    t1r4
    τ
    chooseL
    chooseL
    t2r4
    τ
    Figure: SQu(Gx
    )
    Definition (Sum-of-squares Product)
    Let SQu(G) = Su
    sq
    , sI
    sq
    , Actssq, →sq, Lsq be an LT S s.t. . . .
    Arias, Knapik, Penczek, and Petrucci Modular Analysis of Tree-Topology Models October 2022 9 / 20

    View Slide

  21. Sum-of-Squares Product SQu(G)
    r0
    r1 r2
    r3
    r4
    ?open
    ?chooseL
    ?open
    ?chooseR
    ?chooseL
    beep
    R
    s0
    !open
    M1
    t0 t1 t2
    τ τ
    !chooseL
    !chooseR M2
    Figure: ST Gx
    sI
    sq
    s0r0
    t0r0
    s0r1
    open
    t1r0
    τ
    t2r0
    τ
    t0r1
    open
    t1r1
    τ
    t0r2
    chooseL
    s0r2
    chooseL
    s0r3
    open
    t0r3
    open
    beep
    beep
    t1r3
    τ
    beep
    t2r3
    τ
    beep
    t1r2
    τ
    t2r2
    τ
    t2r1
    τ
    t0r4
    chooseR
    s0r4
    chooseR
    t1r4
    τ
    chooseL
    chooseL
    t2r4
    τ
    Figure: SQu(Gx
    )
    Definition (Sum-of-squares Product)
    Let SQu(G) = Su
    sq
    , sI
    sq
    , Actssq, →sq, Lsq be an LT S s.t. . . .
    Arias, Knapik, Penczek, and Petrucci Modular Analysis of Tree-Topology Models October 2022 9 / 20

    View Slide

  22. Sum-of-Squares Product SQu(G)
    r0
    r1 r2
    r3
    r4
    ?open
    ?chooseL
    ?open
    ?chooseR
    ?chooseL
    beep
    R
    s0
    !open
    M1
    t0 t1 t2
    τ τ
    !chooseL
    !chooseR M2
    Figure: ST Gx
    sI
    sq
    s0r0
    t0r0
    s0r1
    open
    t1r0
    τ
    t2r0
    τ
    t0r1
    open
    t1r1
    τ
    t0r2
    chooseL
    s0r2
    chooseL
    s0r3
    open
    t0r3
    open
    beep
    beep
    t1r3
    τ
    beep
    t2r3
    τ
    beep
    t1r2
    τ
    t2r2
    τ
    t2r1
    τ
    t0r4
    chooseR
    s0r4
    chooseR
    t1r4
    τ
    chooseL
    chooseL
    t2r4
    τ
    Figure: SQu(Gx
    )
    Definition (Sum-of-squares Product)
    Let SQu(G) = Su
    sq
    , sI
    sq
    , Actssq, →sq, Lsq be an LT S s.t. . . .
    Arias, Knapik, Penczek, and Petrucci Modular Analysis of Tree-Topology Models October 2022 9 / 20

    View Slide

  23. Sum-of-Squares Product SQu(G)
    r0
    r1 r2
    r3
    r4
    ?open
    ?chooseL
    ?open
    ?chooseR
    ?chooseL
    beep
    R
    s0
    !open
    M1
    t0 t1 t2
    τ τ
    !chooseL
    !chooseR M2
    Figure: ST Gx
    sI
    sq
    s0r0
    t0r0
    s0r1
    open
    t1r0
    τ
    t2r0
    τ
    t0r1
    open
    t1r1
    τ
    t0r2
    chooseL
    s0r2
    chooseL
    s0r3
    open
    t0r3
    open
    beep
    beep
    t1r3
    τ
    beep
    t2r3
    τ
    beep
    t1r2
    τ
    t2r2
    τ
    t2r1
    τ
    t0r4
    chooseR
    s0r4
    chooseR
    t1r4
    τ
    chooseL
    chooseL
    t2r4
    τ
    Figure: SQu(Gx
    )
    Definition (Sum-of-squares Product)
    Let SQu(G) = Su
    sq
    , sI
    sq
    , Actssq, →sq, Lsq be an LT S s.t. . . .
    Arias, Knapik, Penczek, and Petrucci Modular Analysis of Tree-Topology Models October 2022 9 / 20

    View Slide

  24. Sum-of-Squares Product SQu(G)
    r0
    r1 r2
    r3
    r4
    ?open
    ?chooseL
    ?open
    ?chooseR
    ?chooseL
    beep
    R
    s0
    !open
    M1
    t0 t1 t2
    τ τ
    !chooseL
    !chooseR M2
    Figure: ST Gx
    sI
    sq
    s0r0
    t0r0
    s0r1
    open
    t1r0
    τ
    t2r0
    τ
    t0r1
    open
    t1r1
    τ
    t0r2
    chooseL
    s0r2
    chooseL
    s0r3
    open
    t0r3
    open
    beep
    beep
    t1r3
    τ
    beep
    t2r3
    τ
    beep
    t1r2
    τ
    t2r2
    τ
    t2r1
    τ
    t0r4
    chooseR
    s0r4
    chooseR
    t1r4
    τ
    chooseL
    chooseL
    t2r4
    τ
    Figure: SQu(Gx
    )
    Definition (Sum-of-squares Product)
    Let SQu(G) = Su
    sq
    , sI
    sq
    , Actssq, →sq, Lsq be an LT S s.t. . . .
    Arias, Knapik, Penczek, and Petrucci Modular Analysis of Tree-Topology Models October 2022 9 / 20

    View Slide

  25. Sum-of-Squares Product SQu(G)
    r0
    r1 r2
    r3
    r4
    ?open
    ?chooseL
    ?open
    ?chooseR
    ?chooseL
    beep
    R
    s0
    !open
    M1
    t0 t1 t2
    τ τ
    !chooseL
    !chooseR M2
    Figure: ST Gx
    sI
    sq
    s0r0
    t0r0
    s0r1
    open
    t1r0
    τ
    t2r0
    τ
    t0r1
    open
    t1r1
    τ
    t0r2
    chooseL
    s0r2
    chooseL
    s0r3
    open
    t0r3
    open
    beep
    beep
    t1r3
    τ
    beep
    t2r3
    τ
    beep
    t1r2
    τ
    t2r2
    τ
    t2r1
    τ
    t0r4
    chooseR
    s0r4
    chooseR
    t1r4
    τ
    chooseL
    chooseL
    t2r4
    τ
    Figure: SQu(Gx
    )
    Definition (Sum-of-squares Product)
    Let SQu(G) = Su
    sq
    , sI
    sq
    , Actssq, →sq, Lsq be an LT S s.t. . . .
    Arias, Knapik, Penczek, and Petrucci Modular Analysis of Tree-Topology Models October 2022 9 / 20

    View Slide

  26. Sum-of-Squares Product SQu(G)
    r0
    r1 r2
    r3
    r4
    ?open
    ?chooseL
    ?open
    ?chooseR
    ?chooseL
    beep
    R
    s0
    !open
    M1
    t0 t1 t2
    τ τ
    !chooseL
    !chooseR M2
    Figure: ST Gx
    sI
    sq
    s0r0
    t0r0
    s0r1
    open
    t1r0
    τ
    t2r0
    τ
    t0r1
    open
    t1r1
    τ
    t0r2
    chooseL
    s0r2
    chooseL
    s0r3
    open
    t0r3
    open
    beep
    beep
    t1r3
    τ
    beep
    t2r3
    τ
    beep
    t1r2
    τ
    t2r2
    τ
    t2r1
    τ
    t0r4
    chooseR
    s0r4
    chooseR
    t1r4
    τ
    chooseL
    chooseL
    t2r4
    τ
    Figure: SQu(Gx
    )
    Definition (Sum-of-squares Product)
    Let SQu(G) = Su
    sq
    , sI
    sq
    , Actssq, →sq, Lsq be an LT S s.t. . . .
    Arias, Knapik, Penczek, and Petrucci Modular Analysis of Tree-Topology Models October 2022 9 / 20

    View Slide

  27. Sum-of-Squares Product SQu(G)
    r0
    r1 r2
    r3
    r4
    ?open
    ?chooseL
    ?open
    ?chooseR
    ?chooseL
    beep
    R
    s0
    !open
    M1
    t0 t1 t2
    τ τ
    !chooseL
    !chooseR M2
    Figure: ST Gx
    sI
    sq
    s0r0
    t0r0
    s0r1
    open
    t1r0
    τ
    t2r0
    τ
    t0r1
    open
    t1r1
    τ
    t0r2
    chooseL
    s0r2
    chooseL
    s0r3
    open
    t0r3
    open
    beep
    beep
    t1r3
    τ
    beep
    t2r3
    τ
    beep
    t1r2
    τ
    t2r2
    τ
    t2r1
    τ
    t0r4
    chooseR
    s0r4
    chooseR
    t1r4
    τ
    chooseL
    chooseL
    t2r4
    τ
    Figure: SQu(Gx
    )
    Definition (Sum-of-squares Product)
    Let SQu(G) = Su
    sq
    , sI
    sq
    , Actssq, →sq, Lsq be an LT S s.t. . . .
    Arias, Knapik, Penczek, and Petrucci Modular Analysis of Tree-Topology Models October 2022 9 / 20

    View Slide

  28. Sum-of-Squares Product SQu(G)
    r0
    r1 r2
    r3
    r4
    ?open
    ?chooseL
    ?open
    ?chooseR
    ?chooseL
    beep
    R
    s0
    !open
    M1
    t0 t1 t2
    τ τ
    !chooseL
    !chooseR M2
    Figure: ST Gx
    sI
    sq
    s0r0
    t0r0
    s0r1
    open
    t1r0
    τ
    t2r0
    τ
    t0r1
    open
    t1r1
    τ
    t0r2
    chooseL
    s0r2
    chooseL
    s0r3
    open
    t0r3
    open
    beep
    beep
    t1r3
    τ
    beep
    t2r3
    τ
    beep
    t1r2
    τ
    t2r2
    τ
    t2r1
    τ
    t0r4
    chooseR
    s0r4
    chooseR
    t1r4
    τ
    chooseL
    chooseL
    t2r4
    τ
    Figure: SQu(Gx
    )
    Definition (Sum-of-squares Product)
    Let SQu(G) = Su
    sq
    , sI
    sq
    , Actssq, →sq, Lsq be an LT S s.t. . . .
    Arias, Knapik, Penczek, and Petrucci Modular Analysis of Tree-Topology Models October 2022 9 / 20

    View Slide

  29. Sum-of-Squares Product SQu(G)
    r0
    r1 r2
    r3
    r4
    ?open
    ?chooseL
    ?open
    ?chooseR
    ?chooseL
    beep
    R
    s0
    !open
    M1
    t0 t1 t2
    τ τ
    !chooseL
    !chooseR M2
    Figure: ST Gx
    sI
    sq
    s0r0
    t0r0
    s0r1
    open
    t1r0
    τ
    t2r0
    τ
    t0r1
    open
    t1r1
    τ
    t0r2
    chooseL
    s0r2
    chooseL
    s0r3
    open
    t0r3
    open
    beep
    beep
    t1r3
    τ
    beep
    t2r3
    τ
    beep
    t1r2
    τ
    t2r2
    τ
    t2r1
    τ
    t0r4
    chooseR
    s0r4
    chooseR
    t1r4
    τ
    chooseL
    chooseL
    t2r4
    τ
    Figure: SQu(Gx
    )
    Definition (Sum-of-squares Product)
    Let SQu(G) = Su
    sq
    , sI
    sq
    , Actssq, →sq, Lsq be an LT S s.t. . . .
    Arias, Knapik, Penczek, and Petrucci Modular Analysis of Tree-Topology Models October 2022 9 / 20

    View Slide

  30. Sum-of-Squares Product SQu(G)
    r0
    r1 r2
    r3
    r4
    ?open
    ?chooseL
    ?open
    ?chooseR
    ?chooseL
    beep
    R
    s0
    !open
    M1
    t0 t1 t2
    τ τ
    !chooseL
    !chooseR M2
    Figure: ST Gx
    sI
    sq
    s0r0
    t0r0
    s0r1
    open
    t1r0
    τ
    t2r0
    τ
    t0r1
    open
    t1r1
    τ
    t0r2
    chooseL
    s0r2
    chooseL
    s0r3
    open
    t0r3
    open
    beep
    beep
    t1r3
    τ
    beep
    t2r3
    τ
    beep
    t1r2
    τ
    t2r2
    τ
    t2r1
    τ
    t0r4
    chooseR
    s0r4
    chooseR
    t1r4
    τ
    chooseL
    chooseL
    t2r4
    τ
    Figure: SQu(Gx
    )
    Definition (Sum-of-squares Product)
    Let SQu(G) = Su
    sq
    , sI
    sq
    , Actssq, →sq, Lsq be an LT S s.t. . . .
    Arias, Knapik, Penczek, and Petrucci Modular Analysis of Tree-Topology Models October 2022 9 / 20

