DBtk: a toolkit for directed bigraphs

DBtk: a Toolkit for Directed Bigraphs

Giorgio Bacci Davide Grohmann Marino Miculan

Department of Mathematics and Computer ScienceUniversity of Udine, Italy

CALCO-tools, September, 6 2009

1 / 40

Outline

Part 1 - Introduction to (Directed) Bigraphical Frameworks

+ Directed Bigraphs

+ Directed Bigraphical Reactive Systems (DBRS)

+ How to derive LTSs from (DB)RSs

Part 2 - DBtk: a toolkit for Directed Bigraphs

+ Basic structures and operations

+ Algebra

+ Compilation and Drawing

+ Matching (Simulation via a DBRS)

+ RPO/IPO construction for LTSs upon DBRSs

2 / 40

Outline


+ Directed Bigraphs





+ Algebra




2 / 40

Bigraphical Models

+ Bigraphical models are an emerging framework forconcurrency and mobility.

+ Long term aim: “to express as much as possible of worldwidedistributed computing in one mathematical model” (Milner2001).

+ Bigraphs aim to be such a framework, i.e., unifying model forcomputations based on communications and locality.

+ Many calculi have been represented in bigraphs: CCS,π-calculus, Petri nets, . . .

+ . . . but they can be applied also for dealing with SystemsBiology! (as shown yesterday at MeCBIC 2009)

3 / 40

Example of a Directed Bigraph

A directed bigraph has nodes, edges and links.

0

v0

v1

v2

01

x y

e0

w

e1

1

v3

v4

2

e2

e3

z

u

Nodes

Each node v0, v1, . . . , has an arity (i.e. a set of ports);

4 / 40



0

v0

v1

v2

01

x y

e0

w

e1

1

v3

v4

2

e2

e3

z

u

Edges

Edges e0, e1, . . . represent global resources.

4 / 40



0

v0

v1

v2

01

x y

e0

w

e1

1

v3

v4

2

e2

e3

z

u

Ports

Ports “ask for connections” to resources.

4 / 40



0

v0

v1

v2

01

x y

e0

w

e1

1

v3

v4

2

e2

e3

z

u

Placing

Nodes can be nested, instead edges are not subject to positions.Sites are holes which can be fitted by roots of another bigraph.

4 / 40



0

v0

v1

v2

01

x y

e0

w

e1

1

v3

v4

2

e2

e3

z

u

Links

Links describes the access/request of resources by nodes.

4 / 40



0

v0

v1

v2

01

x y

e0

w

e1

1

v3

v4

2

e2

e3

z

u

Names

Names x , y , z , . . . are “channels” through which controls can giveor request access to resources.

4 / 40

directed bigraph = place graph + directed link graph

directed bigraph

G : 〈3, ({w}, {u})〉 → 〈2, ({x , z}, {y})〉

0

v0

v1

v2

01

x y

e0

w

e1

1

v3

v4

2

e2

e3

z

u

place graph

GP : 3→ 2

0

v0 v2

v1 1

0

1

v3

v4 2

directed link graph

GL : ({w}, {u})→ ({x , z}, {y})

v0 v1

v2

x y

e0

w

e1

v3

v4

e2

e3

z

u

5 / 40

Dynamics: Reaction rules

A signature is a typing over nodes.

door key

A reaction rule is a pair of (parametric) bigraphs.

x x

key is atomic

door is not atomic

Reaction application: matching a rule with a sub-graph.

x y

6 / 40



door key


x x

key is atomic

door is not atomic


x y

6 / 40



door key


x x

key is atomic

door is not atomic


x y

6 / 40



door key


x x

key is atomic

door is not atomic


x y

6 / 40



door key


x x

key is atomic

door is not atomic


x y

6 / 40



door key


x x

key is atomic

door is not atomic


x y

door is active

6 / 40



door key


x x

key is atomic

door is not atomic


x y

door is active

6 / 40



door key


x x

key is atomic

door is not atomic


x y

door is passive

6 / 40



door key


x x

key is atomic

door is not atomic


x y

door is passive

6 / 40



door key


x x

key is atomic

door is not atomic


x y

door is passive

6 / 40



door key


x x

key is atomic

door is not atomic


x y

door is passive

6 / 40



door key


x x

key is atomic

door is not atomic


x y

door is passive

6 / 40

Why another graph-based (meta)model?

