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Formal Models for Distributed Negotiations: Transactions

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Models and Languages for Coordination and Orchestration ... (up to weak barbed congruence) 8. Roberto Bruni _at_ IMT Lucca. 13 April 2005 ... – PowerPoint PPT presentation

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Title: Formal Models for Distributed Negotiations: Transactions


1
Models and Languages for Coordination and
Orchestration IMT- Institutions Markets
Technologies - Alti Studi Lucca
Nominal Calculi for Transactions JOIN
Roberto Bruni Dipartimento di Informatica
Università di Pisa
2
Contents
  • Introduction
  • Join calculus Examples
  • Join and ?
  • Join as Petri nets

3
Contents
  • Introduction
  • Join calculus Examples
  • Join and ?
  • Join as Petri nets

4
Motivation (back to 1995)
  • Mismatch
  • concurrent process calculi
  • CCS, ?-calculus few constructs, based on
    rendez-vous (atomic non-local interactions)
  • programming languages for distributed and mobile
    systems
  • Actors, Obliq separate primitives for
    transmission and synchronization (RPC and
    semaphores) and a much larger set of constructs
    (imperative primitives)
  • Join calculus has been devised to bridge this gap
  • process calculus presentation
  • basis for a practical programming language design

5
Features (as process calculus)
  • Based on an elementary model of concurrency
  • reflexive chemical abstract machine generic
    CHAM imposing locality adding reflection
  • locality only linear reaction patterns allowed
  • each molecule or reaction rule is associated to a
    single reaction site
  • reflection reactions can generate new kinds of
    molecules together with their defining reaction
    rules
  • computational completeness of the model
  • more effective than generic CHAM
  • molecules travel to their reaction site, instead
    of having to blindly mix and match
  • ?-calculus as sequential deterministic subset

6
Features (as distributed programming language)
  • Extends a higher-order functional language
  • parallelism in expressions (fork calls)
  • parallelism in function patterns (join patterns)
  • jointly defined function provide the same
    capabilities as synchronous channels or
    concurrent objects
  • join patterns are more consistent with lexical
    scope
  • static binding of function calls to the code
  • as opposed to dynamic binding of messages to
    receptors
  • Distributed implementations
  • JoCaml ( http//join.inria.fr ), Polyphonic C

7
Join calculus vs. ? calculus
  • Join is essentially ? with restrictions on
    communication patterns
  • Join combines restriction, reception and
    replication in a single receptor definition
  • not available separately
  • asynchronous calculus, continuation passing style
  • asynchrony forces one to create and send
    continuations
  • Nevertheless, join and ? have the same expressive
    power
  • demonstrated by fully abstract encodings in each
    direction
  • (up to weak barbed congruence)

8
Join Calculus vs Petri Nets
  • We shall see that the join-calculus is the
    natural higher order extension of Petri nets
  • places as ports / channels
  • tokens carry values
  • names of places are also admissible values
  • firing can generate fresh pieces of nets
  • new places
  • new transitions

9
Contents
  • Introduction
  • Join calculus Examples
  • Join and ?
  • Join as Petri nets

10
Chemical Abstract MachineMolecules and Membranes
  • States are called solutions s
  • Multisets of molecules m1,,mn
  • data and rules (reflexive CHAM)
  • Hierarchical structure via membranes
  • group solutions into molecules
  • allow nesting
  • e.g. s1 , s2 , s3, s4

multiset union
11
Chemical Abstract MachineChemical Rules
  • Evolution (chemical rules)

12
Chemical Abstract MachineChemical Rules
  • Evolution (chemical rules)
  • Heating / cooling ? (reversible)
  • Structural equivalence

13
Chemical Abstract MachineChemical Rules
  • Evolution (chemical rules)
  • Heating / cooling ? (reversible)
  • Structural equivalence
  • Reactions ?
  • Transitions

14
Chemical Abstract MachineChemical Rules
  • Evolution (chemical rules)
  • Heating / cooling ? (reversible)
  • Structural equivalence
  • Reactions ?
  • Transitions

15
Join Calculus in One Slide
  • Syntax
  • P,Q 0 x?y? def D in P PQ
  • D,E J? P D?E
  • J,K x?y? JK
  • Operational semantics (CHAM Style)
  • 0 ?
  • PQ ? P,Q
  • D?E ? D,E
  • def D in P ? D?dn(D) , P?dn(D) (range
    ?dn(D) globally fresh)
  • J? P, J? ? J? P, P?

