Formal Models for Distributed Negotiations Concurrent Languages Translation

1
Formal Models forDistributed NegotiationsConcurr
ent Languages Translation
XVII Escuela de Ciencias Informaticas (ECI 2003),
Buenos Aires, July 21-26 2003
Roberto Bruni Dipartimento di Informatica
Università di Pisa
2
Process Description Languages

• In concurrency there have been two mainstream
  paradigms
• Petri-net like
• Process Description Languages (PDL)
• Simplified concurrent programming languages
• Primitives resembles conceptual activities
• Focus on certain aspects of interaction
• Not mere mathematical abstraction
• Inspiration of real programming languages
• Role analogous to that of ?-calculus for
  sequential languages

3
PDL Examples

• CCS Milner / CSP Hoare
• Calculus of Comm. Systems / Comm. Sequential
  Processes
• pi-calculus Milner, Parrow, Walker
• name passing
• ambient calculus Cardelli, Gordon
• mobile environments
• spi-calculus Abadi, Gordon / Security Process
  Algebra Focardi, Gorrieri
• cryptography / security
• join-calculus Fournet, Gonthier
• unique receptor
• Linda Gelernter / KLAIM De Nicola, Ferrari,
  Pugliese
• shared and distributed dataspaces

4
PDL Ingredients

• Processes / Agents encode both states and
  programs
• Mostly based on message passing
• Syntax
• Processes are terms over a signature ?
• e.g. parallel composition, input prefix,
  restriction, nondeterministic choice
• possibly modulo some structural axioms E
• e.g. associativity of parallel composition,
  commutativity of choice
• Operational semantics
• Labeled Transition System (LTS) over suitable
  observable actions
• Defined by SOS inference rules taking advantage
  of the signature
• The transitions of a complex agent are defined in
  terms of the transitions of its constituent
  agents
• Reduction semantics
• Often exploit structural axioms to give
  unconditional reduction

5
Abstract Semantics

• Studying behavioral equivalences is fundamental
• More efficient agents can replace obsolete agents
• Trace equivalence ?
• Set of possible executions
• Bisimilarity ?
• Takes into account the branching structure of the
  LTS
• May / Must testing
• Test agents only under suitable scenarios
• Barbed bisimilarity
• Unlabeled bisimilarity state predicates (barbs)
• Better be congruences!
• SOS formats can guarantee that

6
Traces vs Bisimulation

• Bisimilarity is the largest binary relation ? on
  agents such that if P ? Q then
• if P?P then there exists Q such that Q?Q and
  P ? Q
• vice versa

?
?
?
a.ba.c
a.(bc)
?
a
a
a
b
c
bc
b
c
b
c
0
0
0
0
7
Simple Process Algebra (SPA)

• Syntax
• P 0 a?P a!P P\a PP
• Operational semantics
• Actions a?,a!??

8
Truly Concurrent Semantics

• The abstract semantics we have seen are called
  interleaving
• Actions are performed sequentially
• Petri nets can provide truly concurrent semantics
  in a natural way
• Encoding PDL in finite nets
• is not always possible (expressiveness gap)
• requires complex constructions (combinatorial
  explosion of states / transitions)
• e.g. parallel composition must synchronize all
  pairs of complementary actions

9
Why Zero-Safe Nets

• To exploit Zero-Safe nets to compose the models
  of smaller systems according to the PDL signature
• The encoding must preserve the semantics

NP
NQ
NPQ


a?
a!
a?
a!
?
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The Idea

• Channels are encoded as zero places
• a? and a!
• Input, output and synchronization as transactions
• ina, outa, syna
• Z(a1,,an) Za1 ? ? Zan

a?
a!
Za
ina
outa
syna
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a1,,an-Interfaced Nets

• A-interfaced net I(B,A,f)
• B is a Zero-Safe net
• Aa1,,an
• fZ(A)?B is an injective map
• The agent P is modeled by a chan(P)-interfaced
  net P, where chan(P) denote the non restricted
  channels of P
• We let uP denote the initial marking of P

uP
P
f(Zan)
f(Za1)
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The Encoding I

• Inactive agent 0
• 0 (B0,?,?)
• Input prefix a?P
• If a?chan(P)
• We add a NEW place a?P
• A NEW transition ta?Pa?P?uP?a?
• ua?P a?P
• otherwise
• We add the channel a to the interface
• We add a copy of Za
• We extend the injective mapping in the obvious
  way
• We proceed as before

B0
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The Encoding I

• Inactive agent 0
• 0 (B0,?,?)
• Input prefix a?P
• If a?chan(P)
• We add a NEW place a?P
• A NEW transition ta?Pa?P?uP?a?
• ua?P a?P
• otherwise
• We add the channel a to the interface
• We add a copy of Za
• We extend the injective mapping in the obvious
  way
• We proceed as before