Answer

Bigraphs allow for deriving good Labelled Transition System fromsimple Reactive Systems.Where good means the bisimulation on the LTS is a congruence!

How to do that?

By the Leifer-Milner approach: define and use only the minimalinteractions between the agent and the environment.

Our chosen model: Directed Bigraphs

+ They allow for the definition of LTS using the aboveconstruction.

+ They have some encoding methodologies for representing realworld systems and calculi.

7 / 40

Relative and Idem Pushouts [Leifer, Milner 2000]

The “minimality” can be elegantly expressed as a universalcategorical property.

f1f0

h1h0

g1g0

h

f1f0

h1h0

g1g0 h

j k1

k0

k

f1f0

h1h0

g1g0 id

(1) (2) (3)

Fig. 6. A candidate RPO (1), an RPO (2) and an IPO (3).

add v2 to h0 and connect its port to the edge e1 in f0, to do so, there are twopossibility to target e1: trough the name y or trough the name z.

7 Related and future work

Related work. The closest work is Birkedal et al. [1], where a language and amatching algoritm for binding bigraphs are presented. That work has been ofinspiration for our matching inference system, although we had to change quitemany details both in the rules and in the implementation due to the fact thatwe consider directed bigraphs and not binding bigraphs. Also, as far as we know,DBtk provides the first constructions of RPOs and IPOs for bigraphs.

Future work. The present system is still a prototype, and needs some refinementsto mature to a ready-to-use tool. Among the features that can be added, wemention a graphical user interface (GUI), where the user can “draw” interactivelya bigraph, a reaction rule and ultimately a whole BRSs. Another interestingdevelopment will be to consider stochastic bigraphic reactive systems, where rulesare endowed with rates. We think that the matching procedure given in thispaper can be successfully used within a Gillespie algorithm.

Acknowledgements. We thank our students Raul Pellarini, Patrik Osgnach, Fe-derico Chiabai for their help in developing the code.

References

1. L. Birkedal, T. C. Damgaard, A. J. Glenstrup, and R. Milner. Matching of bigraphs.ENTCS, 175(4):3–19, 2007.

2. D. Grohmann and M. Miculan. Directed bigraphs. In Proc. XXIII MFPS, ENTCS,173:121–137. Elsevier, 2007.

3. D. Grohmann and M. Miculan. Reactive systems over directed bigraphs. InProc. CONCUR 2007, LNCS 4703, pages 380–394. Springer-Verlag, 2007.

4. D. Grohmann and M. Miculan. An algebra for directed bigraphs. In Proceedings ofTERMGRAPH 2007, ENTCS 203(1):49–63. Elsevier, 2008.

5. O. H. Jensen, R. Milner. Bigraphs and transitions. In Proc. POPL, 38–49, 2003.6. J. J. Leifer and R. Milner. Transition systems, link graphs and petri nets. Mathe-

matical Structures in Computer Science, 16(6):989–1047, 2006.7. R. Milner. Bigraphical reactive systems. In Proc. 12th CONCUR, LNCS 2154, pages

16–35. Springer, 2001.

Call g0, g1 a bound for f0, f1 if g0 ◦ f0 = g1 ◦ f1.

1. A relative bound (h0, h1, h) for f0, f1 to g0, g1.

2. A relative pushout (RPO) (h0, h1, h) for f0, f1 to g0, g1: for anyother relative bound (k0, k1, k), there is a unique mediator j .

