Connectors and Concurrency

1
Connectors and Concurrency
Dagstuhl Seminar 04241, September 2003
Roberto Bruni Dipartimento di Informatica
Università di Pisa
joint work with Ugo Montanari
2
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
3
CommUnity, Tiles and Connectors
Dagstuhl Seminar 04241, 6-11 June 2004
Roberto Bruni Dipartimento di Informatica
Università di Pisa
Ongoing Work!
joint work with José Luiz Fiadeiro Ivan
Lanese Antónia Lopes Ugo Montanari
4
Roadmap

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

5
Roadmap

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

6
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

7
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

8
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

9
Roadmap

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

10
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

11
CommUnity, graphically
Denotational Semantics
Program
12
CommUnity Programs
design foo is in x, z out v, n do a true ?
v xz nvx b ngtMIN ? nn-x c
vltMAX ? vnz
input/output channels
a special actions always present ? true ? skip
13
Morphsims
output channels cannot be merged
P1
channels of P1 to channels of P2
we shall focus on actions coordination
(actions names are not important)
actions of P2 to actions of P1
actions of P1 correspond to disjoint sets of
actions of P2
P2
14
Example
design P1 is in out do a b
design P is in out do fap faq
far gap gaq gar
hbs
f,g ? a ? p,q,r h ? b ? s
design P2 is in out do f g h
design P3 is in out do p q r
s
15
Star-Shaped Diagrams
glue
cables no output channels actions are true? skip
roles
16
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

17
Configurations
output interface
input interface
(interfaces can be typed)
18
Configurations
output interface
input interface
parallel composition
functoriality
sequential composition
19
Configurations
output interface
input interface
parallel composition
functoriality symmetries symmetric monoidal
cat
sequential composition
20
Observations
initial interface
final interface
21
Tiles

• Combine horizontal and vertical structures
  through interfaces

22
Tiles

• Compose tiles
• horizontally

23
Tiles

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

symmetric monoidal double cat
24
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)

25
Roadmap

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

26
Notation

• Since
• Initial and Final configuration are always equal
• We shall consider only two kinds of observations
• We use a flat notation for tiles

1
0
27
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

28
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

29
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

30
Standard Decomposition Illustrated

• n output channels
• m actions

P

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

31
Channel Manager
one channel manager for each output variable x of
P
channel manager for x
i
i
i
i
i
i
o
32
Guard Manager
one guard manager for each action a of P
guard manager for a
i
i
i
i
i
i
no output channel
33
Glue
one glue
i
i
i
i
i
i
no output channel
34
Cables (morphisms are obvious)
one cable for each manager
i
i
i
i
i
i
no output channel
35
Scheme of translation
channel manager
action synchronization through connectors
channel fusion
state

channel manager
guard manager

guard manager
36
Tiles for components
actions of managers are mutually exclusive,
but at least one action (e.g. ?) must be executed
37
Roadmap

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

38
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

39
Abstract semantics

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

1
1
1
1
2
2
2
2
3
3
3
3
4
4
40
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
• (not all matrix are interesting)
• 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
41
An example Symmetries
connectors boxes are immaterial
(also identities should be considered)
42
AND-Connectors
43
AND-Tables andNormal Form

• 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

44
co-AND-Connectors
1
0
(as AND-table, but transposed)
?
00
01
10
?
11
45
Mixed-AND-Connectors

46
Mixed-AND-Connectors


47
Mixed-AND-Connectors



48
Mixed-AND-Connectors



49
Mixed-AND-Connectors



50
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/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

51
(co-)HIDE-Connectors
!
!
.
?
0
?
1
52
A Relevant Difference
!
?
!
!
53
A Sample Proof
!
!
54
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/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

55
(co-)MEX-Connectors
56
Mixed-MEX-Connectors

?
57
Mixed-MEX-Connectors

?
58
Mixed-MEX-Connectors

?
?
59
Mixed-MEX-Connectors

?
?
60
Mixed-MEX-Connectors

?
?
61
(co-)ZERO-Connectors

62
Some Axioms About ZERO
0

63
A Sample Proof
0
0
64
Some Axioms About MEX-AND

65
Key Axioms
?
66
Key Axioms

!
67
An Axiom Scheme
!
!


!
!
68
An Axiom Scheme
!
!


!
!
69
FULL-Tables and Normal Form

• 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

70
(co-)ONE-Connector
1

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

71
Roadmap

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

72
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)