    View Slide

  31. Sum-of-Squares Product SQu(G)
    r0
    r1 r2
    r3
    r4
    ?open
    ?chooseL
    ?open
    ?chooseR
    ?chooseL
    beep
    R
    s0
    !open
    M1
    t0 t1 t2
    τ τ
    !chooseL
    !chooseR M2
    Figure: ST Gx
    sI
    sq
    s0r0
    t0r0
    s0r1
    open
    t1r0
    τ
    t2r0
    τ
    t0r1
    open
    t1r1
    τ
    t0r2
    chooseL
    s0r2
    chooseL
    s0r3
    open
    t0r3
    open
    beep
    beep
    t1r3
    τ
    beep
    t2r3
    τ
    beep
    t1r2
    τ
    t2r2
    τ
    t2r1
    τ
    t0r4
    chooseR
    s0r4
    chooseR
    t1r4
    τ
    chooseL
    chooseL
    t2r4
    τ
    Figure: SQu(Gx
    )
    Definition (Sum-of-squares Product)
    Let SQu(G) = Su
    sq
    , sI
    sq
    , Actssq, →sq, Lsq be an LT S s.t. . . .
    Arias, Knapik, Penczek, and Petrucci Modular Analysis of Tree-Topology Models October 2022 9 / 20

    View Slide

  32. Sum-of-Squares Product SQu(G)
    r0
    r1 r2
    r3
    r4
    ?open
    ?chooseL
    ?open
    ?chooseR
    ?chooseL
    beep
    R
    s0
    !open
    M1
    t0 t1 t2
    τ τ
    !chooseL
    !chooseR M2
    Figure: ST Gx
    sI
    sq
    s0r0
    t0r0
    s0r1
    open
    t1r0
    τ
    t2r0
    τ
    t0r1
    open
    t1r1
    τ
    t0r2
    chooseL
    s0r2
    chooseL
    s0r3
    open
    t0r3
    open
    beep
    beep
    t1r3
    τ
    beep
    t2r3
    τ
    beep
    t1r2
    τ
    t2r2
    τ
    t2r1
    τ
    t0r4
    chooseR
    s0r4
    chooseR
    t1r4
    τ
    chooseL
    chooseL
    t2r4
    τ
    Figure: SQu(Gx
    )
    Definition (Sum-of-squares Product)
    Let SQu(G) = Su
    sq
    , sI
    sq
    , Actssq, →sq, Lsq be an LT S s.t. . . .
    Arias, Knapik, Penczek, and Petrucci Modular Analysis of Tree-Topology Models October 2022 9 / 20

    View Slide

  33. Sum-of-Squares Product SQu(G)
    r0
    r1 r2
    r3
    r4
    ?open
    ?chooseL
    ?open
    ?chooseR
    ?chooseL
    beep
    R
    s0
    !open
    M1
    t0 t1 t2
    τ τ
    !chooseL
    !chooseR M2
    Figure: ST Gx
    sI
    sq
    s0r0
    t0r0
    s0r1
    open
    t1r0
    τ
    t2r0
    τ
    t0r1
    open
    t1r1
    τ
    t0r2
    chooseL
    s0r2
    chooseL
    s0r3
    open
    t0r3
    open
    beep
    beep
    t1r3
    τ
    beep
    t2r3
    τ
    beep
    t1r2
    τ
    t2r2
    τ
    t2r1
    τ
    t0r4
    chooseR
    s0r4
    chooseR
    t1r4
    τ
    chooseL
    chooseL
    t2r4
    τ
    Figure: SQu(Gx
    )
    Definition (Sum-of-squares Product)
    Let SQu(G) = Su
    sq
    , sI
    sq
    , Actssq, →sq, Lsq be an LT S s.t. . . .
    Arias, Knapik, Penczek, and Petrucci Modular Analysis of Tree-Topology Models October 2022 9 / 20

    View Slide

  34. Sum-of-Squares Product SQu(G)
    r0
    r1 r2
    r3
    r4
    ?open
    ?chooseL
    ?open
    ?chooseR
    ?chooseL
    beep
    R
    s0
    !open
    M1
    t0 t1 t2
    τ τ
    !chooseL
    !chooseR M2
    Figure: ST Gx
    sI
    sq
    s0r0
    t0r0
    s0r1
    open
    t1r0
    τ
    t2r0
    τ
    t0r1
    open
    t1r1
    τ
    t0r2
    chooseL
    s0r2
    chooseL
    s0r3
    open
    t0r3
    open
    beep
    beep
    t1r3
    τ
    beep
    t2r3
    τ
    beep
    t1r2
    τ
    t2r2
    τ
    t2r1
    τ
    t0r4
    chooseR
    s0r4
    chooseR
    t1r4
    τ
    chooseL
    chooseL
    t2r4
    τ
    Figure: SQu(Gx
    )
    Definition (Sum-of-squares Product)
    Let SQu(G) = Su
    sq
    , sI
    sq
    , Actssq, →sq, Lsq be an LT S s.t. . . .
    Arias, Knapik, Penczek, and Petrucci Modular Analysis of Tree-Topology Models October 2022 9 / 20

    View Slide

  35. Sum-of-Squares Product SQu(G)
    r0
    r1 r2
    r3
    r4
    ?open
    ?chooseL
    ?open
    ?chooseR
    ?chooseL
    beep
    R
    s0
    !open
    M1
    t0 t1 t2
    τ τ
    !chooseL
    !chooseR M2
    Figure: ST Gx
    sI
    sq
    s0r0
    t0r0
    s0r1
    open
    t1r0
    τ
    t2r0
    τ
    t0r1
    open
    t1r1
    τ
    t0r2
    chooseL
    s0r2
    chooseL
    s0r3
    open
    t0r3
    open
    beep
    beep
    t1r3
    τ
    beep
    t2r3
    τ
    beep
    t1r2
    τ
    t2r2
    τ
    t2r1
    τ
    t0r4
    chooseR
    s0r4
    chooseR
    t1r4
    τ
    chooseL
    chooseL
    t2r4
    τ
    Figure: SQu(Gx
    )
    Definition (Sum-of-squares Product)
    Let SQu(G) = Su
    sq
    , sI
    sq
    , Actssq, →sq, Lsq be an LT S s.t. . . .
    Arias, Knapik, Penczek, and Petrucci Modular Analysis of Tree-Topology Models October 2022 9 / 20

    View Slide

  36. Sum-of-Squares Product SQu(G)
    r0
    r1 r2
    r3
    r4
    ?open
    ?chooseL
    ?open
    ?chooseR
    ?chooseL
    beep
    R
    s0
    !open
    M1
    t0 t1 t2
    τ τ
    !chooseL
    !chooseR M2
    Figure: ST Gx
    sI
    sq
    s0r0
    t0r0
    s0r1
    open
    t1r0
    τ
    t2r0
    τ
    t0r1
    open
    t1r1
    τ
    t0r2
    chooseL
    s0r2
    chooseL
    s0r3
    open
    t0r3
    open
    beep
    beep
    t1r3
    τ
    beep
    t2r3
    τ
    beep
    t1r2
    τ
    t2r2
    τ
    t2r1
    τ
    t0r4
    chooseR
    s0r4
    chooseR
    t1r4
    τ
    chooseL
    chooseL
    t2r4
    τ
    Figure: SQu(Gx
    )
    Definition (Sum-of-squares Product)
    Let SQu(G) = Su
    sq
    , sI
    sq
    , Actssq, →sq, Lsq be an LT S s.t. . . .
    Arias, Knapik, Penczek, and Petrucci Modular Analysis of Tree-Topology Models October 2022 9 / 20

    View Slide

  37. Sum-of-Squares Product SQu(G)
    r0
    r1 r2
    r3
    r4
    ?open
    ?chooseL
    ?open
    ?chooseR
    ?chooseL
    beep
    R
    s0
    !open
    M1
    t0 t1 t2
    τ τ
    !chooseL
    !chooseR M2
    Figure: ST Gx
    sI
    sq
    s0r0
    t0r0
    s0r1
    open
    t1r0
    τ
    t2r0
    τ
    t0r1
    open
    t1r1
    τ
    t0r2
    chooseL
    s0r2
    chooseL
    s0r3
    open
    t0r3
    open
    beep
    beep
    t1r3
    τ
    beep
    t2r3
    τ
    beep
    t1r2
    τ
    t2r2
    τ
    t2r1
    τ
    t0r4
    chooseR
    s0r4
    chooseR
    t1r4
    τ
    chooseL
    chooseL
    t2r4
    τ
    Figure: SQu(Gx
    )
    Definition (Sum-of-squares Product)
    Let SQu(G) = Su
    sq
    , sI
    sq
    , Actssq, →sq, Lsq be an LT S s.t. . . .
    Arias, Knapik, Penczek, and Petrucci Modular Analysis of Tree-Topology Models October 2022 9 / 20

    View Slide

  38. Sum-of-Squares Product SQu(G)
    r0
    r1 r2
    r3
    r4
    ?open
    ?chooseL
    ?open
    ?chooseR
    ?chooseL
    beep
    R
    s0
    !open
    M1
    t0 t1 t2
    τ τ
    !chooseL
    !chooseR M2
    Figure: ST Gx
    sI
    sq
    s0r0
    t0r0
    s0r1
    open
    t1r0
    τ
    t2r0
    τ
    t0r1
    open
    t1r1
    τ
    t0r2
    chooseL
    s0r2
    chooseL
    s0r3
    open
    t0r3
    open
    beep
    beep
    t1r3
    τ
    beep
    t2r3
    τ
    beep
    t1r2
    τ
    t2r2
    τ
    t2r1
    τ
    t0r4
    chooseR
    s0r4
    chooseR
    t1r4
    τ
    chooseL
    chooseL
    t2r4
    τ
    Figure: SQu(Gx
    )
    Definition (Sum-of-squares Product)
    Let SQu(G) = Su
    sq
    , sI
    sq
    , Actssq, →sq, Lsq be an LT S s.t. . . .
    Arias, Knapik, Penczek, and Petrucci Modular Analysis of Tree-Topology Models October 2022 9 / 20

    View Slide

  39. Sum-of-Squares Product SQu(G)
    r0
    r1 r2
    r3
    r4
    ?open
    ?chooseL
    ?open
    ?chooseR
    ?chooseL
    beep
    R
    s0
    !open
    M1
    t0 t1 t2
    τ τ
    !chooseL
    !chooseR M2
    Figure: ST Gx
    sI
    sq
    s0r0
    t0r0
    s0r1
    open
    t1r0
    τ
    t2r0
    τ
    t0r1
    open
    t1r1
    τ
    t0r2
    chooseL
    s0r2
    chooseL
    s0r3
    open
    t0r3
    open
    beep
    beep
    t1r3
    τ
    beep
    t2r3
    τ
    beep
    t1r2
    τ
    t2r2
    τ
    t2r1
    τ
    t0r4
    chooseR
    s0r4
    chooseR
    t1r4
    τ
    chooseL
    chooseL
    t2r4
    τ
    Figure: SQu(Gx
    )
    Definition (Sum-of-squares Product)
    Let SQu(G) = Su
    sq
    , sI
    sq
    , Actssq, →sq, Lsq be an LT S s.t. . . .
    Arias, Knapik, Penczek, and Petrucci Modular Analysis of Tree-Topology Models October 2022 9 / 20

    View Slide

  40. Sum-of-Squares Product SQu(G)
    r0
    r1 r2
    r3
    r4
    ?open
    ?chooseL
    ?open
    ?chooseR
    ?chooseL
    beep
    R
    s0
    !open
    M1
    t0 t1 t2
    τ τ
    !chooseL
    !chooseR M2
    Figure: ST Gx
    sI
    sq
    s0r0
    t0r0
    s0r1
    open
    t1r0
    τ
    t2r0
    τ
    t0r1
    open
    t1r1
    τ
    t0r2
    chooseL
    s0r2
    chooseL
    s0r3
    open
    t0r3
    open
    beep
    beep
    t1r3
    τ
    beep
    t2r3
    τ
    beep
    t1r2
    τ
    t2r2
    τ
    t2r1
    τ
    t0r4
    chooseR
    s0r4
    chooseR
    t1r4
    τ
    chooseL
    chooseL
    t2r4
    τ
    Figure: SQu(Gx
    )
    Definition (Sum-of-squares Product)
    Let SQu(G) = Su
    sq
    , sI
    sq
    , Actssq, →sq, Lsq be an LT S s.t. . . .
    Arias, Knapik, Penczek, and Petrucci Modular Analysis of Tree-Topology Models October 2022 9 / 20

    View Slide

  41. Sum-of-Squares Product SQu(G)
    r0
    r1 r2
    r3
    r4
    ?open
    ?chooseL
    ?open
    ?chooseR
    ?chooseL
    beep
    R
    s0
    !open
    M1
    t0 t1 t2
    τ τ
    !chooseL
    !chooseR M2
    Figure: ST Gx
    sI
    sq
    s0r0
    t0r0
    s0r1
    open
    t1r0
    τ
    t2r0
    τ
    t0r1
    open
    t1r1
    τ
    t0r2
    chooseL
    s0r2
    chooseL
    s0r3
    open
    t0r3
    open
    beep
    beep
    t1r3
    τ
    beep
    t2r3
    τ
    beep
    t1r2
    τ
    t2r2
    τ
    t2r1
    τ
    t0r4
    chooseR
    s0r4
    chooseR
    t1r4
    τ
    chooseL
    chooseL
    t2r4
    τ
    Figure: SQu(Gx
    )
    Definition (Sum-of-squares Product)
    Let SQu(G) = Su
    sq
    , sI
    sq
    , Actssq, →sq, Lsq be an LT S s.t. . . .
    Arias, Knapik, Penczek, and Petrucci Modular Analysis of Tree-Topology Models October 2022 9 / 20