processes
definitions
resembles funct. prog. let f(x)E in F (same
scoping discipline)
patterns
heating and cooling
reaction
16
JOIN An Example
defined name
  • A process P
  • P ? z?x,z? def x?y? ? z?y,x? in x?v?
  • P as a solution
  • z?x,z? , w?y? ? z?y,w? , w?v?
  • A reaction
  • z?x,z? , w?y? ? z?y,w? , w?v? ?
  • z?x,z? , w?y? ? z?y,w? , z?v,w?

bound name
free name
free name
bound name
received name
membrane
extrusion
17
Continuation Passing Style I
  • The form of definitions resembles very much
  • let f(x)E in E (typical of functional
    programming)
  • e.g. same scoping discipline
  • Asynchrony forces us to create and send
    continuations in join
  • e.g. encoding untyped ?-calculus
  • Mv sends the value of M on v
  • a value is a process serving requests
  • a request must supply two names
  • x (channel for requests for the value of the
    argument)
  • w (to eventually return a value)

18
Continuation Passing Style II
  • Call-by-name
  • xv v?x?
  • ?x.Mv def k?x,w? ? Mw in v?k?
  • MNv def y?p? ? Np
  • in def q?c? ? c?y,v? in Mq
  • Parallel call-by-value
  • xv v?x?
  • ?x.Mv def k?x,w? ? Mw in v?k?
  • MNv def q?c?p?y? ? c?y,v? in MqNp

19
Call-by-Name
  • Strategy leftmost order, no reduction under ?
  • Reductions are entirely sequential
  • The image of the translation is exactly the
    deterministic subset of Join (no parallel
    composition, no conjunction)
  • xv x?v?
  • ?x.Mv def k?x,w? ? Mw in v?k?
  • MNv def y?p? ? Np
  • in def q?c? ? c?y,v? in Mq

20
Call-by-Name Example
  • (?x.M)Nv def y?p? ? Np
    in def q?c? ? c?y,v? in ?x.Mq
  • (?x.M)Nv def y?p? ? Np
    in def q?c? ? c?y,v? in
    def k?x,w? ? Mw in q?k?
  • (?x.M)Nv ? def y?p? ? Np
    in def q?c? ? c?y,v? in
    def k?x,w? ? Mw in k?y,v?
  • (?x.M)Nv ? def y?p? ? Np
    in def q?c? ? c?y,v?
    in def k?x,w? ? Mw in My/xv

21
Call-by-Name Example
  • (?x.M)Nv def y?p? ? Np
    in def q?c? ? c?y,v? in ?x.Mq
  • (?x.M)Nv def y?p? ? Np
    in def q?c? ? c?y,v? in
    def k?x,w? ? Mw in q?k?
  • (?x.M)Nv ? def y?p? ? Np
    in def q?c? ? c?y,v? in
    def k?x,w? ? Mw in k?y,v?
  • (?x.M)Nv ? def y?p? ? Np
    in def q?c? ? c?y,v?
    in def k?x,w? ? Mw in My/xv

22
Call-by-Name Example
  • (?x.M)Nv def y?p? ? Np
    in def q?c? ? c?y,v? in ?x.Mq
  • (?x.M)Nv def y?p? ? Np
    in def q?c? ? c?y,v? in
    def k?x,w? ? Mw in q?k?
  • (?x.M)Nv ? def y?p? ? Np
    in def q?c? ? c?y,v? in
    def k?x,w? ? Mw in k?y,v?
  • (?x.M)Nv ? def x?p? ? Np
    in Mv

23
Parallel Call-by-Value
  • Strategy again no reduction under ?, but in
    (TU), T and U can be evaluated in parallel
  • Confluent, but non deterministic
  • xv v?x?
  • ?x.Mv def k?x,w? ? Mw in v?k?
  • MNv def q?c?p?y? ? c?y,v? in MqNp

24
Call-by-Value Example
  • (?x.M)Nv def q?c?p?y? ? c?y,v? in ?x.Mq
    Np
  • (?x.M)Nv def q?c?p?y? ? c?y,v?
    in Np def k?x,w? ? Mw
    in q?k?
  • (?x.M)Nv ? def q?c?p?y? ? c?y,v?
    in p?z? def k?x,w? ? Mw
    in q?k?
  • (?x.M)Nv ? def q?c?p?y? ? c?y,v?
    in def k?x,w? ? Mw
    in k?z,v?
  • (?x.M)Nv ? def q?c?p?y? ? c?y,v?
    in def k?x,w? ? Mw
    in Mz/xv

