Introduction Directed Bigraphs RPO and IPO Algebra Applications Introduction Directed Bigraphs RPO and IPO Algebra Applications Labelled transition systems Labelled transition systems are relations of the form a − → Q P From Reactions to Observations: the Directed Bigraphical Model where P , Q are systems (processes, programs with state, etc. . . ) and a is a label , that is an observation LTS are used for defining the behaviour of calculi/systems Davide Grohmann, Marino Miculan because they endorse most important techniques for verifying properties (e.g., model checking ) and observational University of Udine equivalence (e.g., bisimulations ) IOC Tallinn, March 15, 2007 the labels should be enough to describe faithfully the aspects we are observing, still not too many to be impractible to use. In general good LTS are difficult to describe, and often many ad hoc choices can be done (compare e.g. CCS, π -calculus and Ambients). Davide Grohmann, Marino Miculan From Reactions to Observations: the Directed Bigraphical Model Davide Grohmann, Marino Miculan From Reactions to Observations: the Directed Bigraphical Model Introduction Directed Bigraphs RPO and IPO Algebra Applications Introduction Directed Bigraphs RPO and IPO Algebra Applications Reactions systems Labelled Transition Systems from Reaction Systems? Semantics can be also specified by reaction (or “reduction”) Principle rules, which are pairs “(redex, reactum)”. For instance: What can be observed about a process P are its interactions with the surrounding environment. (5 + 3 , 8) written as 5 + 3 − → 8 (( λ x . M ) N , M { N / x } ) written as ( λ x . M ) N − → M { N / x } Since a reaction system defines completely the behaviour of a system, it contains also the informations about interactions, A reaction system (RS) is specified by a set R of such rules, although hidden. and possibly a family of active contexts where redexes have to be found in order to fire the rule. Problem ( l , r ) ∈ R Given a reaction system, is it possible to derive a “good” LTS? C [ l ] − → C [ r ] By “good” we intend that Only a silent, “internal” state chage. the induced bisimulation must be a congruence No interaction with the surrounding environment, thus no labels should be not too many (otherwise it is difficult to use observation is specified. in practice) RS are much easier to state than LTS, but are not as useful! Davide Grohmann, Marino Miculan From Reactions to Observations: the Directed Bigraphical Model Davide Grohmann, Marino Miculan From Reactions to Observations: the Directed Bigraphical Model
Introduction Directed Bigraphs RPO and IPO Algebra Applications Introduction Directed Bigraphs RPO and IPO Algebra Applications Ad hoc solutions The “sledgehammer” approach Sometimes it can be done ad hoc, e.g, CCS: from reaction rule Define the labels of LTS as the contexts which may fire a rule L ( P ) − → Q a . P | ¯ a . Q − → P | Q L P − → Q we guess the transitions More formally: a a ¯ L → P ′ → Q ′ − − P Q a L � a � � D ◦ r � � a � α means L ◦ a = D ◦ r α. P − → P τ → P ′ | Q ′ P | Q − where ( r, r � ) is a ground reaction rule a D r because we recognize labels as the (minimal) interaction with the Proposition surrounding contexts. The bisimulation induced by the contextual LTS is a congruence. Ad hoc solutions are difficult, error prone and require lot of work and experience. (Cf. the plethora of LTS and bisimulations for But there are infinite labels for each process π -calculus) And also labels which do not carry any information about P , Aim i.e., when the redex occurs in L and shares nothing with P . We look for a general, uniform way for deriving LTS from RS. How to restrict the set of labels to only those really relevant? that is, “minimal” contexts? Davide Grohmann, Marino Miculan From Reactions to Observations: the Directed Bigraphical Model Davide Grohmann, Marino Miculan From Reactions to Observations: the Directed Bigraphical Model Introduction Directed Bigraphs RPO and IPO Algebra Applications Introduction Directed Bigraphs RPO and IPO Algebra Applications Relative and Idempotent Pushouts (Leifer, Milner, 2000) Labelled transition systems from IPOs The “minimality” can be elegantely expressed as a universal is such that, for some r, r � and D : a L �λ a � A transition categorical property. ( r, r � : J ) is a ground reaction rule (1) (2) (3) L • g 0 g 1 Write � g 0 g 1 f for f 0 , f 1 . D is active and λ = width ( D )( width ( J )) h k h g 1 id • g 0 a D k 0 ( L, D ) is an IPO for ( a, r ) j • h 0 h 1 k 1 g 0 r g 1 g a bound for � a � = D ◦ r � Call � f if h 0 h 1 • f 0 f 1 f 1 f 0 f 1 g 0 ◦ f 0 = g 1 ◦ f 1 . f 0 Remarkably, the bisimulation induced by IPO LTS is the same of contextual LTS. (1) A relative bound ( � h, h ) for � f to � g . Notice that only contexts which form an IPO for the rule are considered as labels. Thus if the reaction takes place (2) A relative pushout (RPO) ( � h, h ) for � f to � g : For any other relative “outside” a , it means that the redex r appears in L and hence bound ( � k, k ) , there is a unique mediator j . the square cannot be “minimal” g for � g, id ) is an RPO for � (3) An idem pushout (IPO) � f : ( � f to � g . Davide Grohmann, Marino Miculan From Reactions to Observations: the Directed Bigraphical Model Davide Grohmann, Marino Miculan From Reactions to Observations: the Directed Bigraphical Model
Introduction Directed Bigraphs RPO and IPO Algebra Applications Introduction Directed Bigraphs RPO and IPO Algebra Applications The Plan: Metamodels with RPOs Bigraphical Models For reaching our Aim (“general methodologies for turning RS into Long term aim:“to express as much as possible of worldwide LTS”), we need to find general metamodels with RPO and IPO distributed computing in one mathematical model.” constructions Bigraphs (Milner 2001) aim to be a unifying model of A category where RPO exist and can be calculated computations based on communications and locality. Conditions for establishing when a span � A has IPOs, and how Fundamental: they have RPO and IPO constructions to calculate these IPOs References: Encoding metodologies, that is, how to represent calculi and Pure bigraphs: structure and dynamics, R.Milner. (2005) systems (with reaction semantics) in these categories. Bigraphs and mobile processes (revised), O.-H.Jensen and R.Milner. (2003) Then we obtain an “reduced” LTS (whose bisimulation is a congruence) automatically. Davide Grohmann, Marino Miculan From Reactions to Observations: the Directed Bigraphical Model Davide Grohmann, Marino Miculan From Reactions to Observations: the Directed Bigraphical Model Introduction Directed Bigraphs RPO and IPO Algebra Applications Introduction Directed Bigraphs RPO and IPO Algebra Applications Example How a system evolves: a set of local reaction rules The ovals and circles are the nodes; the places, M which are nested, A pattern . . . message are the interiors of K K L nodes, while the L links (the thin lines) A key A connect ports that lock admin lie on the periphery of each node . . . and how it reconfigures A A REACTION RULE Davide Grohmann, Marino Miculan From Reactions to Observations: the Directed Bigraphical Model Davide Grohmann, Marino Miculan From Reactions to Observations: the Directed Bigraphical Model
Introduction Directed Bigraphs RPO and IPO Algebra Applications Introduction Directed Bigraphs RPO and IPO Algebra Applications Result of the reaction A bigraph = a place graph + a link graph bigraph y 1 y 0 G : � m, X � →� n, Y � 0 1 v 2 2 v 0 0 v 1 v 3 M 1 A pattern . . . place graph link graph K G P : m → n x 0 x 1 G L : X → Y L roots ... ...outer names 0 1 y 0 y 1 A A v 0 v 2 v 2 v 3 v 0 v 3 v 1 v 1 . . . and how it reconfigures sites ... ...inner names 0 1 2 x 0 x 1 A A REACTION RULE Davide Grohmann, Marino Miculan From Reactions to Observations: the Directed Bigraphical Model Davide Grohmann, Marino Miculan From Reactions to Observations: the Directed Bigraphical Model Introduction Directed Bigraphs RPO and IPO Algebra Applications Introduction Directed Bigraphs RPO and IPO Algebra Applications Output Linear Link Graph Input Linear Link Graph ’OLG y w x z ’ILG An algorithm for the An algorithm for the construction of RPOs; construction of (G)RPOs, as v 0 consistency conditions on e an instance of general the existence of bounds; construction for an algorithm for the ILC ( PLGraphs ). construction of IPOs. [P. Soboci´ nski. Deriving process e [O. H. Jensen and R. Milner. v 0 congruences from reaction rules. Bigraphs and mobile processes PhD thesis, University of Aarhus, (revised). Technical Report, 2004] y x z w University of Cambridge, 2004] Davide Grohmann, Marino Miculan From Reactions to Observations: the Directed Bigraphical Model Davide Grohmann, Marino Miculan From Reactions to Observations: the Directed Bigraphical Model
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