    View Slide

  42. Sum-of-Squares Product SQu(G)
    r0
    r1 r2
    r3
    r4
    ?open
    ?chooseL
    ?open
    ?chooseR
    ?chooseL
    beep
    R
    s0
    !open
    M1
    t0 t1 t2
    τ τ
    !chooseL
    !chooseR M2
    Figure: ST Gx
    sI
    sq
    s0r0
    t0r0
    s0r1
    open
    t1r0
    τ
    t2r0
    τ
    t0r1
    open
    t1r1
    τ
    t0r2
    chooseL
    s0r2
    chooseL
    s0r3
    open
    t0r3
    open
    beep
    beep
    t1r3
    τ
    beep
    t2r3
    τ
    beep
    t1r2
    τ
    t2r2
    τ
    t2r1
    τ
    t0r4
    chooseR
    s0r4
    chooseR
    t1r4
    τ
    chooseL
    chooseL
    t2r4
    τ
    Figure: SQu(Gx
    )
    Definition (Sum-of-squares Product)
    Let SQu(G) = Su
    sq
    , sI
    sq
    , Actssq, →sq, Lsq be an LT S s.t. . . .
    Arias, Knapik, Penczek, and Petrucci Modular Analysis of Tree-Topology Models October 2022 9 / 20

    View Slide

  43. Sum-of-Squares Product SQu(G)
    r0
    r1 r2
    r3
    r4
    ?open
    ?chooseL
    ?open
    ?chooseR
    ?chooseL
    beep
    R
    s0
    !open
    M1
    t0 t1 t2
    τ τ
    !chooseL
    !chooseR M2
    Figure: ST Gx
    sI
    sq
    s0r0
    t0r0
    s0r1
    open
    t1r0
    τ
    t2r0
    τ
    t0r1
    open
    t1r1
    τ
    t0r2
    chooseL
    s0r2
    chooseL
    s0r3
    open
    t0r3
    open
    beep
    beep
    t1r3
    τ
    beep
    t2r3
    τ
    beep
    t1r2
    τ
    t2r2
    τ
    t2r1
    τ
    t0r4
    chooseR
    s0r4
    chooseR
    t1r4
    τ
    chooseL
    chooseL
    t2r4
    τ
    Figure: SQu(Gx
    )
    Definition (Sum-of-squares Product)
    Let SQu(G) = Su
    sq
    , sI
    sq
    , Actssq, →sq, Lsq be an LT S s.t. . . .
    Arias, Knapik, Penczek, and Petrucci Modular Analysis of Tree-Topology Models October 2022 9 / 20

    View Slide

  44. Sum-of-Squares Product SQu(G)
    r0
    r1 r2
    r3
    r4
    ?open
    ?chooseL
    ?open
    ?chooseR
    ?chooseL
    beep
    R
    s0
    !open
    M1
    t0 t1 t2
    τ τ
    !chooseL
    !chooseR M2
    Figure: ST Gx
    sI
    sq
    s0r0
    t0r0
    s0r1
    open
    t1r0
    τ
    t2r0
    τ
    t0r1
    open
    t1r1
    τ
    t0r2
    chooseL
    s0r2
    chooseL
    s0r3
    open
    t0r3
    open
    beep
    beep
    t1r3
    τ
    beep
    t2r3
    τ
    beep
    t1r2
    τ
    t2r2
    τ
    t2r1
    τ
    t0r4
    chooseR
    s0r4
    chooseR
    t1r4
    τ
    chooseL
    chooseL
    t2r4
    τ
    Figure: SQu(Gx
    )
    Definition (Sum-of-squares Product)
    Let SQu(G) = Su
    sq
    , sI
    sq
    , Actssq, →sq, Lsq be an LT S s.t. . . .
    Arias, Knapik, Penczek, and Petrucci Modular Analysis of Tree-Topology Models October 2022 9 / 20

    View Slide

  45. Sum-of-Squares Product SQu(G)
    r0
    r1 r2
    r3
    r4
    ?open
    ?chooseL
    ?open
    ?chooseR
    ?chooseL
    beep
    R
    s0
    !open
    M1
    t0 t1 t2
    τ τ
    !chooseL
    !chooseR M2
    Figure: ST Gx
    sI
    sq
    s0r0
    t0r0
    s0r1
    open
    t1r0
    τ
    t2r0
    τ
    t0r1
    open
    t1r1
    τ
    t0r2
    chooseL
    s0r2
    chooseL
    s0r3
    open
    t0r3
    open
    beep
    beep
    t1r3
    τ
    beep
    t2r3
    τ
    beep
    t1r2
    τ
    t2r2
    τ
    t2r1
    τ
    t0r4
    chooseR
    s0r4
    chooseR
    t1r4
    τ
    chooseL
    chooseL
    t2r4
    τ
    Figure: SQu(Gx
    )
    Definition (Sum-of-squares Product)
    Let SQu(G) = Su
    sq
    , sI
    sq
    , Actssq, →sq, Lsq be an LT S s.t. . . .
    Arias, Knapik, Penczek, and Petrucci Modular Analysis of Tree-Topology Models October 2022 9 / 20

    View Slide

  46. Sum-of-Squares Product SQu(G)
    r0
    r1 r2
    r3
    r4
    ?open
    ?chooseL
    ?open
    ?chooseR
    ?chooseL
    beep
    R
    s0
    !open
    M1
    t0 t1 t2
    τ τ
    !chooseL
    !chooseR M2
    Figure: ST Gx
    sI
    sq
    s0r0
    t0r0
    s0r1
    open
    t1r0
    τ
    t2r0
    τ
    t0r1
    open
    t1r1
    τ
    t0r2
    chooseL
    s0r2
    chooseL
    s0r3
    open
    t0r3
    open
    beep
    beep
    t1r3
    τ
    beep
    t2r3
    τ
    beep
    t1r2
    τ
    t2r2
    τ
    t2r1
    τ
    t0r4
    chooseR
    s0r4
    chooseR
    t1r4
    τ
    chooseL
    chooseL
    t2r4
    τ
    Figure: SQu(Gx
    )
    Definition (Sum-of-squares Product)
    Let SQu(G) = Su
    sq
    , sI
    sq
    , Actssq, →sq, Lsq be an LT S s.t. . . .
    Arias, Knapik, Penczek, and Petrucci Modular Analysis of Tree-Topology Models October 2022 9 / 20

    View Slide

  47. Sum-of-Squares Product SQu(G)
    r0
    r1 r2
    r3
    r4
    ?open
    ?chooseL
    ?open
    ?chooseR
    ?chooseL
    beep
    R
    s0
    !open
    M1
    t0 t1 t2
    τ τ
    !chooseL
    !chooseR M2
    Figure: ST Gx
    sI
    sq
    s0r0
    t0r0
    s0r1
    open
    t1r0
    τ
    t2r0
    τ
    t0r1
    open
    t1r1
    τ
    t0r2
    chooseL
    s0r2
    chooseL
    s0r3
    open
    t0r3
    open
    beep
    beep
    t1r3
    τ
    beep
    t2r3
    τ
    beep
    t1r2
    τ
    t2r2
    τ
    t2r1
    τ
    t0r4
    chooseR
    s0r4
    chooseR
    t1r4
    τ
    chooseL
    chooseL
    t2r4
    τ
    Figure: SQu(Gx
    )
    Definition (Sum-of-squares Product)
    Let SQu(G) = Su
    sq
    , sI
    sq
    , Actssq, →sq, Lsq be an LT S s.t. . . .
    Arias, Knapik, Penczek, and Petrucci Modular Analysis of Tree-Topology Models October 2022 9 / 20

    View Slide

  48. Sum-of-Squares Product SQu(G)
    r0
    r1 r2
    r3
    r4
    ?open
    ?chooseL
    ?open
    ?chooseR
    ?chooseL
    beep
    R
    s0
    !open
    M1
    t0 t1 t2
    τ τ
    !chooseL
    !chooseR M2
    Figure: ST Gx
    sI
    sq
    s0r0
    t0r0
    s0r1
    open
    t1r0
    τ
    t2r0
    τ
    t0r1
    open
    t1r1
    τ
    t0r2
    chooseL
    s0r2
    chooseL
    s0r3
    open
    t0r3
    open
    beep
    beep
    t1r3
    τ
    beep
    t2r3
    τ
    beep
    t1r2
    τ
    t2r2
    τ
    t2r1
    τ
    t0r4
    chooseR
    s0r4
    chooseR
    t1r4
    τ
    chooseL
    chooseL
    t2r4
    τ
    Figure: SQu(Gx
    )
    Definition (Sum-of-squares Product)
    Let SQu(G) = Su
    sq
    , sI
    sq
    , Actssq, →sq, Lsq be an LT S s.t. . . .
    Arias, Knapik, Penczek, and Petrucci Modular Analysis of Tree-Topology Models October 2022 9 / 20

    View Slide

  49. Sum-of-Squares Product SQu(G)
    r0
    r1 r2
    r3
    r4
    ?open
    ?chooseL
    ?open
    ?chooseR
    ?chooseL
    beep
    R
    s0
    !open
    M1
    t0 t1 t2
    τ τ
    !chooseL
    !chooseR M2
    Figure: ST Gx
    sI
    sq
    s0r0
    t0r0
    s0r1
    open
    t1r0
    τ
    t2r0
    τ
    t0r1
    open
    t1r1
    τ
    t0r2
    chooseL
    s0r2
    chooseL
    s0r3
    open
    t0r3
    open
    beep
    beep
    t1r3
    τ
    beep
    t2r3
    τ
    beep
    t1r2
    τ
    t2r2
    τ
    t2r1
    τ
    t0r4
    chooseR
    s0r4
    chooseR
    t1r4
    τ
    chooseL
    chooseL
    t2r4
    τ
    Figure: SQu(Gx
    )
    Definition (Sum-of-squares Product)
    Let SQu(G) = Su
    sq
    , sI
    sq
    , Actssq, →sq, Lsq be an LT S s.t. . . .
    Arias, Knapik, Penczek, and Petrucci Modular Analysis of Tree-Topology Models October 2022 9 / 20

    View Slide

  50. Sum-of-Squares Product SQu(G)
    r0
    r1 r2
    r3
    r4
    ?open
    ?chooseL
    ?open
    ?chooseR
    ?chooseL
    beep
    R
    s0
    !open
    M1
    t0 t1 t2
    τ τ
    !chooseL
    !chooseR M2
    Figure: ST Gx
    sI
    sq
    s0r0
    t0r0
    s0r1
    open
    t1r0
    τ
    t2r0
    τ
    t0r1
    open
    t1r1
    τ
    t0r2
    chooseL
    s0r2
    chooseL
    s0r3
    open
    t0r3
    open
    beep
    beep
    t1r3
    τ
    beep
    t2r3
    τ
    beep
    t1r2
    τ
    t2r2
    τ
    t2r1
    τ
    t0r4
    chooseR
    s0r4
    chooseR
    t1r4
    τ
    chooseL
    chooseL
    t2r4
    τ
    Figure: SQu(Gx
    )
    Definition (Sum-of-squares Product)
    Let SQu(G) = Su
    sq
    , sI
    sq
    , Actssq, →sq, Lsq be an LT S s.t. . . .
    Arias, Knapik, Penczek, and Petrucci Modular Analysis of Tree-Topology Models October 2022 9 / 20

    View Slide

  51. Sum-of-Squares Product SQu(G)
    r0
    r1 r2
    r3
    r4
    ?open
    ?chooseL
    ?open
    ?chooseR
    ?chooseL
    beep
    R
    s0
    !open
    M1
    t0 t1 t2
    τ τ
    !chooseL
    !chooseR M2
    Figure: ST Gx
    sI
    sq
    s0r0
    t0r0
    s0r1
    open
    t1r0
    τ
    t2r0
    τ
    t0r1
    open
    t1r1
    τ
    t0r2
    chooseL
    s0r2
    chooseL
    s0r3
    open
    t0r3
    open
    beep
    beep
    t1r3
    τ
    beep
    t2r3
    τ
    beep
    t1r2
    τ
    t2r2
    τ
    t2r1
    τ
    t0r4
    chooseR
    s0r4
    chooseR
    t1r4
    τ
    chooseL
    chooseL
    t2r4
    τ
    Figure: SQu(Gx
    )
    Definition (Sum-of-squares Product)
    Let SQu(G) = Su
    sq
    , sI
    sq
    , Actssq, →sq, Lsq be an LT S s.t. . . .
    Arias, Knapik, Penczek, and Petrucci Modular Analysis of Tree-Topology Models October 2022 9 / 20