3. A idem pushout (IPO) g0, g1 for f0, g1: (g0, g1, id) is an RPOfor f0, f1 to g0, g1.

8 / 40

Relative Pushouts: a simple example

Processes

P ::= a | P1|P2

Contexts

C ::= − | C |P | P|C

a|b b|c

−|c |d |e a| − |d |e

−|c a|−

−|d |e−|c |d a| − |d

−|e

−|d

9 / 40


Processes

P ::= a | P1|P2

Contexts

C ::= − | C |P | P|C

a|b b|c

−|c |d |e a| − |d |e

−|c a|−

−|d |e−|c |d a| − |d

−|e

−|d

9 / 40


Processes

P ::= a | P1|P2

Contexts

C ::= − | C |P | P|C

a|b b|c

−|c |d |e a| − |d |e

−|c a|−

−|d |e−|c |d a| − |d

−|e

−|d

9 / 40


Processes

P ::= a | P1|P2

Contexts

C ::= − | C |P | P|C

a|b b|c

−|c |d |e a| − |d |e

−|c a|−

−|d |e−|c |d a| − |d

−|e

−|d

9 / 40


Processes

P ::= a | P1|P2

Contexts

C ::= − | C |P | P|C

a|b b|c

−|c |d |e a| − |d |e

−|c a|−

−|d |e−|c |d a| − |d

−|e

−|d

9 / 40

Idem Pushouts: a simple example

Processes

P ::= a | P1|P2

Contexts

C ::= − | C |P | P|C

a|b b|c

−|c a|−

−|c a|−

−

10 / 40

Idem Pushouts: a simple example

Processes

P ::= a | P1|P2

Contexts

C ::= − | C |P | P|C

a|b b|c

−|c a|−

−|c a|−

−

10 / 40

LTSs from RSs and IPOs

Given a reactive system (by means of a set of reaction rules)

an IPO transition aL−→ a′ is defined when

there exist a reaction rule (r , r ′) and anactive context D, such that (L,D) is anIPO for (a, r) and a′ = D ◦ r ′.

ar

L

D

r ′

a′

Labels are only the contexts which form an IPO, that is, theminimal completion context for a that allows a reaction to takeplace, that is L ◦ a = D ◦ r (and then L ◦ a rewrites into D ◦ r ′).

Theorem (Leifer, Milner 2000)

The bisimilarity derived from an IPO LTS is a congruence.

11 / 40

Outline


+ Directed Bigraphs





+ Algebra




12 / 40

Architecture of DBtk

Core Module

Language(DBL)

Drawing (SVG)LTS: RPOsand IPOs

Matchingcompiler

decompiler

visualizer

verification

simulation

13 / 40


Core Module

Language(DBL)


Matchingcompiler

decompiler

visualizer

verification

simulation

13 / 40

Core Module

The Core Module is divided in two sub-modules.

Data structures and operations

+ Data structures (i.e., objects) for representing directedbigraphs (and also place graphs and directed link graphs);.

+ Procedure for tensoring bigraphs by juxtaposing them.

+ Procedure for composing bigraphs by putting the roots of the“lower” one inside the holes of the “upper” one, and bypasting the same names.

Algebra

+ Constructor for elementary bigraphs.

+ Normalization procedure, which takes a bigraph and yields itback in discrete normal form.

+ Derived operations.

14 / 40

Core Module






Algebra




14 / 40

Tensor product

G0 : 〈0, (∅, ∅)〉 → 〈1, ({w}, {x})〉

v0

w

e0

x

v1

e1v2

G1 : 〈0, (∅, ∅)〉 → 〈1, (∅, {z , y})〉

yz

v3

G0 ⊗ G1 : 〈0, (∅, ∅)〉 → 〈2, ({w}, {x , z , y})〉

v0

w

e0

yzx

v1

e1v2

v3

API in Java:public class DirectedBigraph {

...

public DirectedBigraph

tensor(DirectedBigraph) {...}

...

