Connectors and Concurrency joint work with Ugo Montanari Roberto Bruni Dipartimento di Informatica Università di Pisa Dagstuhl Seminar #04241, September

Connectors and Concurrency

joint work withUgo Montanari

Roberto BruniDipartimento di Informatica Università di Pisa

Dagstuhl Seminar #04241, September 2003

CommUnity, Tiles and Connectors

Connectors and Concurrency

Connectors modeling various kind of distributed systems and

causal histories Tile Logic

combining the use of connectors in space and time Connectors + Tiles = Dynamic Connectors

Concurrent Causal Semantics Case study:

Basic process algebra for which tile bisimilarity coincides with causal weak bisimilarity of Darondeau & Degano

Dynamic connectors for concurrency, TCS 281(1-2):131-176, 2002


joint work withJosé Luiz FiadeiroIvan LaneseAntónia LopesUgo Montanari

Roberto BruniDipartimento di Informatica Università di Pisa

Dagstuhl Seminar #04241, 6-11 June 2004

Ongoing

Work

!


Roadmap

Motivation Background

CommUnity Tiles

From CommUnity to Tiles Connectors Concluding remarks


Roadmap


CommUnity Tiles



Motivation I Categorical Approach

objects are system components (architectural elements)

morphisms express simulation, refinement, … complex systems are modeled as diagrams composition via universal construction (colimit) suited for systems with shared resources

memory, action, channels


Motivation II Categorical Approach Algebraic Approach

initial algebra where constants are basic processes operations compose smaller processes into larger ones structural axioms collapse structurally equivalent

processes operational semantics (SOS style) abstract semantics (bisimilarity) suited for message passing calculi

communication encoded in the labels of the LTS


Motivation III Categorical Approach Algebraic Approach Reconcile two selected representatives

CommUnity categorical approach applied to program design

Tile Model co-existence of horizontal (space) and vertical (time)

dimensions Advantages

transfer of concepts and techniques


Roadmap


CommUnity Tiles



CommUnity Prototype architectural description

language conceptual distinction between

computation coordination

(in communicating distributed systems) System configurations as diagrams

Components compute locally Interactions as architectural connectors


CommUnity, graphically

Program Colimit

Morphisms“Denotational Semantics”

s


CommUnity Programs

design foo isin x, zout v, ndo a: true v:= x+z | n:=v+x[] b: n>MIN n:=n-x[] c: v<MAX v:=n+z

input/output channels

actionsguards concurrent assignments

a special actions always present: : true skip


Morphsims

channels of P1 to channels of P2

actions of P2 to actions of P1

P1

P2

actions of P1

correspond todisjoint sets of actions of P2

output channels cannot be merged

(actions names are not important)

we shall focus on actions coordination


Exampledesign P1 isin … out …do a …[] b …

design P2 isin … out …do f …[] g …[] h …

design P3 isin … out …do p …[] q …[] r …[] s …

f,g a p,q,r

h b s

design P isin … out …do f|a|p …[] f|a|q …[] f|a|r …

[] g|a|p …[] g|a|q …[] g|a|r …

[] h|b|s …


Star-Shaped Diagrams

roles

cablesno output channels

actions are true skip

glue


Tile Model Operational and Abstract Semantics of

Open Concurrent Systems Compositional in Space and Time Specification Formats

Depend on chosen connectors Congruence results

Category based


parallelcomposition

Configurations

inputinterface

outputinterface

sequentialcomposition

(interfaces can be typed)


Configurations

inputinterface

outputinterface

parallelcomposition


functoriality


Configurations

inputinterface

outputinterface

parallelcomposition


functoriality+

symmetries=

symmetric monoidal cat


Observationsinitial

interface

finalinterface

concurrent

computation


Tiles Combine horizontal and vertical

structures through interfaces

initial configuration

final configuration

trigger effect


Tiles Compose tiles

horizontally


Tiles Compose tiles

horizontally (also vertically and in parallel)

symmetric monoidal double cat


Operational and Abstract Semantics

Structural equivalence Axioms on configurations (e.g. symmetries)

LTS states = configurations transitions = tiles labels = (trigger,effect) pairs

Abstract semantics tile bisimilarity (congruence results for suitable formats)


Roadmap


CommUnity Tiles



Notation Since:

Initial and Final configuration are always equal We shall consider only two kinds of observations

We use a “flat” notation for tiles1 0


From CommUnity to Tiles Aim:

To define the operational and abstract semantics of CommUnity diagrams by translating them to tiles

First step Standard decomposition of programs and diagrams

identify basic building blocks make the translation easier

Second step From standard diagrams to tiles

basic programs to basic boxes cables, glues and morphisms modeled as connectors


Results The translation of a diagram is tile bisimilar to

the translation of its colimit [IFIP-TCS 2004] Corollary: Diagrams with the same colimit are

mapped to tile bisimilar configurations Colimit axiomatization [Ongoing Work!]

add suitable axioms for connectors axioms bisimulate existence of normal forms … hopefully!

the translation of a diagram is equal up-to-axioms to the translation of its colimit