    View Slide

  52. Sum-of-Squares Product SQu(G)
    r0
    r1 r2
    r3
    r4
    ?open
    ?chooseL
    ?open
    ?chooseR
    ?chooseL
    beep
    R
    s0
    !open
    M1
    t0 t1 t2
    τ τ
    !chooseL
    !chooseR M2
    Figure: ST Gx
    sI
    sq
    s0r0
    t0r0
    s0r1
    open
    t1r0
    τ
    t2r0
    τ
    t0r1
    open
    t1r1
    τ
    t0r2
    chooseL
    s0r2
    chooseL
    s0r3
    open
    t0r3
    open
    beep
    beep
    t1r3
    τ
    beep
    t2r3
    τ
    beep
    t1r2
    τ
    t2r2
    τ
    t2r1
    τ
    t0r4
    chooseR
    s0r4
    chooseR
    t1r4
    τ
    chooseL
    chooseL
    t2r4
    τ
    Figure: SQu(Gx
    )
    Definition (Sum-of-squares Product)
    Let SQu(G) = Su
    sq
    , sI
    sq
    , Actssq, →sq, Lsq be an LT S s.t. . . .
    Arias, Knapik, Penczek, and Petrucci Modular Analysis of Tree-Topology Models October 2022 9 / 20

    View Slide

  53. Sum-of-Squares Product SQu(G)
    r0
    r1 r2
    r3
    r4
    ?open
    ?chooseL
    ?open
    ?chooseR
    ?chooseL
    beep
    R
    s0
    !open
    M1
    t0 t1 t2
    τ τ
    !chooseL
    !chooseR M2
    Figure: ST Gx
    sI
    sq
    s0r0
    t0r0
    s0r1
    open
    t1r0
    τ
    t2r0
    τ
    t0r1
    open
    t1r1
    τ
    t0r2
    chooseL
    s0r2
    chooseL
    s0r3
    open
    t0r3
    open
    beep
    beep
    t1r3
    τ
    beep
    t2r3
    τ
    beep
    t1r2
    τ
    t2r2
    τ
    t2r1
    τ
    t0r4
    chooseR
    s0r4
    chooseR
    t1r4
    τ
    chooseL
    chooseL
    t2r4
    τ
    Figure: SQu(Gx
    )
    Definition (Sum-of-squares Product)
    Let SQu(G) = Su
    sq
    , sI
    sq
    , Actssq, →sq, Lsq be an LT S s.t. . . .
    Arias, Knapik, Penczek, and Petrucci Modular Analysis of Tree-Topology Models October 2022 9 / 20

    View Slide

  54. Sum-of-Squares Product SQu(G)
    r0
    r1 r2
    r3
    r4
    ?open
    ?chooseL
    ?open
    ?chooseR
    ?chooseL
    beep
    R
    s0
    !open
    M1
    t0 t1 t2
    τ τ
    !chooseL
    !chooseR M2
    Figure: ST Gx
    sI
    sq
    s0r0
    t0r0
    s0r1
    open
    t1r0
    τ
    t2r0
    τ
    t0r1
    open
    t1r1
    τ
    t0r2
    chooseL
    s0r2
    chooseL
    s0r3
    open
    t0r3
    open
    beep
    beep
    t1r3
    τ
    beep
    t2r3
    τ
    beep
    t1r2
    τ
    t2r2
    τ
    t2r1
    τ
    t0r4
    chooseR
    s0r4
    chooseR
    t1r4
    τ
    chooseL
    chooseL
    t2r4
    τ
    Figure: SQu(Gx
    )
    Definition (Sum-of-squares Product)
    Let SQu(G) = Su
    sq
    , sI
    sq
    , Actssq, →sq, Lsq be an LT S s.t. . . .
    Arias, Knapik, Penczek, and Petrucci Modular Analysis of Tree-Topology Models October 2022 9 / 20

    View Slide

  55. Sum-of-Squares Product SQu(G)
    r0
    r1 r2
    r3
    r4
    ?open
    ?chooseL
    ?open
    ?chooseR
    ?chooseL
    beep
    R
    s0
    !open
    M1
    t0 t1 t2
    τ τ
    !chooseL
    !chooseR M2
    Figure: ST Gx
    sI
    sq
    s0r0
    t0r0
    s0r1
    open
    t1r0
    τ
    t2r0
    τ
    t0r1
    open
    t1r1
    τ
    t0r2
    chooseL
    s0r2
    chooseL
    s0r3
    open
    t0r3
    open
    beep
    beep
    t1r3
    τ
    beep
    t2r3
    τ
    beep
    t1r2
    τ
    t2r2
    τ
    t2r1
    τ
    t0r4
    chooseR
    s0r4
    chooseR
    t1r4
    τ
    chooseL
    chooseL
    t2r4
    τ
    Figure: SQu(Gx
    )
    Definition (Sum-of-squares Product)
    Let SQu(G) = Su
    sq
    , sI
    sq
    , Actssq, →sq, Lsq be an LT S s.t. . . .
    Arias, Knapik, Penczek, and Petrucci Modular Analysis of Tree-Topology Models October 2022 9 / 20

    View Slide

  56. Sum-of-Squares Product SQu(G)
    r0
    r1 r2
    r3
    r4
    ?open
    ?chooseL
    ?open
    ?chooseR
    ?chooseL
    beep
    R
    s0
    !open
    M1
    t0 t1 t2
    τ τ
    !chooseL
    !chooseR M2
    Figure: ST Gx
    sI
    sq
    s0r0
    t0r0
    s0r1
    open
    t1r0
    τ
    t2r0
    τ
    t0r1
    open
    t1r1
    τ
    t0r2
    chooseL
    s0r2
    chooseL
    s0r3
    open
    t0r3
    open
    beep
    beep
    t1r3
    τ
    beep
    t2r3
    τ
    beep
    t1r2
    τ
    t2r2
    τ
    t2r1
    τ
    t0r4
    chooseR
    s0r4
    chooseR
    t1r4
    τ
    chooseL
    chooseL
    t2r4
    τ
    Figure: SQu(Gx
    )
    Definition (Sum-of-squares Product)
    Let SQu(G) = Su
    sq
    , sI
    sq
    , Actssq, →sq, Lsq be an LT S s.t. . . .
    Arias, Knapik, Penczek, and Petrucci Modular Analysis of Tree-Topology Models October 2022 9 / 20

    View Slide

  57. Sum-of-Squares Product SQu(G)
    r0
    r1 r2
    r3
    r4
    ?open
    ?chooseL
    ?open
    ?chooseR
    ?chooseL
    beep
    R
    s0
    !open
    M1
    t0 t1 t2
    τ τ
    !chooseL
    !chooseR M2
    Figure: ST Gx
    sI
    sq
    s0r0
    t0r0
    s0r1
    open
    t1r0
    τ
    t2r0
    τ
    t0r1
    open
    t1r1
    τ
    t0r2
    chooseL
    s0r2
    chooseL
    s0r3
    open
    t0r3
    open
    beep
    beep
    t1r3
    τ
    beep
    t2r3
    τ
    beep
    t1r2
    τ
    t2r2
    τ
    t2r1
    τ
    t0r4
    chooseR
    s0r4
    chooseR
    t1r4
    τ
    chooseL
    chooseL
    t2r4
    τ
    Figure: SQu(Gx
    )
    Definition (Sum-of-squares Product)
    Let SQu(G) = Su
    sq
    , sI
    sq
    , Actssq, →sq, Lsq be an LT S s.t. . . .
    Arias, Knapik, Penczek, and Petrucci Modular Analysis of Tree-Topology Models October 2022 9 / 20

    View Slide

  58. Sum-of-Squares Product SQu(G)
    r0
    r1 r2
    r3
    r4
    ?open
    ?chooseL
    ?open
    ?chooseR
    ?chooseL
    beep
    R
    s0
    !open
    M1
    t0 t1 t2
    τ τ
    !chooseL
    !chooseR M2
    Figure: ST Gx
    sI
    sq
    s0r0
    t0r0
    s0r1
    open
    t1r0
    τ
    t2r0
    τ
    t0r1
    open
    t1r1
    τ
    t0r2
    chooseL
    s0r2
    chooseL
    s0r3
    open
    t0r3
    open
    beep
    beep
    t1r3
    τ
    beep
    t2r3
    τ
    beep
    t1r2
    τ
    t2r2
    τ
    t2r1
    τ
    t0r4
    chooseR
    s0r4
    chooseR
    t1r4
    τ
    chooseL
    chooseL
    t2r4
    τ
    Figure: SQu(Gx
    )
    Definition (Sum-of-squares Product)
    Let SQu(G) = Su
    sq
    , sI
    sq
    , Actssq, →sq, Lsq be an LT S s.t. . . .
    Arias, Knapik, Penczek, and Petrucci Modular Analysis of Tree-Topology Models October 2022 9 / 20

    View Slide

  59. Sum-of-Squares Product SQu(G)
    r0
    r1 r2
    r3
    r4
    ?open
    ?chooseL
    ?open
    ?chooseR
    ?chooseL
    beep
    R
    s0
    !open
    M1
    t0 t1 t2
    τ τ
    !chooseL
    !chooseR M2
    Figure: ST Gx
    sI
    sq
    s0r0
    t0r0
    s0r1
    open
    t1r0
    τ
    t2r0
    τ
    t0r1
    open
    t1r1
    τ
    t0r2
    chooseL
    s0r2
    chooseL
    s0r3
    open
    t0r3
    open
    beep
    beep
    t1r3
    τ
    beep
    t2r3
    τ
    beep
    t1r2
    τ
    t2r2
    τ
    t2r1
    τ
    t0r4
    chooseR
    s0r4
    chooseR
    t1r4
    τ
    chooseL
    chooseL
    t2r4
    τ
    Figure: SQu(Gx
    )
    Definition (Sum-of-squares Product)
    Let SQu(G) = Su
    sq
    , sI
    sq
    , Actssq, →sq, Lsq be an LT S s.t. . . .
    Arias, Knapik, Penczek, and Petrucci Modular Analysis of Tree-Topology Models October 2022 9 / 20

    View Slide

  60. Sum-of-Squares Product SQu(G)
    r0
    r1 r2
    r3
    r4
    ?open
    ?chooseL
    ?open
    ?chooseR
    ?chooseL
    beep
    R
    s0
    !open
    M1
    t0 t1 t2
    τ τ
    !chooseL
    !chooseR M2
    Figure: ST Gx
    sI
    sq
    s0r0
    t0r0
    s0r1
    open
    t1r0
    τ
    t2r0
    τ
    t0r1
    open
    t1r1
    τ
    t0r2
    chooseL
    s0r2
    chooseL
    s0r3
    open
    t0r3
    open
    beep
    beep
    t1r3
    τ
    beep
    t2r3
    τ
    beep
    t1r2
    τ
    t2r2
    τ
    t2r1
    τ
    t0r4
    chooseR
    s0r4
    chooseR
    t1r4
    τ
    chooseL
    chooseL
    t2r4
    τ
    Figure: SQu(Gx
    )
    Definition (Sum-of-squares Product)
    Let SQu(G) = Su
    sq
    , sI
    sq
    , Actssq, →sq, Lsq be an LT S s.t. . . .
    Arias, Knapik, Penczek, and Petrucci Modular Analysis of Tree-Topology Models October 2022 9 / 20

    View Slide

  61. Sum-of-Squares Product SQu(G)
    r0
    r1 r2
    r3
    r4
    ?open
    ?chooseL
    ?open
    ?chooseR
    ?chooseL
    beep
    R
    s0
    !open
    M1
    t0 t1 t2
    τ τ
    !chooseL
    !chooseR M2
    Figure: ST Gx
    sI
    sq
    s0r0
    t0r0
    s0r1
    open
    t1r0
    τ
    t2r0
    τ
    t0r1
    open
    t1r1
    τ
    t0r2
    chooseL
    s0r2
    chooseL
    s0r3
    open
    t0r3
    open
    beep
    beep
    t1r3
    τ
    beep
    t2r3
    τ
    beep
    t1r2
    τ
    t2r2
    τ
    t2r1
    τ
    t0r4
    chooseR
    s0r4
    chooseR
    t1r4
    τ
    chooseL
    chooseL
    t2r4
    τ
    Figure: SQu(Gx
    )
    Definition (Sum-of-squares Product)
    Let SQu(G) = Su
    sq
    , sI
    sq
    , Actssq, →sq, Lsq be an LT S s.t. . . .
    Arias, Knapik, Penczek, and Petrucci Modular Analysis of Tree-Topology Models October 2022 9 / 20

