1. A MODULAR LTS FOR OPEN REACTIVE SYSTEMS
Uni Pisa
F. Gadducci, G.V.Monreale and U. Montanari

2. Outline Process calculi (Open) Reactive Systems A Few Problems
Some Solutions
Conclusions

3. A process calculus: CCS
Expresses how the system can interact with the environment
Syntax
P ::= M, P1| P2
M ::= 0, a.P, a.P, .P, M1+M2  P Q
Labelled Transition System
Expresses the behaviour of the system in terms of the behaviour of its components
Behavioral Equivalence
Compositional
A symmetric relation R is a bisimulation if whenever P R Q
P P’ implies Q Q’ and P’ R Q’
Bisimilarity ∼ is the largest bisimulation  

4. Reduction semantics for CCS
Elegant and natural Describes the behaviour of the system as a whole
Structural Congruence
P|Q  Q|P P|(Q|R)  (P|Q)|R P|0  P
M+N  N+M M+(N+O)  (M+N)+O M+0 = M
Reduction Relation
Not Compositional
(a.P+M) | (a.Q+N) P | Q .P+M P
Closed wrt. structural congruence and parallel operator
a.0
is equivalent to
b.0
a.0
|a.0
is not equivalent to
b.0
|a.0

5. Preliminary question... Reactive Systems
How to derive LTSs from reduction semantics such that their bisimulation is a congruence?
Reactive Systems
[Leifer and Milner 2000]

6. (Open) Reactive Systems
Categories model the state space of formalisms whose
operational semantics is provided by reduction rules
Terms
Contexts
Reduction Rules
Structural rules C
(Open Terms) l r P C
< I2 , I2> I1 I1 I1 I2 I2
I I2 C’
Redution Relation
Reactive context
r;D
P  l;D
a.0 | a |0
a.- | a |-
-|P
b.-
a.0 | a.0 |P |0 |P
a.- | a.- |P |- |P
b.(a.0 | a.0 )
b.(a.- | a.- )

7. Open Saturated bisimilarity ~SAT is a congruence
Open Saturated LTS
Any context C
P Q iff
X;P;C Q X S
Any instantiation
Open Saturated bisimilarity ~SAT is a congruence
Examples: CCS
-|a.-
<a.-,->
-|-
<-,-> S
-|a.-|P
<a.-, ->
Q|-|P
Redundant
<Q,-> S

8. G-Lux LTS [Klin, Sassone, Sobocinski 2005]
P Q iff X;P;C Q
Minimal context C X L
Most general instantiation I6 I6 I1 I2 I3
GIPB
P;C
l;D X Y C D I6 I1 I4 I5
GIPO C D
X;P
Y;l I4 I5 P
G-LUX l I2 I3
Examples: CCS X Y I1
-|a.-
<a.-,->
-|-
<-,-> L
-|a.-|P
Not minimal
<a.-, ->
Q|-|P
<Q,-> L

9. Problems Lux bisimilarity a congruence under restrictive conditions
Redundancy
Infinitary and flat presentation 1
- |a.P|a.Q
a.R|- 1 1
-|a.P|a.Q
a.-
G-LUX
a.-|a.-
a.-
a.R|P|Q L R 2 1 R
<P,Q>

10. Bisimilarity is not a congruence
We propose (elsewhere) a suitable notion of barbed bisimilarity for open RSs which is
more expressive
able to recast a variety of observational, bisimulation- based equivalence
efficiently characterizable

11. Redundancy
We propose an alternative way to derive LTSs

12. GIPO-GIPB LTS Examples: CCS P Q iff X;P;C Q Minimal context
Most general instantiation I6 I7 I4 I5
GIPO C D P2 l2
Examples: CCS
-|a.-
<a.-,->
-|- P l
<-,-> GG I1 I2 I3
GIPB P1 l1 X Y
-|a.P|a.Q
a.-
Not available R GG

13. Infinitary and flat presentation
We propose a SOS-like presentation via an encoding into tile systems

14. 2-Category of Interactionsl C C I1 I2 I2 I3 I1 I2 r
C ‘ C
It might be not reactive l’ l C X I2 I3 l I1 I2 I3 I1 C r X r

15. C-Square Double Categoryid C1 C1 I1 I2 I1 I2 I1 I2 I1 I2
GIPB
GIPO C4 C3 l1 C4 C3 l2 C4 C3 C4 C3 I3 I4 I3 I4 I3 I4 I3 I4 C2 C2 r C2 C2
Reactive context l2 id r P1 id P2 l1
I1’ I1 I1 I5
GIPO l2
GIPB P1
Y; r l1 l2 X C X
I3’ I3 I4 I6 r C D
P1;P2
Y; r;D Y

16. Example GIPO GIPO GIPB GIPB <-,-> -|- <a.-,-> -|-1 2 2 2 2 2
GIPO
GIPO
GIPB
GIPB
<a.-,->
- | a.-
- | a.- -
<-,->
- | a.- 1 2 2 1 1 1
<P,->
<-,->
- | - - -
-|a.-
<-,->
-|-
<a.-,->
-|a.-
<a.-,->
-|-
<-,->
-|a.-
<a.P,->
-|- -

17. Conclusions
We addressed some issues concerning the adequacy of the technique based on LUX in deriving LTSs for open RSs
We studied three problems:
The induced bisimilarity in not always a congruence
The derived LTS is redundant
It has a flat and infinitary presentation
Future work
Finding a suitable notion of bisimilarity on the GIPO-GIPB LTS
Extending our framework by considering an automatically derived notion of barb for reactive systems