25
Example Cell Abstraction
get?k? s?v? ? k?v? s?v?
  • A cell s contains the value v
  • To get the value
  • send a message on port get
  • the parameter k is the return address, where the
    value v will be sent to

26
Example Cell Abstraction
get?k? s?v? ? k?v? s?v?
set?m,k? s?v? ? k?? s?m?
  • A cell s contains the value v
  • To set the value
  • send a message on port set
  • the parameter m is the new value for s
  • k is the return address (for confirmation)

27
Example Cell Abstraction
get?k? s?v? ? k?v? s?v?
def ? in s?n?
set?m,k? s?v? ? k?? s?m?
  • The initial value in s is n
  • But get, set and s are locally bound by def
  • get and set must be extruded, otherwise no one
    can use them
  • instead, s is kept private (encapsulation)

28
Example Cell Abstraction
get?k? s?v? ? k?v? s?v?
def ? in s?n? c?get,set?
set?m,k? s?v? ? k?? s?m?
  • get, set are extruded on public channel c
  • But c should be known only by the owner of the
    cell

29
Example Cell Abstraction
def create?n,c? ?

in
get?k? s?v? ? k?v? s?v?
def ? in s?n? c?get,set?
set?m,k? s?v? ? k?? s?m?
  • A message to create triggers the outermost def
  • Three fresh names for s, get and set are
    allocated
  • the initial value of s is the first parameter n
  • get and set are sent back to the second argument
    c
  • instead s will never be extruded
  • Invariant
  • in every configuration there is exactly one
    message on s

30
SIXTH HOMEWORK
  • Guess the meaning of
  • def x?u? ? y?u? in P
  • def y?u? ? x?u? in def x?u? ? y?u? in P
  • def s?? ? P ? s?? ? Q in s??
  • def c?? ? Pc?? in Qc??

31
Contents
  • Introduction
  • Join calculus Examples
  • Join and ?
  • Join as Petri nets

32
Core Join Calculus
  • Syntax
  • a unique syntactic category
  • P,Q x?u? def x?u?y?v? ? Q in P PQ
  • Operational semantics
  • CHAM Style
  • (but also LTS is defined)
  • The core join calculus has the same expressive
    power as the full join-calculus
  • via a fully-abstract encoding

33
Full abstraction
  • Two process calculi with equivalences ?1 and ?2
  • The first is more expressive than the second if
    we can find a fully abstract encoding .2?1
  • i.e. an encoding such that
  • P ?2 Q iff P2?1 ?1 Q2?1
  • The two calculi have the same expressive power if
    each one is more expressive than the other
  • (If one is a sub-calculus of the other, then one
    implication is obvious)

34
What is Observable?
  • Communication
  • on internal names (no)
  • on free names (yes)
  • Internal steps
  • countable strong semantics (no)
  • immaterial weak semantics (yes)
  • Equivalence
  • reflexive, symmetric and transitive (yes)
  • closed under contexts congruence (yes)

35
Basic Observations
  • Processes interact with the outside
  • by extruding names on free ports
  • by waiting for answers (via enclosed definitions)
  • Processes are distinguished on the basis of their
    ability to emit messages on their free ports
  • weak asynchronous output barb ?x
  • P?x iff
  • x is a free name in P
  • and ?u such that P ? Q,x?u?

36
Remarks on Barbs
  • Two processes P and Q such that
  • ?u with P?x but ?(Q?x)
  • cannot be reasonably identified!
  • Barbs are just elementary experiments
  • barbs do not count reductions (ok)
  • barbs do not observe branching (uhm)
  • barbs do not observe message reception (uhm)

37
Closure Under Reductions
  • Reductions are mute transitions
  • i.e. only trivial labels are present
  • P?P' can be read as P???P'
  • In ordinary (strong) bisimulation
  • if P?Q and P?P', then ?Q'?P' s.t. Q?Q'
  • (and vice versa)
  • In weak bisimulation
  • if P?Q and P ? P', then ?Q'?P' s.t. Q ? Q'
  • (and vice versa)

38
Closure Under Contexts
  • If P?Q we expect that P and Q can be used
    interchangeably in any larger process
  • but P ? a?b? and Q ? a?c? look equivalent when
    taken in isolation
  • no reduction, a unique barb ?a
  • however, they are not equivalent in the context
  • def a?x? ? x?? in .
  • as in fact
  • def a?x? ? x?? in P ? b?? (i.e. def a?x? ? x?? in
    P ?b)
  • def a?x? ? x?? in Q ? c?? (i.e. def a?x? ? x?? in
    Q ?c)