    View Slide

  62. Sum-of-Squares Product SQu(G)
    r0
    r1 r2
    r3
    r4
    ?open
    ?chooseL
    ?open
    ?chooseR
    ?chooseL
    beep
    R
    s0
    !open
    M1
    t0 t1 t2
    τ τ
    !chooseL
    !chooseR M2
    Figure: ST Gx
    sI
    sq
    s0r0
    t0r0
    s0r1
    open
    t1r0
    τ
    t2r0
    τ
    t0r1
    open
    t1r1
    τ
    t0r2
    chooseL
    s0r2
    chooseL
    s0r3
    open
    t0r3
    open
    beep
    beep
    t1r3
    τ
    beep
    t2r3
    τ
    beep
    t1r2
    τ
    t2r2
    τ
    t2r1
    τ
    t0r4
    chooseR
    s0r4
    chooseR
    t1r4
    τ
    chooseL
    chooseL
    t2r4
    τ
    Figure: SQu(Gx
    )
    Definition (Sum-of-squares Product)
    Let SQu(G) = Su
    sq
    , sI
    sq
    , Actssq, →sq, Lsq be an LT S s.t. . . .
    Arias, Knapik, Penczek, and Petrucci Modular Analysis of Tree-Topology Models October 2022 9 / 20

    View Slide

  63. Sum-of-Squares Product SQu(G)
    r0
    r1 r2
    r3
    r4
    ?open
    ?chooseL
    ?open
    ?chooseR
    ?chooseL
    beep
    R
    s0
    !open
    M1
    t0 t1 t2
    τ τ
    !chooseL
    !chooseR M2
    Figure: ST Gx
    sI
    sq
    s0r0
    t0r0
    s0r1
    open
    t1r0
    τ
    t2r0
    τ
    t0r1
    open
    t1r1
    τ
    t0r2
    chooseL
    s0r2
    chooseL
    s0r3
    open
    t0r3
    open
    beep
    beep
    t1r3
    τ
    beep
    t2r3
    τ
    beep
    t1r2
    τ
    t2r2
    τ
    t2r1
    τ
    t0r4
    chooseR
    s0r4
    chooseR
    t1r4
    τ
    chooseL
    chooseL
    t2r4
    τ
    Figure: SQu(Gx
    )
    Definition (Sum-of-squares Product)
    Let SQu(G) = Su
    sq
    , sI
    sq
    , Actssq, →sq, Lsq be an LT S s.t. . . .
    Arias, Knapik, Penczek, and Petrucci Modular Analysis of Tree-Topology Models October 2022 9 / 20

    View Slide

  64. Sum-of-Squares Product SQu(G)
    r0
    r1 r2
    r3
    r4
    ?open
    ?chooseL
    ?open
    ?chooseR
    ?chooseL
    beep
    R
    s0
    !open
    M1
    t0 t1 t2
    τ τ
    !chooseL
    !chooseR M2
    Figure: ST Gx
    sI
    sq
    s0r0
    t0r0
    s0r1
    open
    t1r0
    τ
    t2r0
    τ
    t0r1
    open
    t1r1
    τ
    t0r2
    chooseL
    s0r2
    chooseL
    s0r3
    open
    t0r3
    open
    beep
    beep
    t1r3
    τ
    beep
    t2r3
    τ
    beep
    t1r2
    τ
    t2r2
    τ
    t2r1
    τ
    t0r4
    chooseR
    s0r4
    chooseR
    t1r4
    τ
    chooseL
    chooseL
    t2r4
    τ
    Figure: SQu(Gx
    )
    Definition (Sum-of-squares Product)
    Let SQu(G) = Su
    sq
    , sI
    sq
    , Actssq, →sq, Lsq be an LT S s.t. . . .
    Arias, Knapik, Penczek, and Petrucci Modular Analysis of Tree-Topology Models October 2022 9 / 20

    View Slide

  65. Sum-of-Squares Product SQu(G)
    At any given moment, SQu(G) traces only the interactions between
    the root and one of its children.
    No information about the post-synchronisation states of a component
    needs to be preserved.
    Arias, Knapik, Penczek, and Petrucci Modular Analysis of Tree-Topology Models October 2022 10 / 20

    View Slide

  66. Sum-of-Squares Product SQu(G)
    At any given moment, SQu(G) traces only the interactions between
    the root and one of its children.
    No information about the post-synchronisation states of a component
    needs to be preserved.
    The sum-of-squares product has a size in O(n · m2)1.
    The size of a representation of a state is small, as it records only local
    states of at most two components of the network.
    1n is the number of children of the root, each with a state space of size m
    Arias, Knapik, Penczek, and Petrucci Modular Analysis of Tree-Topology Models October 2022 10 / 20

    View Slide

  67. Sum-of-Squares Product SQu(G)
    At any given moment, SQu(G) traces only the interactions between
    the root and one of its children.
    No information about the post-synchronisation states of a component
    needs to be preserved.
    The sum-of-squares product has a size in O(n · m2)1.
    The size of a representation of a state is small, as it records only local
    states of at most two components of the network.
    SQu(G) of sync-memoryless tree topologies preserves reachability
    (EFp), but not liveness (EGp)2.
    1n is the number of children of the root, each with a state space of size m
    2Theorem 1 and Proposition 1 in the paper.
    Arias, Knapik, Penczek, and Petrucci Modular Analysis of Tree-Topology Models October 2022 10 / 20

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  68. Sum-of-Squares Product SQu(G)
    In certain hierarchical systems such as ADTrees, the main goal is the
    reachability of the root’s location.
    Optimisation : remove deadlocks and livelocks that halt the root’s
    evolution3.
    These states can be removed without affecting the reachability of a
    location of the root.
    3This can be done in polynomial time.
    Arias, Knapik, Penczek, and Petrucci Modular Analysis of Tree-Topology Models October 2022 11 / 20

    View Slide

  69. Sum-of-Squares Product SQu(G)
    sI
    sq
    s0r0
    t0r0
    s0r1
    open
    s0r2
    s0r3
    open
    beep
    t1r0
    τ
    t2r0
    τ
    t0r1
    t1r1
    τ
    t2r1
    τ
    t0r2
    chooseL
    t0r4
    chooseR
    t1r4
    τ
    t2r4
    τ
    chooseL
    t1r2
    τ
    t2r2
    τ
    t0r3
    beep
    t1r3
    τ
    beep
    t2r3
    τ
    beep
    s0r4
    chooseL
    chooseR
    chooseL
    open
    open
    Figure: SQu(Gx
    )
    s0r0
    s0r3
    s0r2
    open
    beep
    t0r4
    t2r1
    t0r1 t1r1
    t0r3
    t1r3
    t2r3
    t1r4
    τ
    τ
    chooseR
    τ
    τ
    beep
    beep
    beep
    τ
    chooseL
    chooseL
    open
    open
    sI
    sq
    Figure: Reduced
    SQu(Gx
    )
    Arias, Knapik, Penczek, and Petrucci Modular Analysis of Tree-Topology Models October 2022 12 / 20

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  70. Sum-of-Squares Product SQu(G)
    Sum-of-squares product for ST of any height:
    We recursively perform reduction for two-level trees in a bottom-up
    manner.
    If the topology is live-reset, then each reduction is followed by
    applying cmpl(SQu(G)) to the computed sum-of-squares.
    • The operation cmpl transforms SQu(G) into a live-reset LT S.
    No additional operations are needed for sync-deadlock topologies.
    Arias, Knapik, Penczek, and Petrucci Modular Analysis of Tree-Topology Models October 2022 13 / 20

    View Slide

  71. Sum-of-Squares Product SQu(G)
    Sum-of-squares product for ST of any height:
    We recursively perform reduction for two-level trees in a bottom-up
    manner.
    If the topology is live-reset, then each reduction is followed by
    applying cmpl(SQu(G)) to the computed sum-of-squares.
    • The operation cmpl transforms SQu(G) into a live-reset LT S.
    No additional operations are needed for sync-deadlock topologies.
    SQu(G) for any tree height has a size in O(nh · mh+1)4.
    SQu(G) of any tree height preserves reachability5.
    4h is the height of the tree.
    5Theorem 2 in the paper.
    Arias, Knapik, Penczek, and Petrucci Modular Analysis of Tree-Topology Models October 2022 13 / 20

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  72. General Sum-of-Squares GSQ(G)
    r0 r1
    ?chooseL ?chooseF
    ?chooseR
    Rz
    s0 s1
    τ
    !chooseL
    Mz
    1
    t0 t1
    !chooseF
    !chooseR
    Mz
    2
    Figure: A non
    live-reset ST Gz
    sI
    gsq
    s0r0s0t0
    t0r0s0t0
    s1r0s0t0
    τ
    chooseL
    chooseL
    t1r1s0t1
    chooseF
    s0r1s0t1
    chooseF
    chooseR
    chooseR
    s1r1s0t1
    τ
    Figure: GSQ(Gz
    )
    Definition (General Sum-of-squares Product)
    Let GSQ(G) = Sgsq, sI
    gsq
    , Actsgsq, →gsq, Lgsq be an LT S s.t. . . .
    Arias, Knapik, Penczek, and Petrucci Modular Analysis of Tree-Topology Models October 2022 14 / 20

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  73. General Sum-of-Squares GSQ(G)
    r0 r1
    ?chooseL ?chooseF
    ?chooseR
    Rz
    s0 s1
    τ
    !chooseL
    Mz
    1
    t0 t1
    !chooseF
    !chooseR
    Mz
    2
    Figure: A non
    live-reset ST Gz
    sI
    gsq
    s0r0s0t0
    t0r0s0t0
    s1r0s0t0
    τ
    chooseL
    chooseL
    t1r1s0t1
    chooseF
    s0r1s0t1
    chooseF
    chooseR
    chooseR
    s1r1s0t1
    τ
    Figure: GSQ(Gz
    )
    Definition (General Sum-of-squares Product)
    Let GSQ(G) = Sgsq, sI
    gsq
    , Actsgsq, →gsq, Lgsq be an LT S s.t. . . .
    Arias, Knapik, Penczek, and Petrucci Modular Analysis of Tree-Topology Models October 2022 14 / 20

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  74. General Sum-of-Squares GSQ(G)
    r0 r1
    ?chooseL ?chooseF
    ?chooseR
    Rz
    s0 s1
    τ
    !chooseL
    Mz
    1
    t0 t1
    !chooseF
    !chooseR
    Mz
    2
    Figure: A non
    live-reset ST Gz
    sI
    gsq
    s0r0s0t0
    t0r0s0t0
    s1r0s0t0
    τ
    chooseL
    chooseL
    t1r1s0t1
    chooseF
    s0r1s0t1
    chooseF
    chooseR
    chooseR
    s1r1s0t1
    τ
    Figure: GSQ(Gz
    )
    Definition (General Sum-of-squares Product)
    Let GSQ(G) = Sgsq, sI
    gsq
    , Actsgsq, →gsq, Lgsq be an LT S s.t. . . .
    Arias, Knapik, Penczek, and Petrucci Modular Analysis of Tree-Topology Models October 2022 14 / 20

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  75. General Sum-of-Squares GSQ(G)
    r0 r1
    ?chooseL ?chooseF
    ?chooseR
    Rz
    s0 s1
    τ
    !chooseL
    Mz
    1
    t0 t1
    !chooseF
    !chooseR
    Mz
    2
    Figure: A non
    live-reset ST Gz
    sI
    gsq
    s0r0s0t0
    t0r0s0t0
    s1r0s0t0
    τ
    chooseL
    chooseL
    t1r1s0t1
    chooseF
    s0r1s0t1
    chooseF
    chooseR
    chooseR
    s1r1s0t1
    τ
    Figure: GSQ(Gz
    )
    Definition (General Sum-of-squares Product)
    Let GSQ(G) = Sgsq, sI
    gsq
    , Actsgsq, →gsq, Lgsq be an LT S s.t. . . .
    Arias, Knapik, Penczek, and Petrucci Modular Analysis of Tree-Topology Models October 2022 14 / 20

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  76. General Sum-of-Squares GSQ(G)
    r0 r1
    ?chooseL ?chooseF
    ?chooseR
    Rz
    s0 s1
    τ
    !chooseL
    Mz
    1
    t0 t1
    !chooseF
    !chooseR
    Mz
    2
    Figure: A non
    live-reset ST Gz
    sI
    gsq
    s0r0s0t0
    t0r0s0t0
    s1r0s0t0
    τ
    chooseL
    chooseL
    t1r1s0t1
    chooseF
    s0r1s0t1
    chooseF
    chooseR
    chooseR
    s1r1s0t1
    τ
    Figure: GSQ(Gz
    )
    Definition (General Sum-of-squares Product)
    Let GSQ(G) = Sgsq, sI
    gsq
    , Actsgsq, →gsq, Lgsq be an LT S s.t. . . .
    Arias, Knapik, Penczek, and Petrucci Modular Analysis of Tree-Topology Models October 2022 14 / 20

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  77. General Sum-of-Squares GSQ(G)
    r0 r1
    ?chooseL ?chooseF
    ?chooseR
    Rz
    s0 s1
    τ
    !chooseL
    Mz
    1
    t0 t1
    !chooseF
    !chooseR
    Mz
    2
    Figure: A non
    live-reset ST Gz
    sI
    gsq
    s0r0s0t0
    t0r0s0t0
    s1r0s0t0
    τ
    chooseL
    chooseL
    t1r1s0t1
    chooseF
    s0r1s0t1
    chooseF
    chooseR
    chooseR
    s1r1s0t1
    τ
    Figure: GSQ(Gz
    )
    Definition (General Sum-of-squares Product)
    Let GSQ(G) = Sgsq, sI
    gsq
    , Actsgsq, →gsq, Lgsq be an LT S s.t. . . .
    Arias, Knapik, Penczek, and Petrucci Modular Analysis of Tree-Topology Models October 2022 14 / 20

