ICFEM2022

Jaime Arias Almeida

October 27, 2022

Research

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ICFEM2022

Jaime Arias Almeida

October 27, 2022

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Transcript

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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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

View Slide
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
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

View Slide
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
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

View Slide
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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

View Slide
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

View Slide
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

View Slide
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

View Slide
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

View Slide
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

View Slide
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

View Slide
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

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

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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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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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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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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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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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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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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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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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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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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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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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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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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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