Complete Axioms for Stateless Connectors joint work with Roberto Bruni and Ugo Montanari Dipartimento di Informatica Università di Pisa Ivan Lanese Dipartimento

Complete Axioms for Stateless Connectors

joint work withRoberto Bruni and Ugo MontanariDipartimento di Informatica Università di Pisa

Ivan LaneseDipartimento di Informatica Università di Pisa

CALCO 2005, Swansea, Wales, UK, 3-6 September 2005

CALCO 2005, 3-6 September, Swansea, Wales, UK Bruni, Lanese, Montanari

Roadmap Why connectors? The tile model Stateless connectors Axiomatization of synch-

connectors Adding mutual exclusion Concluding remarks





Interaction and connectors Modern systems are huge

composed by different entities that collaborate to reach a common goal

interactions are performed at some well-specified interfaces…

…and are managed by connectors Connectors allow separation

between computation and coordination


Coordination via connectors Connectors useful to

ensure compatibility among independently developed components

allow to reuse them allow run-time reconfiguration

Connectors exist at different levels of abstraction (architecture, applications, …)


Which connectors? We follow the algebraic approach

system as term in an algebra We propose an algebra of simple

stateless connectors for synchronization and mutual exclusion expressive enough to model the architectural

connectors of CommUnity [IFIP TCS 04] build on symmetric monoidal categories and

P-monoidal categories related to Stefanescu’s flow algebras and REO

connectors





The tile model Operational and

observational semantics of open concurrent systems compositional in space

and time Category based


parallelcomposition

Configurations

inputinterface

outputinterface

sequentialcomposition


Configurations

inputinterface

outputinterface

parallelcomposition


functoriality


Configurations

inputinterface

outputinterface

parallelcomposition


functoriality+

symmetries=

symmetric monoidal cat


Observations

initialinterface

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


Tiles as LTS Structural equivalence

axioms on configurations (e.g. symmetries) LTS

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

Observational semantics tile trace equivalence/bisimilarity congruence results for suitable formats





Connectors Connectors to express synchronization and

mutual exclusion constraints on local choices

Possible outcomes: tick (1, action performed) or untick (0, action forbidden)

Operational semantics via tiles and observational semantics via tile bisimilarity

Denotational semantics via tick-tables (boolean matrices)

Complete axiomatization of connectors and reduction to normal form


Basic connectors

! !

0 0

Symmetry

Duplicator

Bang

Mex

Zero


Notation Only two kinds of allowed observations Initial and final states always coincide

(since connectors are stateless) Thus we can use a “flat” notation for

tiles

1 0


Operational semantics Tiles specify the behaviours of

basic connectors When composed, connectors must

agree on the observation at the interfaces


Basic tiles (I)

Dual connectors have dual tiles


Basic tiles (II)

!

!

0


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

Denotations are just matrixes n inputs 2n rows m outputs 2m columns dual is transposition sequential composition is matrix multiplication parallel composition is matrix expansion

cells are filled with empty/copies of matrices

…

0101

0010

…

…111001…

Denotational semantics

12

3

4

12

3

12

3

12

3

4

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


Denotational semantics

1

0

111001001

0

10

11

10

01

00

11100100

1

0

.

!

1

0

11100100

1

0

.

0


Semantic correspondance Tile bisimilarity coincides with tile

trace equivalence (stateless property)

Two connectors are tile bisimilar iff they have the same associated tick-tables

Tile bisimilarity is a congruence





Axiomatization We want to find a complete axiomatization

for the bisimilarity of connectors Synch-connectors (without mex and zero)

symmetries, duplicators and bangs form a gs-monoidal category

adding dual connectors we get a P-monoidal category

No simple known axiomatization works for mex, but we show an axiomatization for the full class of connectors


Gs-monoidal axioms

=

=

=

=!


Additional P-monoidal axioms

=

=

=!! .


Synch-tables

Entry with empty domain is enabled Entries are closed under (domains)

union intersection difference complementation

Base: set of minimal (non empty) entries w.r.t. domain intersection

Each synch-table is determined by its base


Normal form

Sort connectors

!

!

… …

…

…

Central points (correspond to cells of

the base)

Hiding connectors

directly connected to central points

Central points are connected to at least one external interface


Properties

All the axioms bisimulate (correctness)

Each connector can be transformed in normal form using the axioms

Bijective correspondance between synch-tables and connectors in normal form





Adding mex and zero

Synch-connectors are not expressive enough (only synchronization)

Adding mex and zero to express mutual exclusion constraints and enforce inactivity

Just mex has to be inserted: zero and dual connectors can be derived

Mex and zero form a gs-monoidal category


Obtaining zero connector

=

1

0

10

1

0

11100100

01

00

10

11

10

=x

= !def

1

0

.0

0


Obtaining comex connector

=!

!

!

Hiding and synchronization allow to flip wires


Looking for axiomatization of mex

=



=



=



=



=


Key axioms


Key axioms

= !

! = !

!

!


Some axioms about mex-dup

=

=


Some axioms about zero

=

0

0

= =0

0=0

0

=0

!0! 0 = . =

0


A sample proof

00 = ! 0

= !

0

0

= .

!=0


Additional axioms

= !


An axiom scheme

!

!

!… …


An axiom scheme

!

…!


Entry with empty domain is enabled All the tables with that property can

be expressed Generalized sorted and normal form

Full tables


Full tables

Zeros directly connected to free

variables

……

……

0

0… …

0

0

!

!!

!


Full tables

Hiding connected to roots of mex or to central points

……

……

0

0… …

0

0

!

!!

!


Full tables

Each hidden variable is connected to at most two central

points

……

……

0

0… …

0

0

!

!!

!


Full tables

At most one path between a

central point and a variable

……

……

0

0… …

0

0

!

!!

!


Full tables

No hidden variables are

connected to the same central points

……

……

0

0… …

0

0

!

!!

!


Full tables

No two central points have the

same set of variables

……

……

0

0… …

0

0

!

!!

!


Full tables

Each central point is

connected to at least a free

variable

……

……

0

0… …

0

0

!

!!

!


Full tables

Each pair of central points share at least

a variable

……

……

0

0… …

0

0

!

!!

!


Full tables

Hidden variables attached to roots of mex are on the left

……

……

0

0… …

0

0

!

!!

!


Properties

Full extension of the properties of synch- connectors all the axioms bisimulate each connector can be transformed in

normal form using the axioms bijective correspondance between tables

and connectors in normal form More complex axiomatization and

normalization





Conclusions

Full correspondences between observational semantics denotational semantics equivalence classes modulo axioms

Normalization allows to find a standard representative


Axiomatization and colimits

In [IFIP TCS 04] connectors used to model CommUnity

Translation of a diagram is isomorphic to the translation of the colimit

Now: translation of a diagram is equal up to the axioms to the translation of the colimit

Furthermore normalization allows to algebraically compute the colimit


Comparison with REO connectors

REO connectors add directionality and data flow

For synchronization purposes the two kinds of connectors are almost equivalent

REO connectors allow some state (buffers) and some priority among configurations (LossySync)

Algebraic theory of REO connectors less developed (as far as we know)


Future work

Open problem: does a finite axiomatization exist? maybe Wan Fokkink techniques

Extend the results to larger classes of connectors actions ruled by a synchronization

algebra (instead of just 0 and 1) REO connectors probabilistic connectors


Documents

Complete Axioms for Stateless Connectors joint work with Roberto Bruni and Ugo Montanari Dipartimento di Informatica Università di Pisa Ivan Lanese Dipartimento