    View Slide

  78. General Sum-of-Squares GSQ(G)
    r0 r1
    ?chooseL ?chooseF
    ?chooseR
    Rz
    s0 s1
    τ
    !chooseL
    Mz
    1
    t0 t1
    !chooseF
    !chooseR
    Mz
    2
    Figure: A non
    live-reset ST Gz
    sI
    gsq
    s0r0s0t0
    t0r0s0t0
    s1r0s0t0
    τ
    chooseL
    chooseL
    t1r1s0t1
    chooseF
    s0r1s0t1
    chooseF
    chooseR
    chooseR
    s1r1s0t1
    τ
    Figure: GSQ(Gz
    )
    Definition (General Sum-of-squares Product)
    Let GSQ(G) = Sgsq, sI
    gsq
    , Actsgsq, →gsq, Lgsq be an LT S s.t. . . .
    Arias, Knapik, Penczek, and Petrucci Modular Analysis of Tree-Topology Models October 2022 14 / 20

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  79. General Sum-of-Squares GSQ(G)
    r0 r1
    ?chooseL ?chooseF
    ?chooseR
    Rz
    s0 s1
    τ
    !chooseL
    Mz
    1
    t0 t1
    !chooseF
    !chooseR
    Mz
    2
    Figure: A non
    live-reset ST Gz
    sI
    gsq
    s0r0s0t0
    t0r0s0t0
    s1r0s0t0
    τ
    chooseL
    chooseL
    t1r1s0t1
    chooseF
    s0r1s0t1
    chooseF
    chooseR
    chooseR
    s1r1s0t1
    τ
    Figure: GSQ(Gz
    )
    Definition (General Sum-of-squares Product)
    Let GSQ(G) = Sgsq, sI
    gsq
    , Actsgsq, →gsq, Lgsq be an LT S s.t. . . .
    Arias, Knapik, Penczek, and Petrucci Modular Analysis of Tree-Topology Models October 2022 14 / 20

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  80. General Sum-of-Squares GSQ(G)
    r0 r1
    ?chooseL ?chooseF
    ?chooseR
    Rz
    s0 s1
    τ
    !chooseL
    Mz
    1
    t0 t1
    !chooseF
    !chooseR
    Mz
    2
    Figure: A non
    live-reset ST Gz
    sI
    gsq
    s0r0s0t0
    t0r0s0t0
    s1r0s0t0
    τ
    chooseL
    chooseL
    t1r1s0t1
    chooseF
    s0r1s0t1
    chooseF
    chooseR
    chooseR
    s1r1s0t1
    τ
    Figure: GSQ(Gz
    )
    Definition (General Sum-of-squares Product)
    Let GSQ(G) = Sgsq, sI
    gsq
    , Actsgsq, →gsq, Lgsq be an LT S s.t. . . .
    Arias, Knapik, Penczek, and Petrucci Modular Analysis of Tree-Topology Models October 2022 14 / 20

    View Slide

  81. General Sum-of-Squares GSQ(G)
    r0 r1
    ?chooseL ?chooseF
    ?chooseR
    Rz
    s0 s1
    τ
    !chooseL
    Mz
    1
    t0 t1
    !chooseF
    !chooseR
    Mz
    2
    Figure: A non
    live-reset ST Gz
    sI
    gsq
    s0r0s0t0
    t0r0s0t0
    s1r0s0t0
    τ
    chooseL
    chooseL
    t1r1s0t1
    chooseF
    s0r1s0t1
    chooseF
    chooseR
    chooseR
    s1r1s0t1
    τ
    Figure: GSQ(Gz
    )
    Definition (General Sum-of-squares Product)
    Let GSQ(G) = Sgsq, sI
    gsq
    , Actsgsq, →gsq, Lgsq be an LT S s.t. . . .
    Arias, Knapik, Penczek, and Petrucci Modular Analysis of Tree-Topology Models October 2022 14 / 20

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  82. General Sum-of-Squares GSQ(G)
    r0 r1
    ?chooseL ?chooseF
    ?chooseR
    Rz
    s0 s1
    τ
    !chooseL
    Mz
    1
    t0 t1
    !chooseF
    !chooseR
    Mz
    2
    Figure: A non
    live-reset ST Gz
    sI
    gsq
    s0r0s0t0
    t0r0s0t0
    s1r0s0t0
    τ
    chooseL
    chooseL
    t1r1s0t1
    chooseF
    s0r1s0t1
    chooseF
    chooseR
    chooseR
    s1r1s0t1
    τ
    Figure: GSQ(Gz
    )
    Definition (General Sum-of-squares Product)
    Let GSQ(G) = Sgsq, sI
    gsq
    , Actsgsq, →gsq, Lgsq be an LT S s.t. . . .
    Arias, Knapik, Penczek, and Petrucci Modular Analysis of Tree-Topology Models October 2022 14 / 20

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  83. General Sum-of-Squares GSQ(G)
    r0 r1
    ?chooseL ?chooseF
    ?chooseR
    Rz
    s0 s1
    τ
    !chooseL
    Mz
    1
    t0 t1
    !chooseF
    !chooseR
    Mz
    2
    Figure: A non
    live-reset ST Gz
    sI
    gsq
    s0r0s0t0
    t0r0s0t0
    s1r0s0t0
    τ
    chooseL
    chooseL
    t1r1s0t1
    chooseF
    s0r1s0t1
    chooseF
    chooseR
    chooseR
    s1r1s0t1
    τ
    Figure: GSQ(Gz
    )
    Definition (General Sum-of-squares Product)
    Let GSQ(G) = Sgsq, sI
    gsq
    , Actsgsq, →gsq, Lgsq be an LT S s.t. . . .
    Arias, Knapik, Penczek, and Petrucci Modular Analysis of Tree-Topology Models October 2022 14 / 20

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  84. General Sum-of-Squares GSQ(G)
    r0 r1
    ?chooseL ?chooseF
    ?chooseR
    Rz
    s0 s1
    τ
    !chooseL
    Mz
    1
    t0 t1
    !chooseF
    !chooseR
    Mz
    2
    Figure: A non
    live-reset ST Gz
    sI
    gsq
    s0r0s0t0
    t0r0s0t0
    s1r0s0t0
    τ
    chooseL
    chooseL
    t1r1s0t1
    chooseF
    s0r1s0t1
    chooseF
    chooseR
    chooseR
    s1r1s0t1
    τ
    Figure: GSQ(Gz
    )
    Definition (General Sum-of-squares Product)
    Let GSQ(G) = Sgsq, sI
    gsq
    , Actsgsq, →gsq, Lgsq be an LT S s.t. . . .
    Arias, Knapik, Penczek, and Petrucci Modular Analysis of Tree-Topology Models October 2022 14 / 20

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  85. General Sum-of-Squares GSQ(G)
    r0 r1
    ?chooseL ?chooseF
    ?chooseR
    Rz
    s0 s1
    τ
    !chooseL
    Mz
    1
    t0 t1
    !chooseF
    !chooseR
    Mz
    2
    Figure: A non
    live-reset ST Gz
    sI
    gsq
    s0r0s0t0
    t0r0s0t0
    s1r0s0t0
    τ
    chooseL
    chooseL
    t1r1s0t1
    chooseF
    s0r1s0t1
    chooseF
    chooseR
    chooseR
    s1r1s0t1
    τ
    Figure: GSQ(Gz
    )
    Definition (General Sum-of-squares Product)
    Let GSQ(G) = Sgsq, sI
    gsq
    , Actsgsq, →gsq, Lgsq be an LT S s.t. . . .
    Arias, Knapik, Penczek, and Petrucci Modular Analysis of Tree-Topology Models October 2022 14 / 20

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  86. General Sum-of-Squares GSQ(G)
    r0 r1
    ?chooseL ?chooseF
    ?chooseR
    Rz
    s0 s1
    τ
    !chooseL
    Mz
    1
    t0 t1
    !chooseF
    !chooseR
    Mz
    2
    Figure: A non
    live-reset ST Gz
    sI
    gsq
    s0r0s0t0
    t0r0s0t0
    s1r0s0t0
    τ
    chooseL
    chooseL
    t1r1s0t1
    chooseF
    s0r1s0t1
    chooseF
    chooseR
    chooseR
    s1r1s0t1
    τ
    Figure: GSQ(Gz
    )
    Definition (General Sum-of-squares Product)
    Let GSQ(G) = Sgsq, sI
    gsq
    , Actsgsq, →gsq, Lgsq be an LT S s.t. . . .
    Arias, Knapik, Penczek, and Petrucci Modular Analysis of Tree-Topology Models October 2022 14 / 20

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  87. General Sum-of-Squares GSQ(G)
    r0 r1
    ?chooseL ?chooseF
    ?chooseR
    Rz
    s0 s1
    τ
    !chooseL
    Mz
    1
    t0 t1
    !chooseF
    !chooseR
    Mz
    2
    Figure: A non
    live-reset ST Gz
    sI
    gsq
    s0r0s0t0
    t0r0s0t0
    s1r0s0t0
    τ
    chooseL
    chooseL
    t1r1s0t1
    chooseF
    s0r1s0t1
    chooseF
    chooseR
    chooseR
    s1r1s0t1
    τ
    Figure: GSQ(Gz
    )
    Definition (General Sum-of-squares Product)
    Let GSQ(G) = Sgsq, sI
    gsq
    , Actsgsq, →gsq, Lgsq be an LT S s.t. . . .
    Arias, Knapik, Penczek, and Petrucci Modular Analysis of Tree-Topology Models October 2022 14 / 20

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  88. General Sum-of-Squares GSQ(G)
    r0 r1
    ?chooseL ?chooseF
    ?chooseR
    Rz
    s0 s1
    τ
    !chooseL
    Mz
    1
    t0 t1
    !chooseF
    !chooseR
    Mz
    2
    Figure: A non
    live-reset ST Gz
    sI
    gsq
    s0r0s0t0
    t0r0s0t0
    s1r0s0t0
    τ
    chooseL
    chooseL
    t1r1s0t1
    chooseF
    s0r1s0t1
    chooseF
    chooseR
    chooseR
    s1r1s0t1
    τ
    Figure: GSQ(Gz
    )
    Definition (General Sum-of-squares Product)
    Let GSQ(G) = Sgsq, sI
    gsq
    , Actsgsq, →gsq, Lgsq be an LT S s.t. . . .
    Arias, Knapik, Penczek, and Petrucci Modular Analysis of Tree-Topology Models October 2022 14 / 20

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  89. General Sum-of-Squares GSQ(G)
    In the general case, both the root’s state and post-synchronisation
    state of its children have to be preserved.
    Arias, Knapik, Penczek, and Petrucci Modular Analysis of Tree-Topology Models October 2022 15 / 20

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  90. General Sum-of-Squares GSQ(G)
    In the general case, both the root’s state and post-synchronisation
    state of its children have to be preserved.
    GSQ(G) also preserves reachability6.
    Reduction or good performance is not guaranteed .
    6Theorem 3 in the paper.
    Arias, Knapik, Penczek, and Petrucci Modular Analysis of Tree-Topology Models October 2022 15 / 20

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  91. General Sum-of-Squares GSQ(G)
    In the general case, both the root’s state and post-synchronisation
    state of its children have to be preserved.
    GSQ(G) also preserves reachability6.
    Reduction or good performance is not guaranteed .
    A recursive construction preserving reachability can be used to deal
    with trees of any height7.
    6Theorem 3 in the paper.
    7Theorem 4 in the paper.
    Arias, Knapik, Penczek, and Petrucci Modular Analysis of Tree-Topology Models October 2022 15 / 20

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  92. General Sum-of-Squares GSQ(G)
    In the general case, both the root’s state and post-synchronisation
    state of its children have to be preserved.
    GSQ(G) also preserves reachability6.
    Reduction or good performance is not guaranteed .
    A recursive construction preserving reachability can be used to deal
    with trees of any height7.
    The memory needed to preserve GSQ(G) is often much smaller
    than the memory needed to hold the synchronous product of the
    entire network.
    6Theorem 3 in the paper.
    7Theorem 4 in the paper.
    Arias, Knapik, Penczek, and Petrucci Modular Analysis of Tree-Topology Models October 2022 15 / 20

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  93. Experiments
    Our approach has been implemented in the open-source tool LTR,
    written in C: https://bit.ly/3F2xadA
    Two benchmarks were used:
    • Case studies and scalable ADTrees from [Petrucci et al., 2019]8.
    • 210 random live-reset tree networks of depths 1-3.
    Files to reproduce the results: https://bit.ly/3ysE20i.
    8Timeout: 30 minutes
    Arias, Knapik, Penczek, and Petrucci Modular Analysis of Tree-Topology Models October 2022 16 / 20