}

G = G0.tensor(G1);

15 / 40

Composition

H : 〈2, ({w}, {x , z , y})〉 → 〈1, ({y}, ∅)〉

v4

v5

w

y

y

e5

x z

e4

G : 〈0, (∅, ∅)〉 → 〈2, ({w}, {x , z , y})〉

v0

w

e0

yzx

v1

e1v2

v3

H ◦ G : 〈0, (∅, ∅)〉 → 〈1, ({y}, ∅)〉

v0

v5

v5

e0

y

e5

v1

e1v2 e4

v3

API in Java:public class DirectedBigraph {

...

public DirectedBigraph

compose(DirectedBigraph) {...}

...

}

F = H.compose(G);

16 / 40

Core Module






Algebra




17 / 40

Elementary bigraphs

Barren root

1merge 0(special case of merge)

Merge

mergemerge 2(generalized to merge n)

0 1

Swap

γ@[1,0](generalized to @[3,4,1,0,2])

1 0

Closure

HNxy

x X y(generalized to [x,w] X [y,z]) y

x

Substitution

MyX

y/[x 1,...,x n](also y/x and y/)

y

x1 x2 . . . xn

. . .

Fusion

OYx

[y 1,...,y n]\x(also y\x and \x) x

y1 y2 . . . ym. . .

Ion

K~x~y (l)

K[-y 1,...,-y n,+x 1,...,+x m](the names in the list can be written in any order)

y1 y2 . . . yn

x1 x2 . . .xm. . .

. . .

18 / 40

Discrete Normal Form

A generic bigraph

x4 x0 x7 x6 x1

y1 y0

e0

e1

0v0

v1

x2 x3 x5

e2y2

v2

v3

v4

API in Javapublic class DirectedBigraphAlgebra {

...

public static NormalForm

toNormalForm(DIrectedBigraph) {...}

...

}

NF = DirectedBigraphAlgebra.toNormalForm(G);

Its Discrete Normal Form

x5 x4 x0 x7 x6 x1 x2 x3

w2 w0 w1 w5 w4 w3 w6 w8 w10 w9 w7 w11

e0 e1 e2

w2 w0 w1

z2 z0 z1

w5 w4 w3 w6 w8 w10 w9 w7 w11

z3

0v0

v1

v2

v3

v4

z2 z0 z1 z3

y2 y1 y0

19 / 40


Core Module

Language(DBL)


Matchingcompiler

decompiler

visualizer

verification

simulation

20 / 40

The Directed Bigraphical Language (DBL)

Signature

Signature [ACTIVITY NAME:#PORTS, ...];

Bigraphs

They are described in a functional-stylelanguage, which allows to compose (o)or tensor (*) elementary bigraphs.

y1

e3

e2

e0

e1

v4v5

v0

v1

v2v3

An example

Signature [active t4:1,atomic t5:1,atomic t2:2];let a = merge 1 | X[x,z] | (y1X[a,y] * t4[-w]) o

(w\ w * merge 1 | t5[+a] * y/y) inlet b = (((wX[b,c] * merge 1) o (b/b * t4[+c]) o

t2[+b,-d] | t2[+x,-e]) o ([d,e]X)) inlet c = t2 [+z,+y] in a o (b * c)

21 / 40


Core Module

Language(DBL)


Matchingcompiler

decompiler

visualizer

verification

simulation

22 / 40

Compiling from DBL into native data structures

What DBtk provides:

+ Expressions in DBL can be compiled in the internalrepresentation of bigraphs.

+ Internal data structures can be decompiled into DBLexpressions.

How they are implemented:

+ The compiler is constructed using Lex and Yacc tools.

+ The decompiler recovers the algrebraic DBL expression indiscrete normal form.