B0

uP
P
f(Za)
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The Encoding I
a?P

• Inactive agent 0
• 0 (B0,?,?)
• Input prefix a?P
• If a?chan(P)
• We add a NEW place a?P
• A NEW transition ta?Pa?P?uP?a?
• ua?P a?P
• otherwise
• We add the channel a to the interface
• We add a copy of Za
• We extend the injective mapping in the obvious
  way
• We proceed as before

B0

Ba?P
uP
P
f(Za)
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The Encoding I
a?P

• Inactive agent 0
• 0 (B0,?,?)
• Input prefix a?P
• If a?chan(P)
• We add a NEW place a?P
• A NEW transition ta?Pa?P?uP?a?
• ua?P a?P
• otherwise
• We add the channel a to the interface
• We add a copy of Za
• We extend the injective mapping in the obvious
  way
• We proceed as before

B0
ta?P

Ba?P
uP
P
f(Za)
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The Encoding I
a?P

• Inactive agent 0
• 0 (B0,?,?)
• Input prefix a?P
• If a?chan(P)
• We add a NEW place a?P
• A NEW transition ta?Pa?P?uP?a?
• ua?P a?P
• otherwise
• We add the channel a to the interface
• We add a copy of Za
• We extend the injective mapping in the obvious
  way
• We proceed as before

B0
ta?P

Ba?P
uP
P
f(Za)
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The Encoding II

• Output prefix a!P
• Analogous to input
• Restriction P\a
• Let P(B,A,f)
• P\a(B,A,f)
• B is obtained from B by removing transitions
  f(ina) and f(outa), if present
• A A-a
• f fZ(A)
• uP\a uP

uP
P
BP\a
f(Za)
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The Encoding III

• Parallel composition P1P2
• Let P1(B1,A1,f1) and P2(B2,A2,f2)
• Let Z Z(A1) ? Z(A2)
• P1P2(B,A,f)
• A A1 ? A2
• B is the union of B1 and B2 where f1(Z) and f2(Z)
  are collapsed
• f f1 ? f2
• uP1P2 uP1 ? uP2

uP1
uP2


P1
P2
f1(Z)
BP1P2
f2(Z)
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Example a?0
a?0
ta?0
syna
ina
outa
0
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Example b?a?0
b?a?0
tb?a?0
synb
inb
outb
a?0
ta?0
syna
ina
outa
0
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Example a!0
a!0
ta!0
syna
ina
outa
0
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Example b?a?0a!0
b?a?0
a!0
tb?a?0
synb
inb
outb
a?0
ta?0
ta!0
syna
ina
outa
0
0
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Example (b?a?0a!0)\a
b?a?0
a!0
tb?a?0
synb
inb
outb
a?0
ta?0
ta!0
0
0
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Abstract net
b?a?0
a!0
b?a?0
a!0
tb?a?0
b?
synb
inb
outb
a?0
a?0
?
ta?0
ta!0
0
0
0
0
Proposition The abstract net of P under the
CTPh and ITPh coincide
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Semantic Correspondence

• Proposition
• Each transaction of P(B,A,f) contains at most
  one occurrence of transitions in f(A)
• We can associate unambiguous labels to
  transactions
• ?(?) a? if a?chan(P) and f(ina) is fired in
  transaction ?
• ?(?) a! if a?chan(P) and f(outa) is fired in
  transaction ?
• ?(?) ? otherwise
• Theorem
• P is bisimilar to A(P)
• (in the interleaving sense)
• matching labels in the LTS via the labeling ? of
  transactions

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About Restriction

• The restriction operator P\a hides channel a from
  external observers
• Then a has just local scope
• No interaction on a is possible with the
  environment
• It is natural to consider equivalent two
  processes that differ just for the renaming of
  restricted names
• For such P and Q, we write P ?res Q
• Two A-interfaced nets (B,A,f) and (B,A,f) are
  isomorphic if there exists a ZS net homomorphism
  ?B?B that respects interfaces
• i.e. ?(f(x)) f(x)
• Proposition
• If P ?res Q then P is isomorphic to Q

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About Choice
?
?
P ? Q
P ? Q
?
?

PR ? Q
RP ? Q






NP
NQ
NPQ


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Recap

• We have seen
• Short introduction to PDL
• Operational and abstract semantics
• Encoding of SPA in ZSN
• Truly concurrent semantics
• Correspondence theorem

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References

• Zero-safe nets comparing the collective and
  individual token approaches (Information and
  Computation 156(1-2)46-89, Academic Press 2000)
• R. Bruni, U. Montanari