Complete Axioms for Stateless Connectors

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Complete Axioms for Stateless Connectors
CALCO 2005, Swansea, Wales, UK, 3-6 September 2005
Ivan Lanese Dipartimento di Informatica
Università di Pisa
joint work with Roberto Bruni and Ugo
Montanari Dipartimento di Informatica Università
di Pisa
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Roadmap

• Why connectors?
• The tile model
• Stateless connectors
• Axiomatization of synch-connectors
• Adding mutual exclusion
• Concluding remarks

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Roadmap

• Why connectors?
• The tile model
• Stateless connectors
• Axiomatization of synch-connectors
• Adding mutual exclusion
• Concluding remarks

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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

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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, )

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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 Stefanescus flow algebras and REO
  connectors

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Roadmap

• Why connectors?
• The tile model
• Stateless connectors
• Axiomatization of synch-connectors
• Adding mutual exclusion
• Concluding remarks

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The tile model

• Operational and observational semantics of open
  concurrent systems
• compositional in space and time
• Category based

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Configurations
output interface
input interface
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Configurations
output interface
input interface
parallel composition
functoriality
sequential composition
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Configurations
output interface
input interface
parallel composition
functoriality symmetries symmetric monoidal
cat
sequential composition
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Observations
initial interface
final interface
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Tiles

• Combine horizontal and vertical structures
  through interfaces

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Tiles

• Compose tiles
• horizontally

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Tiles

• Compose tiles
• horizontally
• (also vertically and in parallel)

symmetric monoidal double cat
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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

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Roadmap

• Why connectors?
• The tile model
• Stateless connectors
• Axiomatization of synch-connectors
• Adding mutual exclusion
• Concluding remarks

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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

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Basic connectors

• Symmetry

Duplicator
Bang
Mex
Zero
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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

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Operational semantics

• Tiles specify the behaviours of basic connectors
• When composed, connectors must agree on the
  observation at the interfaces

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Basic tiles (I)
Dual connectors have dual tiles
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Basic tiles (II)
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Denotational semantics

• 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

1
1
1
1
2
2
2
2
3
3
3
3
4
4
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Denotational semantics
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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

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Roadmap

• Why connectors?
• The tile model
• Stateless connectors
• Axiomatization of synch-connectors
• Adding mutual exclusion
• Concluding remarks

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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

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Gs-monoidal axioms
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Additional P-monoidal axioms
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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

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Normal form
Central points (correspond to cells of the base)

• Sort connectors

Hiding connectors directly connected to central
points
Central points are connected to at least one
external interface
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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

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Roadmap

• Why connectors?
• The tile model
• Stateless connectors
• Axiomatization of synch-connectors
• Adding mutual exclusion
• Concluding remarks

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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

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Obtaining zero connector

def

!
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Obtaining comex connector
!

!
!

• Hiding and synchronization allow to flip wires

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Looking for axiomatization of mex

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Looking for axiomatization of mex

?
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Looking for axiomatization of mex

?
?
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Looking for axiomatization of mex

?
?
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Looking for axiomatization of mex

?
?
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Key axioms
?
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Key axioms

!
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Some axioms about mex-dup

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Some axioms about zero
0

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A sample proof
0
0
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Additional axioms

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An axiom scheme
!


!
!
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An axiom scheme
!

!
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Full tables

• Entry with empty domain is enabled
• All the tables with that property can be
  expressed
• Generalized sorted and normal form

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Full tables
Zeros directly connected to free variables
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Full tables
Hiding connected to roots of mex or to central
points
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Full tables
Each hidden variable is connected to at most two
central points
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Full tables
At most one path between a central point and a
variable
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Full tables
No hidden variables are connected to the same
central points
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Full tables
No two central points have the same set of
variables
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Full tables
Each central point is connected to at least a
free variable
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Full tables
Each pair of central points share at least a
variable
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Full tables
Hidden variables attached to roots of mex are on
the left
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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

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Roadmap

• Why connectors?
• The tile model
• Stateless connectors
• Axiomatization of synch-connectors
• Adding mutual exclusion
• Concluding remarks

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Conclusions

• Full correspondences between
• observational semantics
• denotational semantics
• equivalence classes modulo axioms
• Normalization allows to find a standard
  representative

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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

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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)

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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

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Thanks!
Questions?