23 / 40


Core Module

Language(DBL)


Matchingcompiler

decompiler

visualizer

verification

simulation

24 / 40

Graphical Visualization

DBtk allows to represent bigraphs in the XML-based SVG format,which is the W3C open standard for vector graphics.

y1

e3

e2

e0

e1

v4v5

v0

v1

v2v3

0

v0

v1 v5

v2

v3 v4

y1

e3

e0

e1

e2

The translation method takes adirected bigraph data structureand returns a string containingits SVG representation.

API in Javapublic class Drawing {

...

public static String

dbg2svg(DirectedBigraph) {...}

...

}

System.out.println(dbg2svg(G));

(Fruitful in combination with the compiler: bigraphs are defined inDBL, compiled to the data structure and then converted in SVG.)

25 / 40


Core Module

Language(DBL)


Matchingcompiler

decompiler

visualizer

verification

simulation

26 / 40

How to apply a rule to an agent?

Agent

58 4. Matching

in quanto gli archi e le sostituzioni che non possono risiedere in d possono invece

risiedere all’interno del conteso C che compone con d attraverso i link Z+ lasciati

intatti dall’identita.

Sia a l’agente mostrato in Figura 4.1. Esso rappresenta una software-house dislo-

cata in due edifici distinti all’interno dei quali lavorano con i loro computer i dipen-

denti. I computer sono connessi tra loro attraverso una rete domestica che permette

la condivisione dei documenti (intesa come condivisione di parte del file-system, e

quindi di cartelle). In questa rappresentazione la condivisione dei documenti e resa

esplicita dal fatto che le cartelle sono connesse tra di loro attraverso dei link (archi o

nomi). I nodi folder rappresentano una copia locale della cartella condivisa e possono

contenere dei documenti (modificati in locale). Nell’esempio preso in considerazione,

quando due cartelle sono collegate ad un arco (e in figura), esse condividono in modo

esclusivo quella cartella, mentre se sono collegate attraverso un nome condiviso (x

in figura), queste ultime possono essere condivise anche da utenti esterni all’azienda.

a =

0

data

folder

computer

building

data

folder

computer

data

folder

data

folder

computer

buildinge

x

Figura 4.1: Agente bigrafico per la rappresentazione di un’azienda con rete condivisa.

Sia R ! R! la regola di reazione mostrata nella fascia centrale di Figura 4.2.

Tale regola codifica la situazione in cui uno di due computer all’interno della stessa

locazione (nel caso dell’agente a, all’interno dello stesso edificio) toglie la condivisione

della propria copia locale della cartella. Questa situazione viene codificata collegando

ad un arco (unario) la cartella, ovvero restringendo la condivisione della cartella ad

un solo utente. Per il redex R e possibile trovare un match all’interno dell’agente

a, ovvero e possiamo scomporre l’agente a in C " (R # id{z}+) " d, per qualche

contesto C e parametro discreto d. In Figura 4.2 e mostrata tale scomposizione,

inoltre viene mostrata la riscrittura dell’agente ovvero l’applicazione della regola di

reazione R ! R!. Si noti che questo e l’unico match possibile del redex in a, in

quanto c’e un solo edificio con almeno due computer.

Reaction rule

4.2. Regole di matching 59

R =

0

0 1 2 3

folder folder

computer computer

y

= R!

0

0 1 2 3

folder folder

computer computer

y

e!

0

data

folder

computer

building

data

folder

computer

building

0e

x

0 1 2 3

data data data

folder

0

data

folder

computer

building

data

folder

computer

building

0e

x

0 1 2 3

data data data

folder

C = = C

d = = d

Figura 4.2: Esempio di match di un redex R nell’agente a e riscrittura dell’agente:a = C ! (R" id{z}+) ! d e a! = C ! (R! " id{z}+) ! d.

