Hoare vs. Milner: Comparing Synchronizations in a Graphical Framework With Mobility

1
Hoare vs. Milner Comparing Synchronizationsin a
Graphical Framework With Mobility
Ugo Montanari Università di Pisa
in collaboration with
Ivan Lanese Università di Pisa
2
Outline

• Graphical Calculi for Distributed Systems
• Synchronized Edge Replacement Systems
• Mobility
• Hoare and Milner Synchronization, with Fusion
• Direct Comparison
• Comparison with Translations
• Conclusions and Future Work

3
Outline

• Graphical Calculi for Distributed Systems
• Synchronized Edge Replacement Systems
• Mobility
• Hoare and Milner Synchronization, with Fusion
• Direct Comparison
• Comparison with Translations
• Conclusions and Future Work

4
Graphical Approach to Distributed Systems

• Motivations
• Intuitive representation of distribution
• Natural concurrent semantics
• No need of structural axioms
• Existing modeling languages, e.g. UML
• Applications to software architectures and ADLs
• Well-developed foundations

5
Graph vs. Term Transformations

• Terms
• LTS defined via SOS rules
• Reduction rules
• Abstract semantics
• Non-interleaving semantics
• Graphs
• Double-pushout derivations
• Concurrent semantics based on shift equivalence
• Synchronized (hyper)edge replacement

6
(Hyper)Graphs

• Edge Atomic item with a label from alphabet LE
  LEnn0,1, with as many (ordered) tentacles
  as the rank of its label.
• Graph A set of nodes and a set of edges such
  that each edge is connected, by its tentacles, to
  its attachment nodes. A set of external nodes,
  identified by distinct names, defines the
  connecting points with the environment.

x
L
L
y
1
M
4
2
3
z
7
A Notation For Graphs

• Edge Atomic item with a label from alphabet LE
  LEnn0,1, with as many (ordered) tentacles
  as the rank of its label.
• Graph A set of nodes and a set of edges such
  that each edge is connected, by its tentacles, to
  its attachment nodes. A set of external nodes,
  identified by distinct names, defines the
  connecting points with the environment.

8
A Notation For Graphs
Well formed judgements for graphs

• Structural Axioms

(AG2) G1G2 G2G1
(AG1) (G1G2)G3 G1(G2G3)
(AG3) G1 nil G1
(AG4) ?x.?y.G ?y.?x.G
(AG6) ?x.G ?y.G y/x if y ? fn(G)
(AG5) ?x.G G if x ? fn(G)
(AG7) ?x.(G1G2 ) (?x. G1) G2 if x ? fn(G2)
9
A Notation For Graphs
Well formed judgements for graphs

• Syntactic Rules

(RG1)
(RG2)
x1,,xn nil
?
x1,,xn L(y1,,ym)
?
?, x G
(RG3)
(RG4)
? G1G2
?
? ? x. G
10
A Notation For Graphs

• Ring Example

11
Outline

• Graphical Calculi for Distributed Systems
• Synchronized Edge Replacement Systems
• Mobility
• Hoare and Milner Synchronization, with Fusion
• Direct Comparison
• Comparison with Translations
• Conclusions and Future Work

12
Edge Replacement Systems

• Productions A context free production rewrites a
  single edge labeled by L into an arbitrary graph
  R. (Notation L ? R)

L
R
H
3
3
4
4
2
2
1
1
13
Edge Replacement Systems

• Productions A context free production rewrites a
  single edge labeled by L into an arbitrary graph
  R. (Notation L ? R)

Rewritings of different edges can be executed
concurrently
14
Synchronized Edge Replacement

• Synchronized rewriting Actions are associated
  to nodes in productions. Each rewrite of an edge
  must match actions with (a number of) its
  adjacent edges and they have to move
  simultaneously

How many edges synchronize depends on the
synchronization policy

• Synchronized rewriting propagates
  synchronization
• all over the graph

15
Synchronized Edge Replacement

• Hoare Synchronization All adjacent edges must
  match the actions on the shared node

• Milner Synchronization Only two of the adjacent
  edges synchronize by matching their complementary
  actions

16
Outline

• Graphical Calculi for Distributed Systems
• Synchronized Edge Replacement Systems
• Mobility
• Hoare and Milner Synchronization, with Fusion
• Direct Comparison
• Comparison with Translations
• Conclusions and Future Work

17
Adding Mobility
18
Transitions as Judgements
Formalization of synchronized rewriting as
judgements
19
Transitions as Judgements
Formalization of synchronized rewriting as
judgements

• Transitions
• are generated from the productions by
  applying the transition rules
• of the chosen synchronization mechanism

20
Synchronization via Unification
Hoare synchronization

• On each node all edges must have the same action
• Synchronization is possible if there is a most
  general unifier of the new nodes

For any R ? ? x A x N (not necessarily a
partial function) ?(R) ?? ?? n(R) is the mgu
of equations (a b) ? (Y Z) with (x,a,Y) and
(x,b,Z) in R where (as usual) ?? z (x,a,Y)
? R, z ?set(Y), z ? ?
21
Example

