CommUnity, Tiles and Connectors

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CommUnity, Tiles and Connectors
Agile Meeting, 8-9 July 2004
Ivan Lanese Dipartimento di Informatica
Università di Pisa
Ongoing Work!
joint work with Roberto Bruni José Luiz
Fiadeiro Antónia Lopes Ugo Montanari
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Roadmap

• Motivation
• From CommUnity to Tiles
• Connectors
• Concluding remarks

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Roadmap

• Motivation
• From CommUnity to Tiles
• Connectors
• Concluding remarks

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General motivation I

• Comparing the categorical and the algebraic
  approach to systems
• Categorical approach
• Algebraic approach

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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

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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)

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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

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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

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Roadmap

• Motivation
• From CommUnity to Tiles
• Connectors
• Concluding remarks

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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

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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!

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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

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Standard Decomposition Illustrated
P
glue
cables
Channel / guard managers
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Standard Decomposition Illustrated

• n output channels
• m actions

P

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

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Channel Manager
one channel manager for each output variable x of
P
channel manager for x
i
i
i
i
i
i
o
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Guard Manager
one guard manager for each action a of P
guard manager for a
i
i
i
i
i
i
no output channel
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Glue
one glue
i
i
i
i
i
i
no output channel
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Cables
one cable for each manager
i
i
i
i
i
i
no output channel
Morphisms are obvious
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Scheme of the translation
channel manager
action synchronization through connectors
channel fusion
state

channel manager
guard manager

guard manager
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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
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Tiles for components
actions of managers are mutually exclusive,
but at least one action (e.g. ?) must be executed
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Roadmap

• Motivation
• From CommUnity to Tiles
• Connectors
• Concluding remarks

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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

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Abstract semantics

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

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1
1
1
2
2
2
2
3
3
3
3
4
4
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Abstract semantics

• 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

1
1
1
1
2
2
2
2
3
3
3
3
4
4
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An example Symmetries
connectors boxes are immaterial
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AND-Connectors
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AND-Tables andNormal Form

• n m 0 or 0 lt 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

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co-AND-Connectors
1
0
(transposition of AND table)
?
00
01
10
?
11
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Mixed-AND-Connectors

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Mixed-AND-Connectors


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Mixed-AND-Connectors



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Mixed-AND-Connectors



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Mixed-AND-Connectors



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Mixed-AND-Tables andNormal Form

• n m 0 or 0 lt 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

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HIDE-Connectors (and co-)
!
!
.
?
0
?
1
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A Relevant Difference
!
?
!
!
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A Sample Proof
!
!
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SYNC-Tables and Normal Form

• n m 0 or 0 lt 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

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MEX-Connectors
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co-MEX-Connectors

• Transposed tables
• Symmetric axioms

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Mixed-MEX-Connectors

?
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Mixed-MEX-Connectors

?
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Mixed-MEX-Connectors

?
?
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Mixed-MEX-Connectors

?
?
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Mixed-MEX-Connectors

?
?
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ZERO-Connectors

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Some Axioms About ZERO
0

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A Sample Proof
0
0
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Some Axioms About MEX-AND

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Key Axioms
?
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Key Axioms

!
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An Axiom Scheme
!
!


n
n

!
!
!
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Finally (almost) FULL-Tables

• n m 0 or 0 lt 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

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FULL-Tables and Normal Form
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ONE-Connector
1

• All tables can now be defined, but
• negation is introduced
• and inconsistent connectors with it!
• ONE is not needed for CommUnity

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Roadmap

• Motivation
• From CommUnity to Tiles
• Connectors
• Concluding remarks

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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

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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?