4.2 Regole di matching

In questa sezione presentiamo una caratterizzazione induttiva per la ricerca di un

match, ovvero una caratterizzazione alternativa alla verifica dell’uguaglianza a =

C ! (R" idZ+) ! d, per induzione sulla struttura di a, R, C e d. Quella che andiamo

a presentare e una caratterizzazione precisa nel senso che e sia corretta che completa,

rispetto alla categoria dei bigrafi astratti. Questa tipologia di approccio ci permette

di analizzare con un buon livello di dettaglio il problema del matching, e ci da le

direttive per la specifica l’algoritmo che andremo a presentare tra poco (sezione 4.3).

Questa caratterizzazione si basa sulla forma normale discreta (DNF) dei bigrafi

diretti che permette di decomporre un qualunque bigrafo in sotto bigrafi piu semplici,

ed allo stesso tempo ci o!re un metodo standard per separare i concetti di localita

e di connettivita come sotto-grafi componibili. Quest’ultima considerazione risulta

molto utile nella definizione della caratterizzazione, in quanto essa riflettera questa

separazione “logica” tra localita e connettivita attraverso una notazione puramente

sintattica. Avvalersi della forma normale risulta comodo anche per la dimostrazione

della completezza e della correttezza della caratterizzazione stessa1 e sono utili anche

1Le dimostrazioni di questi due risultati esulano dalla trattazione di questo lavoro

27 / 40

Match the redex into the agent and replace it bythe reactum4.2. Regole di matching 59

R =

0

0 1 2 3

folder folder

computer computer

y

= R!

0

0 1 2 3

folder folder

computer computer

y

e!

0

data

folder

computer

building

data

folder

computer

building

0

z

e

x

0 1 2 3

data data data

folder

z

0

data

folder

computer

building

data

folder

computer

building

0

z

e

x

0 1 2 3

data data data

folder

z

C = = C

d = = d

Figura 4.2: Esempio di match di un redex R nell’agente a e riscrittura dell’agente:a = C ! (R" id{z}+) ! d e a! = C ! (R! " id{z}+) ! d.

4.2 Regole di matching

In questa sezione presentiamo una caratterizzazione induttiva per la ricerca di un

match, ovvero una caratterizzazione alternativa alla verifica dell’uguaglianza a =

C ! (R" idZ+) ! d, per induzione sulla struttura di a, R, C e d. Quella che andiamo

a presentare e una caratterizzazione precisa nel senso che e sia corretta che completa,

rispetto alla categoria dei bigrafi astratti. Questa tipologia di approccio ci permette

di analizzare con un buon livello di dettaglio il problema del matching, e ci da le

direttive per la specifica l’algoritmo che andremo a presentare tra poco (sezione 4.3).

Questa caratterizzazione si basa sulla forma normale discreta (DNF) dei bigrafi

diretti che permette di decomporre un qualunque bigrafo in sotto bigrafi piu semplici,

ed allo stesso tempo ci o!re un metodo standard per separare i concetti di localita

e di connettivita come sotto-grafi componibili. Quest’ultima considerazione risulta

molto utile nella definizione della caratterizzazione, in quanto essa riflettera questa

separazione “logica” tra localita e connettivita attraverso una notazione puramente

sintattica. Avvalersi della forma normale risulta comodo anche per la dimostrazione

della completezza e della correttezza della caratterizzazione stessa1 e sono utili anche

1Le dimostrazioni di questi due risultati esulano dalla trattazione di questo lavoro

28 / 40

A more formal definition of match

Matching sentence

ωa, ωR , ω′C , ωC ` a,R ↪→ C , d

=a

ωa ⊗ id

d

RidX +

ωR ⊗ id

ω′C ⊗ id

idY +

CidV -

ωC ⊗ id

idZ+

C ′

(context)

R ′

(redex)

a′

(agent)

29 / 40

How the algorithm works?