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  94. Experiments
    Figure: Statespace sizes of SQu(G) for live-reset tree networks.
    Table: Results for security case studies [Petrucci et al., 2019]
    no reductions reduced/original model size (%)
    model
    |S| |T| sos reduction pattern pattern+layer
    forestall 62,689 185,944 4.018 % 9.278 % 1.836 %
    gain admin 51,158,719 364,218,554 0.01 % 2.192 % 0.036 %
    iot dev 3,381 6,860 156.479 % 29.89 % 8.017 %
    treasure hunters 479 1,326 69.418 % 27.535 % 9.03 %
    Arias, Knapik, Penczek, and Petrucci Modular Analysis of Tree-Topology Models October 2022 17 / 20

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  95. Experiments
    Figure: Statespace sizes of SQu(G) for live-reset tree networks.
    Table: Results for security case studies [Petrucci et al., 2019]
    no reductions reduced/original model size (%)
    model
    |S| |T| sos reduction pattern pattern+layer
    forestall 62,689 185,944 4.018 % 9.278 % 1.836 %
    gain admin 51,158,719 364,218,554 0.01 % 2.192 % 0.036 %
    iot dev 3,381 6,860 156.479 % 29.89 % 8.017 %
    treasure hunters 479 1,326 69.418 % 27.535 % 9.03 %
    Arias, Knapik, Penczek, and Petrucci Modular Analysis of Tree-Topology Models October 2022 17 / 20

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  96. Experiments
    Table: Some results for scalable models [Petrucci et al., 2019]
    no reductions reduced/original model size (%)
    model
    |S| |T| sos reduction pattern pattern+layer
    (2, 7, 2, 4) 185 432 68.233 % 31.605 % 19.935 %
    (2, 13, 3, 6) 8,823 35,602 8.241 % 22.485 % 6.044 %
    (2, 15, 3, 8) 34,481 160,096 2.437 % 15.351 % 2.697 %
    (2, 11, 4, 4) 1,825 6,332 163.16 % 40.432 % 28.589 %
    (2, 15, 4, 6) 26,725 124,708 13.464 % 25.726 % 8.204 %
    (2, 17, 4, 8) 103,955 549,762 4.037 % 17.629 % 3.705 %
    (2, 23, 4, 10) 5,417,613 37,414,404 0.079 % 11.828 % 0.722 %
    (2, 13, 5, 4) 5,603 22,774 258.34 % 45.784 % 32.829 %
    (2, 17, 5, 6) 80,687 428,086 22.074 % 29.446 % 9.818 %
    (2, 19, 5, 8) 312,889 1,858,220 6.693 % 20.276 % 4.504 %
    (2, 25, 5, 10) 16,261,031 123,086,630 0.133 % 13.423 % 1.006 %
    (2, 15, 6, 4) 17,065 79,784 51.903 % 37.588 %
    (2, 21, 6, 8) 940,715 6,202,486 23.287 % 5.344 %
    (2, 27, 6, 10) 48,799,477 401,798,336 15.356 %
    (2, 21, 9, 4) 470,483 3,093,598 75.931 % 56.036 %
    (2, 23, 10, 4) 1,415,545 10,225,856 86.268 %
    (3, 10, 2, 6) 3,803 15,598 9.438 % 4.861 % 4.056 %
    (3, 13, 2, 9) 43,387 228,362 0.99 % 1.732 % 0.679 %
    (3, 13, 3, 6) 34,739 186,530 6.438 % 2.83 % 2.443 %
    (3, 16, 3, 9) 392,531 2,577,950 0.703 % 1.003 % 0.489 %
    (3, 16, 4, 6) 314,699 2,097,686 4.532 % 1.676 % 1.455 %
    (3, 22, 5, 9) 31,901,507 293,805,446 0.362 % 0.186 %
    (3, 22, 6, 6) 25,597,115 238,092,350 0.597 % 0.521 %
    (3, 25, 7, 6) 230,505,107 2,450,127,602 0.358 %
    Arias, Knapik, Penczek, and Petrucci Modular Analysis of Tree-Topology Models October 2022 18 / 20

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  97. Experiments
    Table: Some results for scalable models [Petrucci et al., 2019]
    no reductions reduced/original model size (%)
    model
    |S| |T| sos reduction pattern pattern+layer
    (2, 7, 2, 4) 185 432 68.233 % 31.605 % 19.935 %
    (2, 13, 3, 6) 8,823 35,602 8.241 % 22.485 % 6.044 %
    (2, 15, 3, 8) 34,481 160,096 2.437 % 15.351 % 2.697 %
    (2, 11, 4, 4) 1,825 6,332 163.16 % 40.432 % 28.589 %
    (2, 15, 4, 6) 26,725 124,708 13.464 % 25.726 % 8.204 %
    (2, 17, 4, 8) 103,955 549,762 4.037 % 17.629 % 3.705 %
    (2, 23, 4, 10) 5,417,613 37,414,404 0.079 % 11.828 % 0.722 %
    (2, 13, 5, 4) 5,603 22,774 258.34 % 45.784 % 32.829 %
    (2, 17, 5, 6) 80,687 428,086 22.074 % 29.446 % 9.818 %
    (2, 19, 5, 8) 312,889 1,858,220 6.693 % 20.276 % 4.504 %
    (2, 25, 5, 10) 16,261,031 123,086,630 0.133 % 13.423 % 1.006 %
    (2, 15, 6, 4) 17,065 79,784 51.903 % 37.588 %
    (2, 21, 6, 8) 940,715 6,202,486 23.287 % 5.344 %
    (2, 27, 6, 10) 48,799,477 401,798,336 15.356 %
    (2, 21, 9, 4) 470,483 3,093,598 75.931 % 56.036 %
    (2, 23, 10, 4) 1,415,545 10,225,856 86.268 %
    (3, 10, 2, 6) 3,803 15,598 9.438 % 4.861 % 4.056 %
    (3, 13, 2, 9) 43,387 228,362 0.99 % 1.732 % 0.679 %
    (3, 13, 3, 6) 34,739 186,530 6.438 % 2.83 % 2.443 %
    (3, 16, 3, 9) 392,531 2,577,950 0.703 % 1.003 % 0.489 %
    (3, 16, 4, 6) 314,699 2,097,686 4.532 % 1.676 % 1.455 %
    (3, 22, 5, 9) 31,901,507 293,805,446 0.362 % 0.186 %
    (3, 22, 6, 6) 25,597,115 238,092,350 0.597 % 0.521 %
    (3, 25, 7, 6) 230,505,107 2,450,127,602 0.358 %
    Arias, Knapik, Penczek, and Petrucci Modular Analysis of Tree-Topology Models October 2022 18 / 20

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  98. Experiments
    Table: Some results for scalable models [Petrucci et al., 2019]
    no reductions reduced/original model size (%)
    model
    |S| |T| sos reduction pattern pattern+layer
    (2, 7, 2, 4) 185 432 68.233 % 31.605 % 19.935 %
    (2, 13, 3, 6) 8,823 35,602 8.241 % 22.485 % 6.044 %
    (2, 15, 3, 8) 34,481 160,096 2.437 % 15.351 % 2.697 %
    (2, 11, 4, 4) 1,825 6,332 163.16 % 40.432 % 28.589 %
    (2, 15, 4, 6) 26,725 124,708 13.464 % 25.726 % 8.204 %
    (2, 17, 4, 8) 103,955 549,762 4.037 % 17.629 % 3.705 %
    (2, 23, 4, 10) 5,417,613 37,414,404 0.079 % 11.828 % 0.722 %
    (2, 13, 5, 4) 5,603 22,774 258.34 % 45.784 % 32.829 %
    (2, 17, 5, 6) 80,687 428,086 22.074 % 29.446 % 9.818 %
    (2, 19, 5, 8) 312,889 1,858,220 6.693 % 20.276 % 4.504 %
    (2, 25, 5, 10) 16,261,031 123,086,630 0.133 % 13.423 % 1.006 %
    (2, 15, 6, 4) 17,065 79,784 51.903 % 37.588 %
    (2, 21, 6, 8) 940,715 6,202,486 23.287 % 5.344 %
    (2, 27, 6, 10) 48,799,477 401,798,336 15.356 %
    (2, 21, 9, 4) 470,483 3,093,598 75.931 % 56.036 %
    (2, 23, 10, 4) 1,415,545 10,225,856 86.268 %
    (3, 10, 2, 6) 3,803 15,598 9.438 % 4.861 % 4.056 %
    (3, 13, 2, 9) 43,387 228,362 0.99 % 1.732 % 0.679 %
    (3, 13, 3, 6) 34,739 186,530 6.438 % 2.83 % 2.443 %
    (3, 16, 3, 9) 392,531 2,577,950 0.703 % 1.003 % 0.489 %
    (3, 16, 4, 6) 314,699 2,097,686 4.532 % 1.676 % 1.455 %
    (3, 22, 5, 9) 31,901,507 293,805,446 0.362 % 0.186 %
    (3, 22, 6, 6) 25,597,115 238,092,350 0.597 % 0.521 %
    (3, 25, 7, 6) 230,505,107 2,450,127,602 0.358 %
    Arias, Knapik, Penczek, and Petrucci Modular Analysis of Tree-Topology Models October 2022 18 / 20

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  99. Experiments
    Table: Some results for scalable models [Petrucci et al., 2019]
    no reductions reduced/original model size (%)
    model
    |S| |T| sos reduction pattern pattern+layer pattern+sos
    (2, 7, 2, 4) 185 432 68.233 % 31.605 % 19.935 % 17.342 %
    (2, 13, 3, 6) 8,823 35,602 8.241 % 22.485 % 6.044 % 1.085 %
    (2, 15, 3, 8) 34,481 160,096 2.437 % 15.351 % 2.697 % 0.337 %
    (2, 11, 4, 4) 1,825 6,332 163.16 % 40.432 % 28.589 % 9.121 %
    (2, 15, 4, 6) 26,725 124,708 13.464 % 25.726 % 8.204 % 0.831 %
    (2, 17, 4, 8) 103,955 549,762 4.037 % 17.629 % 3.705 % 0.267 %
    (2, 23, 4, 10) 5,417,613 37,414,404 0.079 % 11.828 % 0.722 % 0.005 %
    (2, 13, 5, 4) 5,603 22,774 258.34 % 45.784 % 32.829 % 6.206 %
    (2, 17, 5, 6) 80,687 428,086 22.074 % 29.446 % 9.818 % 0.596 %
    (2, 19, 5, 8) 312,889 1,858,220 6.693 % 20.276 % 4.504 % 0.194 %
    (2, 25, 5, 10) 16,261,031 123,086,630 0.133 % 13.423 % 1.006 % 0.004 %
    (2, 15, 6, 4) 17,065 79,784 51.903 % 37.588 % 4.178 %
    (2, 21, 6, 8) 940,715 6,202,486 23.287 % 5.344 % 0.137 %
    (2, 27, 6, 10) 48,799,477 401,798,336 15.356 % 0.003 %
    (2, 21, 9, 4) 470,483 3,093,598 75.931 % 56.036 % 1.25 %
    (2, 23, 10, 4) 1,415,545 10,225,856 86.268 % 0.834 %
    (3, 10, 2, 6) 3,803 15,598 9.438 % 4.861 % 4.056 % 1.005 %
    (3, 13, 2, 9) 43,387 228,362 0.99 % 1.732 % 0.679 % 0.106 %
    (3, 13, 3, 6) 34,739 186,530 6.438 % 2.83 % 2.443 % 0.257 %
    (3, 16, 3, 9) 392,531 2,577,950 0.703 % 1.003 % 0.489 % 0.029 %
    (3, 16, 4, 6) 314,699 2,097,686 4.532 % 1.676 % 1.455 % 0.061 %
    (3, 22, 5, 9) 31,901,507 293,805,446 0.362 % 0.186 % 0.002 %
    (3, 22, 6, 6) 25,597,115 238,092,350 0.597 % 0.521 % 0.003 %
    (3, 25, 7, 6) 230,505,107 2,450,127,602 0.358 % 0.001 %
    Arias, Knapik, Penczek, and Petrucci Modular Analysis of Tree-Topology Models October 2022 18 / 20

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  100. Concluding Remarks
    Contributions
    Algorithm to compact large tree networks of LT Ss in which the
    components reset or deadlock after synchronising with their parents.
    Arias, Knapik, Penczek, and Petrucci Modular Analysis of Tree-Topology Models October 2022 19 / 20

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  101. Concluding Remarks
    Contributions
    Algorithm to compact large tree networks of LT Ss in which the
    components reset or deadlock after synchronising with their parents.
    An extension of the algorithm for general tree-like synchronisation
    topologies.
    Arias, Knapik, Penczek, and Petrucci Modular Analysis of Tree-Topology Models October 2022 19 / 20

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  102. Concluding Remarks
    Contributions
    Algorithm to compact large tree networks of LT Ss in which the
    components reset or deadlock after synchronising with their parents.
    An extension of the algorithm for general tree-like synchronisation
    topologies.
    Simplified model preserves reachability, but do not preserve liveness.
    Arias, Knapik, Penczek, and Petrucci Modular Analysis of Tree-Topology Models October 2022 19 / 20

