Synchronized
Composition of Labeled
Transition System
Ang Chen
University of Geneva, Switzerland
17.06.2008

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composition test

Labeled Transition
System

Transition System
Transition System. A transition system is a quadruple A =< S, T, α, β >
where
• S is a finite or infinite set of states,
• T is a finite or infinite set of transitions,
• α and β are two mappings from T to S which take each transition t in T
to the two states α(t) and β(t), respectively the source and the target of
the transition t.
4

Labeled Transition
System
Labeled Transition System. A labeled transition system is a quadruple
A =< S, T, A, s0 > where:
• S is a set of states,
• A is a set of actions names (labels), called alphabet
a
• T ⊆ S × A × S, denoted as s − s , a ∈ A, s, s ∈ S, is a transition relation.
→
α : T → S, β : T → S, λ : T → A are mappings from a transition to its
source state, target state, and label, respectively,
• s0 ∈ S is the initial state.
5

Example: Counter
0 3
t4
LTS of counter
t1 t3
1 2
t2
It has 4 states {0, 1, 2, 3}, one action inc, initial state 0, and the following
transitions:
• t1 : 0 → inc → 1, α(t1) = 0, λ(t1) = inc, β(t1) = 1
• t2 : 1 → inc → 2, α(t2) = 1, λ(t2) = inc, β(t2) = 2
• t3 : 2 → inc → 3, α(t3) = 2, λ(t3) = inc, β(t3) = 3
• t4 : 3 → inc → 0, α(t4) = 3, λ(t4) = inc, β(t4) = 0
6

Different models for
the same semantics
t1
P0 P1
0+1=1
x
1+1=2
0 inc
t4 t2 2+1=3
x+1 3+1=0
t3 P2
P3
Place/Transition Net model of the counter APN (CO-OPN) model of the counter
(Places are Safe)
They have different labels (action) set
The P/T Net can be considered as an unfolded APN model
7

LTS: Summary
• LTS: State, Action (names), Transition
• Semantics are what we want to have, i.e. LTS
• The same semantics can be realized by different
DSL with different syntax
• Composition of LTS can be independent of DSL
• Assumption: LTS transitions are instance of Action
(labels). Each LTS transition is always labeled by a
label.
8

Model Composition

Model Composition
• Synonyms
• communication (e.g. in Process Algebra)
• synchronization (e.g. Petri Net)
• transaction (e.g in CO-OPN)
• Composition implies constraints
10

Synchronous Product
of Transition Systems
• André Arnold
• Free Product: no interaction between
components
• Synchronous product
• a subsystem of the free product
• synchronization constraints (implied by
interaction)
11

Arnold’s Synchronous Product
of TS
• Only essential synchronization constraints
are used
e.g.
• program/process P is a TS
• boolean variable B is a TS
• a constraint is :
•“P.read” is synchronized with
“B.get”
12

Synchronous Product of TS
vs. CO-OPN transaction
• “when A sends a message to B, B receives it”
A.out//B.in
A B
B.out//A.ackout
with CO-OPN, we can do more:
“when B receives a message, it send
an acknowledgment to A”
13

Proposition
Use CO-OPN Transactions
operators to specify
synchronization constraints for
model composition
14

Free product of
counters
System A System B
0 3 0 3
inc inc
LTS inc
inc inc inc
1 2 1 2
inc inc
x x
Model 0 Ainc 0 Binc
x+1 x+1
Two simultaneous, independent
counters
15

Free Product
(interleaving run) Bt4
Bt1 Bt2 Bt3
00 01 02 03
x At1
At1 At1 At1
Bt4
0 Ainc
x+1 Bt1 Bt2 Bt3
10 11 12 13
At4
At4 At2 At4 At4
At2 At2 At2
Bt4
x
Bt1 Bt2 Bt3
20 21 22 23
0 Binc
x+1 At3
At3
At3
At3 Bt4
Bt1 Bt2 Bt3
30 31 32 33
LTS of the composed system
16

Composing the
counters: I
SA SB
x
x
0
0
x+1
x+1
y
x
Ainc Binc
y+1
x+1
Ainc//Binc
Ainc+//Binc+
17

Composed LTS: I
At4//Bt4
00 Bt1 Bt2 Bt3
01 02 03
SA SB
x
x
At1Bt2
0
At1
0 At1 At1Bt1 At1 At1Bt3
x+1 At1
x+1 At1Bt4
y
x
Ainc Binc
y+1
x+1 10 12 Bt3
Bt1 Bt2
11 13
Ainc//Binc
At4Bt1 At2Bt3 At2Bt4 At4Bt3
At2 At2
At2
At2
At2Bt1 At2Bt3
At4Bt2
Ainc+//Binc+ 20 Bt1 Bt2 Bt3
21 22 23
At3 At3 At3Bt4 At3
At3Bt1 At3Bt3
At3Bt2 At3
30 Bt1 Bt2 Bt3
31 32 33
18