Idea

1. Creation of a match object for the pair (agent,redex).

2. At each invocation of the method next a new match iscomputed (if it exists).

3. Backtracking is implemented by using two stacks.

6oloclacidomtiroglA.3.4

Matcher Object

Stack trace

Stack current

Figura 4.4: Un oggetto matcher

possibili applicazioni. Sullo stack trace verranno inseriti oggetti di tipo rule dotati

di un metodo !ll current per generare tutte le possibili applicazioni della regola che

verranno salvate sullo stack current come coppie (rule op,matcher) dove rule op e

il tipo dell’oggetto che realizza l’operazione che dovra essere eseguita sulle soluzioni

fornite dal matcher con cui e accoppiata.

In !gura 4.4 gli stack trace e current vengono rappresentati con i due riquadri

rosa ed azzurro rispettivamente. Gli oggetti rule contenuti nel trace vengono rappre-

sentati con dei rombi, mentre le coppie (rule op,matcher) come un rettangolo e ed

un cerchio rosso racchiusi dentro lo stesso elemento. Si noti inoltre che un oggetto

matcher e formato da tutte le componenti racchiuse dentro la linea verde, quindi a

loro volta anche i matcher formano un albero in cui i nodi sono oggetti matcher e i

rami ne indicano i relativi sotto-matcher.

Ora vediamo uno schema generale di funzionamento dell’algoritmo adottato:

viene creato un nuovo matcher con parametri di ingresso i due bigra! agente

API in Javapublic class Matcher {

...

public Matcher(DirectedBigraph,

DirectedBigraph) {...}

...

public Match next() {...}

...

}

30 / 40

Another example with two possible matches - I

Agent

92 4. Matching

In figura 4.6 vengono mostrati gli unici due match presenti nel nostro esempio, in-

fatti alla terza chiamata del metodo next viene restituita un’eccezione No more match.

Si noti come ricomponendo le tre componenti C, R, d, per entrambe le soluzioni, si

ottiene sempre l’agente a, a dimostrazione della correttezza del match ottenuto.

e0

e1

e3

e2

y

a =

0

Figura 4.5: Agente e Redex utilizzati nell’esempio.

Redex

4.5. Esempi 93

R =

0 1

0

yx

C =

0

0

1

e1

e0

e3

e2

z0 z1 z2

y

d =

0

z0 z1 z2

(Primo match)

= R

0 1

0

yx

= C

0

0 1

e3

e0

e1

e2

y

= d

0

(Secondo match)

Figura 4.6: Le due occorrenze di match del redex R sull’agente a di Figura 4.5.

31 / 40

Another example with two possible matches - II

4.5. Esempi 93

R =

0 1

0

yx

C =

0

0

1

e1

e0

e3

e2

z0 z1 z2

y

d =

0

z0 z1 z2

(Primo match)

= R

0 1

0

yx

= C

0

0 1

e3

e0

e1

e2

y

= d

0

(Secondo match)

Figura 4.6: Le due occorrenze di match del redex R sull’agente a di Figura 4.5.

result of the first next() result of the second next()

32 / 40


Core Module

Language(DBL)


Matchingcompiler

decompiler

visualizer

verification

simulation

33 / 40

RPO computation

1. Given a commuting square, i.e., a span and a bound, create aRPO object.

2. The method get yields back the corresponding RPO triple.yields back the RPO triple.

3. Notice that the construction is deterministic so only one RPOcan exit for a given square.

API in Javapublic class DirectedBigraphRPO {

...

public DirectedBigraphRPO(DirectedBigraph,

DirectedBigraph,

DirectedBigraph,


...

public RPO get() {...}

...