39
The Observational Congruence
  • We take the largest equivalence relation ? that
  • is a refinement of output barbs
  • if P?Q then (?x. P?x iff Q?x)
  • is closed under weak reduction
  • if P?Q and P ? P', then ?Q'?P' s.t. Q ? Q'
  • is a congruence w.r.t. definitions and parallel
  • if P?Q then (?D. def D in P ? def D in Q)
  • if P?Q then (?R. PR ? QR)

40
Observational Congruence Examples
  • If fn(P)? then P ? 0
  • If P?Q then P ? Q
  • a?u? ? b?u?
  • a?b? ? a?c?
  • a?b? ? def c?x? ? b?x? in a?c?
  • it is not possible to distinguish between
    different names that exhibit the same external
    behaviour

41
Core Join vs Full Join
  • Expressiveness-preserving simplification of
    syntax
  • recursive binding
  • shift binding variables from definition to
    reception
  • def J ? Q in P becomes def Jb?ã,b'? ? Qb?ã,b'?
    in Pb?ã,b?
  • where ã is the vector of variables in fn(Q)?dn(J)
  • complex definitions
  • n-way join patterns and multiple clauses
    connected by ? as sequences joining two atoms at
    most
  • polyadic messages
  • name tuples are communicated by using auxiliary
    private names

42
Asynchronous ?
  • Syntax
  • P,Q x?u? x(u).P ?u.P !x(u).P PQ
  • Abstract semantics
  • asynchronous barbed congruence
  • ex. x(u).x?u? ? 0
  • ex. equator EQ(x,y) ? !x(u).y?u? !y(v).x?v?
  • Px/y ? Qx/y implies EQ(x,y)P ? EQ(x,y)Q

processes
43
Naïve Encoding Join in ?
  • x?v?j?? x?v?
  • PQj?? Pj?? Qj??
  • def x?u?y?v? ? Q in Pj??
    ?x.?y.(!x(u).y(v). Qj?? Pj?? )
  • In the translation we loose
  • the symmetry between x and y
  • the atomicity of their joint reduction
  • it does not matter, because x and y are
    restricted
  • Not closed under ? contexts!!!
  • if x or y are extruded, then new receptors could
    appear

44
Problems with Full Abstraction of Join in ?
Example
  • Let P ? def x?? ? 0 in a?x? x?? j??
  • and Q ? def y?? ? 0 in a?y? j??
  • the two encoded processes are equivalent
  • P and Q are not
  • Take the ?-context ?a.( a(u).u().b?? . )
  • then ?a.( a(u).u().b?? P ) ?b
  • while ?( ?a.( a(u).u().b?? Q ) ?b )

45
Some kind of Firewall
  • Instead of extruding defined names, we extrude
    dummy synonyms, setting up appropriate relays
  • Rx, xe !x(v).?w.(r?w,v? xe?w?)
  • If x is a free name of P, then we must embed P in
    the firewall for x, defined as
  • Fx. ? ?r.!r(y,ye).Ry,ye ?x.(Rx,xe .j??)

46
Naïve Encoding ? in Join
  • Each ?-channel x is simulated by two ports
  • xo for output (where emitters send values)
  • xi for input (the receiver defines a name k for
    its continuation and sends it as a reception
    offer on xi)
  • x?v???j xo?vo,vi?
  • x(u).P??j def k?vo,vi? ? P??j in xi?k?
  • ?x.P??j def xo?vo,vi?xi?k? ? k?vo,vi? in
    P??j
  • !x(u).P??j def k?vo,vi? ? xi?k? P??j in
    xi?k?
  • PQ??j P??j Q??j
  • Not closed under Join contexts!!!
  • problems with free names and input barbs

47
Problems with Full Abstraction of ? in Join
Examples
  • x?a? x?b? x(u).y?u? ??j
  • cannot reduce because there is no englobing ?x
  • x(u).x?u? ??j
  • exhibits a barb on xi that reveals the presence
    of an input on x
  • A context could provide messages with arbitrary
    i/o-pairs
  • ex. mismatched order xo?vi,vo?
  • ex. mismatched names xo?vo,wi?

