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

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

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  

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

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

(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 <0 I2 , 0 I2> I1 I1 I1 I2 0 I2 I1 I2 C’ Redution Relation Reactive context r;D P  l;D a.0 | a.0 0|0 a.- | a.- -|- -|P b.- a.0 | a.0 |P 0|0 |P a.- | a.- |P -|- |P b.(a.0 | a.0 ) b.(a.- | a.- )

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

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

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>

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

Redundancy We propose an alternative way to derive LTSs

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

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

2-Category of Interactions l 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

C-Square Double Category id 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

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,-> -|- -

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