}

34 / 40

RPO construction example

!4, ({x, y}, {z})"

!2, ({a, b, c, g}, {d, f})"

!3, ({o, p, q}, {l, m, n})" !4, ({k, s, t, u, v, w}, {h})"

!3, ({(o, t), (o, u), (p, t), (p, u), (q, k), v, w}, {[n, h], m})"

0

0

v11

z

l n

1

v22

y

m

2

3

e0

x

q po

0

0

v1

y

s h

1

1

k

x

2

23

e1

z

v w

3

v0

e0

tu

0

0

n m p

[n, h]m (p, t) (p, u)

1

1

2

e1

lq

v w(q, k)

2

v0

o

(o, t)(o, u)

0

0

1

hu

[n, h](p, u) (o, u)

1

2v2

v ws

v wm

2

3

k t

(p, t)(o, t)(q, k)

0

0e2

[n, h]m (o, t) (o, u) w

d ag

1

12

v3

e3

v (p, t) (p, u) (q, k)

bcf

0

0

e1

e2

e3

n m l

d ab c

1

1

2v0

v3

o p q

g f

0

0

1

e2

e3

uw h

ga b c

1

2

3v2v3

v s t k

d f

Fig.7.

An

example

ofan

RP

Oconstruction.

35 / 40

IPO computation

1. Given a span (i.e., two graphs with common inner interface),construct an IPO object.

2. For the fact that many different IPOs can exist for a span, thenon-determinism is managed with stacks as done for thematching.

3. Every time a call to the method next is performed a new IPOis computed (if it is exists).

API in Javapublic class DirectedBigraphIPO {

...

public DirectedBigraphIPO(DirectedBigraph,


...

public IPO next() {...}

...

}

36 / 40

IPO construction example

f0 =

0

v1

x y z

e1

= f1

0

v1 v2

v3

w u t

e1

h0 =

0

x y z

v2v3

0

u x yt zt

= h1

0

w u t

zt ytux

0

!0, (", ")#

!1, ({y, z}, {x})# !1, ({t}, {w, u})#

!1, ({yt, zt}, {x, u})#

f0 =

0

v1

x y z

e1

= f1

0

v1 v2

v3

w u t

e1

h0 =

0

x y z

v2v3

0

u x yt zt

= h1

0

w u t

zt ytux

0

!0, (", ")#

!1, ({y, z}, {x})# !1, ({t}, {w, u})#

!1, ({yt, zt}, {x, u})#

Fig. 8. An example: two IPOs are possible for the span f0, f1, one connects v2

to y and the other v2 to z.

result ofthe first next()

result ofthe second next()

f0 =

0

v1

x y z

e1

= f1

0

v1 v2

v3

w u t

e1

h0 =

0

x y z

v2v3

0

u x yt zt

= h1

0

w u t

zt ytux

0

!0, (", ")#

!1, ({y, z}, {x})# !1, ({t}, {w, u})#

!1, ({yt, zt}, {x, u})#

f0 =

0

v1

x y z

e1

= f1

0

v1 v2

v3

w u t

e1

h0 =

0

x y z

v2v3

0

u x yt zt

= h1

0

w u t

zt ytux

0

!0, (", ")#

!1, ({y, z}, {x})# !1, ({t}, {w, u})#

!1, ({yt, zt}, {x, u})#

Fig. 8. An example: two IPOs are possible for the span f0, f1, one connects v2

to y and the other v2 to z.

37 / 40

Conclusion and Future Works

What we have done

+ APIs for managing directed bigraphical structures, i.e., algebra+ operations.

+ A compiler, a decompiler, and a visualizer in SVG.

+ APIs for computing a Matching between two bigraphs.

+ APIs for calculating RPOs and IPOs.

Future works

+ Integration of the toolkit into Eclipse (www.eclipse.org/).

+ A (stochastic) simulation environment based on matching.

+ Tools for verification and behavioral reasoning, like modelchecking and bisimulations, based on IPO derived LTS.

+ A better language not based on bigraph algebra.

+ A more powerful and intuitive visual interface.

38 / 40

www.eclipse.org/

Why should graphs be depicted only in 2D?

39 / 40

Thank you

DBtk is available at

http://sole.dimi.uniud.it/~davide.grohmann/dbtk/

40 / 40

http://sole.dimi.uniud.it/~davide.grohmann/dbtk/

DBtk: a toolkit for directed bigraphs

Documents