48
Proxies and Firewall
  • A much more complicated notion of firewall is
    needed (omitted here)
  • several pairs can exist for the same channel x
  • any two pairs are interchangeable (message
    merging)
  • a new proxy pair is created whenever a pair of
    names is received from the outside (and they are
    merged) the new pair will be transmitted in
    place of the old one
  • whenever a pair of names is sent to the outside,
    a new firewall is inserted for setting up proxies
    for future incoming messages on this pair

49
Implementability
  • Uniqueness of receptors favour distributed
    implementation of Join

50
Contents
  • Introduction
  • Join calculus Examples
  • Join and ?
  • Join as Petri nets

51
Petri Net Flavors
  • The basic net model can be extended in many ways
  • to model interesting features
  • e.g. read arcs
  • to increase expressiveness
  • e.g. inhibitor arcs
  • Many other variations have been proposed in the
    literature (stochastic, priorities, time, )
  • We survey some of them, as incremental extensions
    (bottom-up), showing that they can also be
    recovered in the other way round (top-down)
    starting from Join

52
Basic Model P/T Petri Nets
b
a
2
3
2
c
d
53
Basic Model P/T Petri Nets
b
a
2
3
2
c
d
54
Colured Nets (also High-Level Nets)
b
a
5
s
1
x
y
x?y
structured data as tokens
yy
x3
6
c
d
55
Coloured Nets (also High-Level Nets)
b
a
5
x
y
x1 ys
x?y
structured data as tokens
yy
x3
4
ss
6
c
d
56
Reconfigurable Nets
network reconfigurability vs static connectivity
b
a
a
c
c
d
x
y
post-sets places depend on fetched values
y
x
x
y
c
d
57
Reconfigurable Nets
network reconfigurability vs static connectivity
b
a
a
d
x
y
post-sets places depend on fetched values
xc yc
y
x
c
c
x
y
c
d
58
Reconfigurable Nets
network reconfigurability vs static connectivity
b
a
d
x
y
post-sets places depend on fetched values
xd ya
y
x
c
c
a
x
y
c
d
59
Dynamic Nets
b
a
a
c
c
d
x
firings can generate new net fragments
y
dynamic control
c
d
N(x,y)
60
From Petri Nets to Dynamic Nets and Back via JOIN
  • The join-calculus is the natural higher order
    extension of Petri nets
  • more and more restrictive type systems recover
    all kinds of nets we have seen
  • Dynamic nets (no restriction)
  • Reconfigurable nets (no definitions inside
    definitions)
  • High level nets (no channel names as messages)
  • Place/Transition nets (no values in messages)

61
Nets as Join Terms
  • Roughly
  • places are channels
  • transitions are definitions
  • tokens are message values
  • nets are join processes
  • different classes of nets corresponds to
    different classes of terms
  • Note that in general a definition can contain
    another definition
  • a reduction will release fresh places and
    transitions
  • fresh transitions can release tokens in
    previously existing places, but they cannot fetch
    tokens from them

62
Type System ?0 I
  • Aim
  • To identify terms that correspond to P/T Petri
    nets
  • Three kinds of judgements
  • - P ??
  • P is ok and contains no def_in_
  • - P ?
  • P is ok
  • - D ?
  • D is ok

63
Type System ?0 II
64
Type System ?1
  • Aim
  • To identify terms that correspond to coloured
    Petri nets
  • Three kinds of judgements (as before)
  • Type environments needed
  • Channels must be kept distinct from messages
  • ? set of channel names
  • ? set of messages
  • ? and ? must be disjoint in ?? - P ?

65
Type System ?1 II
66
Results
  • ?0 characterizes terms that correspond to P/T
    Petri nets
  • ?1 characterizes terms that correspond to
    Coloured nets
  • A third type system ?2 characterizes terms that
    correspond to reconfigurable nets
  • A trivial type system ?3 characterizes terms that
    correspond to dynamic nets
  • All type systems
  • enjoy subject reduction
  • allow the definition of a behaviour preserving
    isomorphism between typeable terms and (the
    corresponding kind of) nets

67
SEVENTH HOMEWORK
  • Define the type system ?2 that characterizes all
    those terms of the join-calculus that correspond
    to reconfigurable nets

68
References
  • The reflexive chemical abstract machine and the
    Join calculus (Proc. POPL96, ACM, pp. 372-385)
  • C. Fournet, G. Gonthier
  • High-level Petri nets as type theories in the
    Join-calculus (Proc. FoSSaCS01, LNCS 2030, pp.
    104-120)
  • M. Buscemi, V. Sassone
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