ICFEM2022

Transcript

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