Composing the
counters: II
SA SB
x
0
0
x+1
y
x
Ainc
y+1
x+1
Ainc//Binc
Ainc+//Binc
19

Composing the
counters: III
SA SB
x
0
0
x+1
y
x Binc
y+1
x+1
Ainc//Binc
Ainc//Binc+
20

Composed LTS: III
At4//Bt4
00 Bt1 Bt2 Bt3
01 02 03
At1Bt2
SA SB At1Bt1 At1Bt3
x At1Bt4
0
0
x+1 10 Bt1 Bt2 Bt3
11 12 13
y
x Binc At4Bt1 At2Bt3 At2Bt4
y+1 At2Bt3
x+1 At2Bt1 At4Bt3
At4Bt2
Ainc//Binc 20 Bt1 Bt2 Bt3
21 22 23
At3Bt4
At3Bt1 At3Bt3
At3Bt2
Ainc//Binc+ 30 Bt1 Bt2 Bt3
31 32 33
21

Composed LTS: IV
SA SB
0
0
y
x Ainc//Binc
y+1
x+1
Ainc//Binc
At4Bt4
LTS At1Bt1 At2Bt2 At3Bt3
00 11 22 33
22

Conditional
Synchronizations I
SA=3, SB=3::Ainc // Binc (At4//Bt4)
SA=3, SB=3::Ainc // Binc
Bt1 Bt2 Bt3
00 01 02 03
SA SB
At1
x!=3 At1 At1 At1
x!=3
0
0
x+1
x+1
3 Bt1 Bt2 Bt3
10 11 12 13
3
Ainc Binc
0
At2
0 At2 At2 At2
SA=3, SB=3::Ainc//Binc
Bt1 Bt2 Bt3
20 21 22 23
Two counters are At3
At3
At3
At3
synchronized at Bt1 Bt2 Bt3
30 31 32 33
each cycle
23

Conditional
Synchronizations II
SA=3, SB=3::Ainc // Binc (At4//Bt4)
SA=3, SB=3::Ainc+ // Binc
Bt1 Bt2 Bt3
00 01 02 03
SA SB
At1
At1 At1 At1
x!=3
x
0
0
x+1
x+1
3 Bt1 Bt2 Bt3
10 11 12 13
3
Ainc Binc
0
0
At4 At2
At4 At2 At4 At2 At4 At2
+
SA=3, SB=3::Ainc //Binc
Bt1 Bt2 Bt3
20 21 22 23
Same as before, but A At3
At3
At3
At3
can resets itself
without synchronizing Bt1 Bt2 Bt3
30 31 32 33
with B +
1). SA=3, SB=3::Ainc // Binc
24

Conditional
Synchronizations
SA=3, SB=3::Ainc // Binc (At4//Bt4)
SA=3, SB=3::Ainc+//Binc+
Bt1 Bt2 Bt3
00 01 02 03
SA SB Bt4
x At1 At1
x At1
At1
0
0
x+1
x+1 Bt1 Bt2 Bt3
10 11 12 13
3
3
Ainc Binc
Bt4
0 At2
At4 At2
At4 At2 At4 At2
0 At4
SA=3, SB=3::Ainc//Binc
Bt1 Bt2 Bt3
20 21 22 23
At3
At3
At3 Bt4
At3
Both A and B can reset Bt1 Bt2 Bt3
30 31 32 33
themselves, or they can Bt4
be synchronized at 0 + +
2). SA=3, SB=3::Ainc //Binc
25

More conditional
synchronizations
At4//Bt4
SA=3::Ainc//Binc+
Bt1 Bt2 Bt3
00 01 02 03
SA SB
At1
At1 At1 At1
x
x!=3
0
0
x+1
x+1 Bt1 Bt2 Bt3
10 11 12 13
x
3
Ainc Binc
x+1
At4Bt3 At2
At4Bt2 At2
At2 At4Bt1 At2
0
SA=3::Ainc//Binc+
Bt1 Bt2 Bt3
20 21 22 23
At3
At3
At3
At3
A triggers B at Bt1 Bt2 Bt3
30 31 32 33
each cycle of A +
1). SA=3::Ainc//Binc
26

More conditional
synchronizations
At4//Bt4
SA=3::Ainc//Binc
00 01 02 03
SA SB At1
At1 At1 At1
x!=3
0
0
x+1 10 11 12 13
x
3
Ainc
At4Bt3 At2
x+1 At4Bt2 At2
At2 At4Bt1 At2
0
SA=3::Ainc//Binc 20 21 22 23
At3
At3
At3
At3
B is driven by A at 30 31 32 33
each cycle of A 2). SA=3::Ainc//Binc
27

Summary
• Use transaction for model composition
• Synchronization constraints: extend transaction
with modifiers +,- to control the composition
• Formalization of these composition operators:
ongoing work
• Formalism Integration with ID-Net: this summer
28