Why Standard Decomposition

Decomposition is part of the translation The translation becomes easier It is necessary to achieve Colimit

Axiomatization in the colimit, conditional assignments are

synchronized according to the diagram the axiomatization of connectors cannot change

the number of computational entities decomposition is necessary to guarantee that the

number of computational entities in the diagram is the same as the number of computational entities in the colimit


Standard Decomposition Illustrated•n output channels•m actions

•n channel managers•m guard managers•n+m cables•1 glue

P


Channel Manager

channelmanager

for x

i

i

i

i

i

i

o

one channel manager for each output variable x of P

actions

one actiontrue x:= …

for eachassignement

input channels

variables neededin assignments

output channelchannel for x


Guard Manager

guardmanager

for a

i

i

i

i

i

i

no output channel

one guard manager for each action a of P

one action

Q skip

input channels

variables neededin the guard


Glue

i

i

i

i

i

i

no output channel

input channels

all channels of P

actions

one actiontrue skip

for eachaction of P

one glue


Cables (morphisms are obvious)

one cable for each manager

i

i

i

i

i

i

no output channel

input channels

one input channelfor each

channel of the role

actions

one actiontrue skip

for eachaction of the role


Scheme of translation

channel manager

channel manager

guard manager

guard manager

……

actionsynchronization

throughconnectors

channelfusion

state


Tiles for components

channelmanager

for x

i

i

i

i

i

i

o

channelmanager

for x

i

i

i

i

i

i

o

channelmanager

for x

i

i

i

i

i

i

o

…

actions of managers are mutually exclusive, butat least one action (e.g. ) must be executed


Roadmap


CommUnity Tiles



Connectors Connectors are used to model diagram

morphisms, cables and glues they express constraints on local choices

Connectors with the same abstract semantics should be identified different ways of modeling the same

constraint How?

Axioms + reduction to normal form


Abstract semantics Connectors can be seen as black boxes

input interface output interface admissible signals on interfaces

12

3

4

12

3

12

3

12

3

4


Connectors can be seen as black boxes input interface output interface admissible signals on interfaces

Abstract semantics is just a matrix n inputs 2n rows m outputs 2m columns (not all matrix are interesting) sequential composition is matrix multiplication parallel composition is matrix expansion

cells are filled with empty/id copies of matrices

…

0101

0010

…

…111001…

Abstract semantics

12

3

4

12

3

12

3

12

3

4

domain is{input 3, outputs 1,2,3}


An example: Symmetries

11

10

01

00

11100100 (also identities should be considered)

=

connectors boxes are immaterial


AND-Connectors

00 01 10 11

0

1

=

=


AND-Tables andNormal Form

n = m = 0 or 0 < n m entry with empty domain is enabled

only one entry with empty input/outputs entries are closed under (domains)

union intersection difference complementation

at most one entry enabled for every row/column exactly one entry for every row

… …


co-AND-Connectors

01

00

10

=

11

10

=

(as AND-table, but transposed)


Mixed-AND-Connectors

=

=



=

=



=

=

=



=

=

=



=

=

=


n = m = 0 or 0 < n,m entry with empty domain is enabled




Mixed-AND-Tables andNormal Form

… ……


(co-)HIDE-Connectors!

!

1

0

.

.

10

!

!

=!

=

!!

=!! .


A Relevant Difference

!

!

!01

00

.

11

10

01

00

.

11

10

!

!

!


A Sample Proof!

!

!

!

!

=

=

=


SYNC-Tables and Normal Form





!

!

… ……

…


(co-)MEX-Connectors

00 01 10 11

0

1

=

=


Mixed-MEX-Connectors

=



=



=



=



=


(co-)ZERO-Connectors

=

1

0

10

1

0

11100100

01

00

10

11

10

=x

= !0def

.

10

0


Some Axioms About ZERO

=

0

0

= =0

0=0

0

=0

!0! 0 = . =

0


A Sample Proof

00 = ! 0

= !

0

0

= .

!=0


Some Axioms About MEX-AND

=

=


Key Axioms


Key Axioms

= !

! = !

!

!


An Axiom Scheme

!

! !

!

… …


An Axiom Scheme

! !

! !

… …






FULL-Tables and Normal Form

!

……

……

!

…

…

…!


(co-)ONE-Connector

.

10

1

All tables can now be defined, but “negation” is introduced… … and “inconsistent” connectors with it!

ONE is not needed for CommUnity

!

1

=def


Roadmap


CommUnity Tiles



Concluding Remarks We have learned that reconciling the

categorical and algebraic approach is not an easy task…

We got some insights on further extensions more labels, weights

Related to constraint solving flownomials (Cazanescu, Stefanescu), bicategories

of relations (Carboni, Walters), gs-monoidal cats (Corradini, Gadducci), sharing graphs (Hasegawa), action structures and named calculi (Milner, Gardner)

Documents

Connectors and Concurrency joint work with Ugo Montanari Roberto Bruni Dipartimento di Informatica Università di Pisa Dagstuhl Seminar #04241, September