22
Synchronization via Unification
Milner synchronization

• On each node at most two edges must have
  actions, and in this case they
  must be complementary
• Synchronization is possible if there is a most
  general unifier of the new nodes

23
Adding Fusion
Synchronized rewriting with mobility and fusion
24
Outline

• Graphical Calculi for Distributed Systems
• Synchronized Edge Replacement Systems
• Mobility
• Hoare and Milner Synchronization, with Fusion
• Direct Comparison
• Comparison with Translations
• Conclusions and Future Work

25
Rewriting Rules, Hoare Synchronization I
26
Rewriting Rules, Hoare Synchronization II
27
Rewriting Rules, Milner Synchronization I
28
Rewriting Rules, Milner Synchronization II
29
Related Work

• Grammars for distributed systems
• Castellani and Montanari, LNCS 1953, 1982,
  Degano and Montanari, JACM 1987
• Graph amalgamation
• Boehm, Fonio and Habel, JCSS, 1987
• CHARM (R for restriction)
• Corradini, Montanari and Rossi, TCS 1994
• Mobile version (w. applications to software
  architectures, only p-I-like mobility, Hoare
  synchronization)
• Hirsch and Montanari, Coordination 2000
• Modeling p-calculus (Milner synchronization)
• Hirsch and Montanari, Concur 2001
• Modeling Ambient calculus Ferrari, Montanari and
  Tuosto, ICTCS 2001
• Modeling Fusion calculus Lanese and Montanari,
  to appear in TCS

30
Outline

• Graphical Calculi for Distributed Systems
• Synchronized Edge Replacement Systems
• Mobility
• Hoare and Milner Synchronization, with Fusion
• Direct Comparison
• Comparison with Translations
• Conclusions and Future Work

31
Expressiveness Measure

• (S1,C1) (S2,C2)
• (i.e. style S1 is more expressive than style S2)
• iff there exists a uniform simulation function f
  such that for all P and G
• C2-behavS2(P)(G) C1-behavS1(f(P))(G)

32
Hoare and Milner, Direct Comparison, I

• (Milner,C1) (Hoare,C2) for all C1 and C2
• i.e. Hoare cannot be uniformely simulated by
  Milner
• The reason is that Milner synchronization style
  is monotone, i.e. in a Milner computation we can
  always add to a graph an additional part which
  stays idle, while Hoare style is not monotone

33
Hoare and Milner, Direct Comparison, II

• (Hoare,C1) (Milner,C2) for all C1 and C2
• i.e. Milner cannot be uniformely simulated by
  Hoare
• The reason is that in Hoare synchronization style
  restriction just hides part of the observation,
  while in Milner style restriction may forbid
  computations

34
Outline

• Graphical Calculi for Distributed Systems
• Synchronized Edge Replacement Systems
• Mobility
• Hoare and Milner Synchronization, with Fusion
• Direct Comparison
• Comparison with Translations
• Conclusions and Future Work

35
Translation via Amoeboids

• Amoeboids are graphs with suitable edge labels
  and corresponding productions which simulate the
  behavior of nodes in a different synchronization
  style
• Function - replaces nodes with amoeboids
  while function --1 replaces amoeboids with
  nodes.
• We always have that (G)-1 G

36
Implementing Hoare with Milner

• H-amoeboids implement broadcasting. C-amoeboids
  saturate nodes with less than 3 tentacles. We
  have rules for every action a (here with arity 2).

We have C-behavH(P)(G) C-behavM(f(P))(G)
-1
37
Implementing Milner with Hoare

• M-amoeboids implement routing. We have rules for
  every action a

and two analogous productions for synchronizing x
with z and y with z. We have only
C-behavM(P)(G) ?? C-behavH(f(P))(G)-1 sin
ce the amoeboids can also synchronize several
pairs in parallel.
38
Outline

• Graphical Calculi for Distributed Systems
• Synchronized Edge Replacement Systems
• Mobility
• Hoare and Milner Synchronization, with Fusion
• Direct Comparison
• Comparison with Translations
• Conclusions and Future Work

39
Conclusions and Future Work

• Graph models with synchronized hyperedge
  replacement allow for more general
  synchronization mechanisms than ordinary process
  algebras, e.g. processes can synchronize at more
  than one channel and with more than one other
  process.
• These extensions are needed for implementing one
  synchronization style into another.
• Reachability in Hoare/Milner synchronization
  styles cannot be simulated uniformely
• No countexample uses mobility, and thus the
  expressivenesses are incomparable even without
  mobility, and mobility does not bridge the gap
• Distributed simulation via amoeboids of Milner
  style routers allows only concurrent pairwise
  synchronization
• Generic synchronization styles and more general
  notions of implementation and refinement
  involving atomicity and bisimilarity can be
  considered see the forthcoming PhD thesis of
  Ivan Lanese