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  103. Concluding Remarks
    Contributions
    Algorithm to compact large tree networks of LT Ss in which the
    components reset or deadlock after synchronising with their parents.
    An extension of the algorithm for general tree-like synchronisation
    topologies.
    Simplified model preserves reachability, but do not preserve liveness.
    Experimental evaluation shows that the method yields extremely
    effective reductions for sync-memoryless networks.
    Arias, Knapik, Penczek, and Petrucci Modular Analysis of Tree-Topology Models October 2022 19 / 20

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  104. Concluding Remarks
    Future Work
    Investigate other classes and topologies that do not need tracing what
    happens after synchronisation.
    Arias, Knapik, Penczek, and Petrucci Modular Analysis of Tree-Topology Models October 2022 20 / 20

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  105. Concluding Remarks
    Future Work
    Investigate other classes and topologies that do not need tracing what
    happens after synchronisation.
    Investigate optimisations for the “vanilla” technique of general
    sum-of-squares, e.g., efficient implementation of the memory gadget.
    Arias, Knapik, Penczek, and Petrucci Modular Analysis of Tree-Topology Models October 2022 20 / 20

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  106. Concluding Remarks
    Future Work
    Investigate other classes and topologies that do not need tracing what
    happens after synchronisation.
    Investigate optimisations for the “vanilla” technique of general
    sum-of-squares, e.g., efficient implementation of the memory gadget.
    Evaluate the prototype tool on real world examples and make better
    comparisons of the results in terms of both time and memory used.
    Arias, Knapik, Penczek, and Petrucci Modular Analysis of Tree-Topology Models October 2022 20 / 20

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  107. Thank You
    Arias, Knapik, Penczek, and Petrucci Modular Analysis of Tree-Topology Models October 2022 21 / 20

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  108. Modular Analysis of Tree-Topology Models
    Jaime Arias1, Michal Knapik2, Wojciech Penczek2, and Laure
    Petrucci1
    1LIPN, CNRS UMR 7030, Universit´
    e Sorbonne Paris Nord
    2Institute of Computer Science, Polish Academy of Sciences
    23rd International Conference on Formal Engineering Methods (ICFEM 2022)
    Madrid, Spain, 24-27 October 2022
    Arias, Knapik, Penczek, and Petrucci Modular Analysis of Tree-Topology Models October 2022 22 / 20

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  109. References I
    Aminof, B., Kupferman, O., and Murano, A. (2012).
    Improved model checking of hierarchical systems.
    Inf. Comput., 210:68–86.
    Andr´
    e, ´
    E., Lime, D., Ramparison, M., and Stoelinga, M. (2021).
    Parametric analyses of attack-fault trees.
    Fundam. Informaticae, 182(1):69–94.
    Arias, J., Budde, C. E., Penczek, W., Petrucci, L., Sidoruk, T., and Stoelinga, M. (2020).
    Hackers vs. security: Attack-defence trees as asynchronous multi-agent systems.
    In ICFEM, volume 12531 of LNCS, pages 3–19. Springer.
    Arias, J., Celerier, J.-M., and Desiante-Catherine, M. (2016).
    Authoring and automatic verification of interactive multimedia scores.
    Journal of New Music Research.
    Baier, C. and Katoen, J. (2008).
    Principles of model checking.
    MIT Press.
    Belardinelli, F., Lomuscio, A., Murano, A., and Rubin, S. (2017).
    Verification of broadcasting multi-agent systems against an epistemic strategy logic.
    In IJCAI, pages 91–97.
    Arias, Knapik, Penczek, and Petrucci Modular Analysis of Tree-Topology Models October 2022 23 / 20

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  110. References II
    Petrucci, L., Knapik, M., Penczek, W., and Sidoruk, T. (2019).
    Squeezing state spaces of (attack-defence) trees.
    In ICECCS, pages 71–80. IEEE.
    van der Aalst, W. M. P. and van Hee, K. M. (2002).
    Workflow Management: Models, Methods, and Systems.
    Cooperative information systems. MIT Press.
    Arias, Knapik, Penczek, and Petrucci Modular Analysis of Tree-Topology Models October 2022 24 / 20

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  111. Tree Synchronisation Topology
    Definition (Labelled Transition System)
    A Labelled Transition System (LT S) is a tuple
    M = S, sI , Acts, →, L where:
    S is a finite set of states;
    sI ∈ S is the initial state;
    Acts is a finite set of action names,
    including a silent action τ;
    → ⊆ S × Acts × S is a transition relation;
    L: S → 2PV is a labelling function where
    PV denotes a set of propositions.
    s0
    s1
    s2
    chooseL
    τ
    τ
    chooseR
    A run in LT S M is an infinite sequence of states and actions
    ρ = s0act0s1act1 . . . s.t. si acti


    → si+1 for all i ≥ 0.
    Arias, Knapik, Penczek, and Petrucci Modular Analysis of Tree-Topology Models October 2022 25 / 20

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  112. Tree Synchronisation Topology
    Definition (Live-Reset LTS)
    A LT S M with initial state s0 is live-reset if for each run
    ρ = s0act0s1act1 . . .:
    ∀i ∈ N if acti ∈ upacts(M), then si+1 = s0.
    Definition (Sync-Deadlock LTS)
    A LT S M is sync-deadlock if for each run ρ = s0act0s1act1 . . .:
    ∀i ∈ N if acti ∈ upacts(M), then for each j > i, actj ∈ locacts(M).
    Arias, Knapik, Penczek, and Petrucci Modular Analysis of Tree-Topology Models October 2022 26 / 20

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  113. Tree Synchronisation Topology
    r0
    r1 r2
    r3
    r4
    ?open
    ?chooseL
    ?open
    ?chooseR
    ?chooseL
    beep
    R
    s0
    !open
    M1
    t0 t1 t2
    !chooseL
    τ
    τ
    !chooseR M2
    Figure: A simple tree synchronisation topology Gx
    .
    locacts(R) = {beep}
    locacts(M2) = τ
    upacts(M2) = {chooseL, chooseR}
    downacts(R) = {open, chooseL, chooseR}
    upacts(R) = downacts(M1) =
    downacts(M2) = locacts(M1) = ∅
    Arias, Knapik, Penczek, and Petrucci Modular Analysis of Tree-Topology Models October 2022 27 / 20

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  114. Sum-of-squares Product SQu(G)
    Definition (Sum-of-squares Product)
    Define SQu(G) = Su
    sq
    , sI
    sq
    , Actssq, →sq, Lsq as an LT S s.t.:
    1. Su
    sq
    = n
    i=1
    (Si × SR) ∪ {sI
    sq
    }.
    2. sI
    sq
    ∈ Su
    sq
    is a fresh initial state.
    3. Actssq = acts(G) ∪ { }, where ∈ acts(G) is a fresh, silent action.
    4. The transition relation →sq is defined as follows:
    4.1 sI
    sq

    →sq
    (sI
    i
    , sI
    R
    ), for all i ∈ {1, . . . , n};
    4.2 if (si
    , sR
    ) act


    → (si
    , sR
    ) is a transition in Mi
    ||R, then also
    (si
    , sR
    ) act


    →sq
    (si
    , sR
    ), for all i ∈ {1, . . . , n};
    4.3 if G is live-reset (resp. sync-deadlock) and (si
    , sR
    ) act


    → (sI
    i
    , sR
    ) is a
    (resp. synchronised) transition in Mi
    ||R, then (si
    , sR
    ) act


    →sq
    (sI
    j
    , sR
    )
    for all j ∈ {1, . . . , n} \ {i}.
    5. Lsq(si , sR) = Li (si ) ∪ LR(sR) ∪ j=i
    Lj (sI
    j
    ), for each (si , sR) ∈ Su
    sq
    .
    Arias, Knapik, Penczek, and Petrucci Modular Analysis of Tree-Topology Models October 2022 28 / 20

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  115. Reachability and Liveness
    Definition (Reachability and liveness)
    For each p ∈ PV we write G |= EFp (resp. G |= EGp) iff there exists
    ρ ∈ Runs(G) s.t. ρ = s0act0s1act1 . . . and p ∈ L(si ) for some (resp. for
    all) i ∈ N.
    Both EF and EG are Computation Tree Logic (CTL) modalities.
    If G |= EFp, then we say that p is reachable in G from the initial state.
    By replacing |= with |=F and Runs with RunsF we obtain the notion
    of one-shot F-reachability and the dual of liveness.
    Arias, Knapik, Penczek, and Petrucci Modular Analysis of Tree-Topology Models October 2022 29 / 20

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  116. Reachability and Liveness
    Theorem (Sum-of-squares Product Preserves Reachability)
    Let G be a sync-memoryless two-level tree ST with root R and p ∈ PV.
    If G is live-reset, then G |= EFp iff SQu(G) |= EFp. If G is sync-deadlock
    and F = downacts(R), then G |=F EFp iff SQu(G) |=F EFp.
    Proposition (Sum-of-squares Does Not Preserve EG)
    There is a live-reset two-level tree ST G s.t. for some p ∈ PV, G |= EGp
    and SQu(G) |= EGp.
    Arias, Knapik, Penczek, and Petrucci Modular Analysis of Tree-Topology Models October 2022 30 / 20

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  117. Root-deadlock
    Definition (Root-deadlock)
    Let G be a ST with Net = {R, M1, . . . , Mn}, root(G) = R and
    children(R) = {M1, . . . , Mn}. We say that a state s of
    Mi ∈ children(R) is in a root-deadlock iff there is no run
    ρ ∈ Runs(Mi , s) s.t. ρ = s0act0s1act1 . . . with acti ∈ acts(R), where
    s0 = s, for some i ∈ N.
    The set of root-deadlocked states of an LT S can be computed in
    polynomial time using either a model checker or conventional graph
    algorithms.
    These states can be removed without affecting the reachability of a
    location of the root.
    Arias, Knapik, Penczek, and Petrucci Modular Analysis of Tree-Topology Models October 2022 31 / 20

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  118. cmpl(SQu(G))
    Definition (cmpl(SQu(G)))
    Let G be a live-reset two-level tree ST with components
    Net = {R, M1, . . . , Mn}. By cmpl(SQu(G)) we denote the result of
    replacing in SQu(G) every transition ((sMi
    , sR), act, (sMi
    , sI
    R
    )) with
    ((sMi
    , sR), act, sI
    sq
    ).
    Arias, Knapik, Penczek, and Petrucci Modular Analysis of Tree-Topology Models October 2022 32 / 20

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  119. Memory Unit
    Definition (postSync(Mi
    ))
    For each i ∈ {1, . . . , n}, let postSync(Mi ) denote the set of all the local
    states of Mi visited immediately after synchronising with R. Formally:
    postSync(Mi ) = {s ∈ Si | ∃act ∈ ActsR ∩ Actsi s.t.
    (s , r ) act


    → (s, r) in Si × SR}.
    Definition (Memory Unit)
    The post-synchronisation memory unit of G is defined as:
    mem(M1, . . . , Mn) = n
    i=1
    ({sI
    i
    } ∪ postSync(Mi )).
    Arias, Knapik, Penczek, and Petrucci Modular Analysis of Tree-Topology Models October 2022 33 / 20

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  120. General Sum-of-Squares GSQ(G)
    Definition (General Sum-of-squares Product)
    Let GSQ(G) = Sgsq, sI
    gsq
    , Actsgsq, →gsq, Lgsq be an LT S s.t.:
    Sgsq = n
    i=1
    (Si × SR) × mem(M1, . . . , Mn) ∪ {sI
    gsq
    }.
    sI
    gsq
    ∈ Sgsq is a fresh initial state.
    Actsgsq = acts(G) ∪ { }, where ∈ acts(G) is a fresh, silent action.
    The transition relation →gsq is defined as follows:
    • For each i ∈ {1, . . . , n}, sI
    gsq

    →gsq
    (sI
    i
    , sI
    R
    , m0
    ), where ∀n
    i=1
    m0
    [i] = sI
    i
    .
    • If si
    act


    →i
    si
    and act ∈ locacts(Mi
    ), then (si
    , sR
    , m) act


    →sq
    (si
    , sR
    , m),
    for each sR
    ∈ SR
    and m ∈ mem(M1
    , . . . , Mn
    ); similarly, if
    sR
    act


    →R
    sR
    and act ∈ locacts(R), then (si
    , sR
    , m) act


    →sq
    (si
    , sR
    , m),
    for each si ∈ Si
    and m ∈ mem(M1
    , . . . , Mn
    ).
    • If act ∈ upacts(Mi
    ), si
    act


    →i
    si
    , and sR
    act


    →R
    sR
    , then
    (si
    , sR
    , m) act


    →sq
    (sj
    , sR
    , m ), where m = m[i/si
    ] and sj
    = m[j] for
    some j ∈ {1, . . . , n}.
    Lgsq(si , sR, m)=Li (si ) ∪ LR(sR) ∪ j=i
    Lj (m[j]), for each
    (si , sR, m)∈Sgsq.
    Arias, Knapik, Penczek, and Petrucci Modular Analysis of Tree-Topology Models October 2022 34 / 20

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