community, tiles and connectors joint work with roberto bruni josé luiz fiadeiro antónia lopes ugo...

CommUnity, Tiles and Connectors

joint work withRoberto BruniJosé Luiz FiadeiroAntónia LopesUgo Montanari

Ivan LaneseDipartimento di Informatica Università di Pisa

Agile Meeting, 8-9 July 2004

Ongoing

Work

!


Roadmap Motivation From CommUnity

to Tiles Connectors Concluding

remarks


General motivation I Comparing the categorical and the algebraic

approach to systems Categorical approach Algebraic approach


General motivation II Comparing the categorical and the algebraic

approach to systems Categorical approach

objects are system components morphisms express simulation, refinement, … complex systems are modeled as diagrams composition via universal construction (colimit)

Algebraic approach


General motivation III Comparing the categorical and the algebraic

approach to systems Categorical approach Algebraic approach

System represented by an algebra constants are basic components operations compose smaller systems into larger ones structural axioms collapse structurally equivalent

systems operational semantics (SOS style) abstract semantics (bisimilarity)


Specific aim Reconcile two selected representatives

CommUnity (categorical) architectural description language distinction between computation and

coordination Tile Model (algebraic)

operational model for concurrent systems co-existence of horizontal (space) and vertical

(time) dimensions


Specific aim: advantages Advantage: transfer of concepts and

techniques Semantics model for CommUnity Observational equivalence of CommUnity

configurations CommUnity-like connectors in the tile

model Separation between computation and

coordination for tiles




remarks


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 (coordination) 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 the translation of a diagram is equal up-to-

axioms to the translation of its colimit axioms bisimulate (correctness) existence of normal forms … hopefully!


Why Standard Decomposition

Decomposition is part of the translation Decomposition is necessary to achieve

colimit axiomatization the axiomatization of connectors cannot

change the number of computational entities

decomposition allows to have the same number of computational entities in the diagram and in the colimit


Standard Decomposition Illustrated

glue

cables

Channel / guard managers

P


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 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

Morphisms are obvious


Scheme of the translation

channel manager

channel manager

guard manager

guard manager

……

actionsynchronization

throughconnectors

channelfusion

state


Notation Since:

In CommUnity initial and final configuration are always equal (apart from values in state)

We shall consider only two kinds of observations action performed / action not performed

We use a “flat” notation for tiles

1 0


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




remarks


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 input/output swap is transposition 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

=

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/output is enabled 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

=

(transpositionof AND table)


Mixed-AND-Connectors

=

=


Mixed-AND-Connectors

=

=

=


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




Mixed-AND-Tables andNormal Form

… ……


HIDE-Connectors (and co-)!

!

1

0

.

.

10

!

!

=!! .

=!

=

!


A Relevant Difference

!

!

!01

00

.

11

10

01

00

.

11

10

!

!

!


A Sample Proof!

!

!

!

!

=

=

=


SYNC-Tables and Normal Form





!

!

… ……

…


MEX-Connectors

00 01 10 11

0

1

=

=


co-MEX-Connectors

Transposed tables Symmetric axioms


Mixed-MEX-Connectors

=


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

!

!

!

……

!

!

……

=n n



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



Finally (almost) FULL-Tables


FULL-Tables and Normal Form

!

……!

!

!

……

……

!

!


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




remarks


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


Future work Still one open problem

is the axiom schema really needed? On the mapping from CommUnity to tiles

locations reconfigurations refining

More in general what happens if we choose other algebraic

and categorical formalisms?

community, tiles and connectors joint work with roberto bruni josé luiz fiadeiro antónia lopes ugo...

Documents

tiles connectors

connectors tiles

connectors cables

connectors glue

community initial

axiomatization of connectors

connectors notation

connectors scheme