The probabilistic asynchronous p-calculus

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The probabilistic asynchronous p-calculus

• Catuscia Palamidessi, INRIA Futurs, France

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Motivations

• Main motivation behind the development
• of the probabilistic asynchronous p-calculus
• To use it as an intermediate language for the
  fully distributed implementation of the
  p-calculus with mixed choice
• Plan of the talk
• The p-calculus,
• Operational semantics
• Expressive power
• The probabilistic asynchronous p-calculus
• Probabilistic automata, operational semantics
• Distributed implementation
• Encoding the p-calculus into the probabilistic
  asynchronous p-calculus

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The p-calculus

• Proposed by Milner, Parrow, Walker 92 as a
  formal language to
• reason about concurrent systems
• Concurrent several processes running in parallel
• Asynchronous every process proceeds at its own
  speed
• Synchonous communication aka handshaking
• Mixed guarded choice input and output guards
  like in CSP and CCS. The implementation of
  guarded choice is aka the binary interaction
  problem
• Dynamic generation of communication channels
• Scope extrusion a channel name can be
  communicated and its scope extended to include
  the recipient

Q
z
z
x
y
R
P
z
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p the p-calculus (w/ mixed choice)

• Syntax
• g x(y) xy t prefixes (input, output,
  silent)
• P Si gi . Pi mixed guarded choice
• P P parallel
• (x) P new name
• recA P recursion
• A procedure name

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Operational semantics

• Transition system P -a? Q
• Rules
• Choice Si gi . Pi gi? Pi
• P -xy? P
• Open ___________________
• (y) P -x(y)? P

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Operational semantics

• Rules (continued)
• P -x(y)? P Q
  -xz? Q
• Com ________________________
• P Q -t? P z/y Q
• P -x(y)? P
  Q -x(z)? Q
• Close _________________________
• P Q -t? (z) (P z/y
  Q)
• P -g? P
• Par _________________ fn(Q)
  and bn(g) disjoint
• Q P -g? Q P

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Features which make p very expressive- and
cause difficulty in its distributed
implementation

• (Mixed) Guarded choice
• Symmetric solution to certain distributed
  problems involving distributed agreement
• Link mobility
• Network reconfiguration
• It allows expressing HO (e.g. l calculus) in a
  natural way
• In combination with guarded choice, it allows
  solving more distributed problems than those
  solvable by guarded choice alone

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The expressive power of p

• Example of distributed agreement
• The leader election problem in a symmetric
  network
• Two symmetric processes must elect one of them as
  the leader
• In a finite amount of time
• The two processes must agree
• A symmetric and fully distributed solution in p,
  using guarded choice
• x.Pwins y.Ploses y.Qwins
  x.Qloses

? Ploses Qwins
? Pwins Qloses
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Example of a network where the leader election
problem cannot be solved by guarded choice alone
For the following network there is no (fully
distributed and symmetric) solution in CCS, or in
CSP
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A solution to the leader election problem in p
winner
winner
looser
winner
looser
looser
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Approaches to the implementation of guarded
choice in literature

• Parrow and Sjodin 92, Knabe 93, Tsai and
  Bagrodia 94 asymmetric solution based on
  introducing an order on processes
• Other asymmetric solutions based on
  differentiating the initial state
• Plenty of centralized solutions
• Joung and Smolka 98 proposed a randomized
  solution to the multiway interaction problem, but
  it works only under an assumption of partial
  synchrony among processes
• Our solution is the first to be fully
  distributed, symmetric, and using no synchronous
  hypotheses.

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State of the art in p

• Formalisms able to express distributed agreement
  are difficult to implement in a distributed
  fashion
• For this reason, the field has evolved towards
  variants of p which retain mobility, but have no
  guarded choice
• One example of such variant is the asynchronous p
  calculus proposed by Honda-Tokoro91, Boudol,
  92 (Asynchronous Asynchronous communication)

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pa the Asynchonous pVersion of Amadio,
Castellani, Sangiorgi 97

• Syntax
• g x(y) t prefixes
• P Si gi . Pi input guarded choice
• xy output action
• P P parallel
• (x) P new name
• recA P recursion
• A procedure name

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Characteristics of pa

• Asynchronous communication
• we cant write a continuation after an output,
• i.e. no xy.P, but only xy P
• so P will proceed without waiting for the actual
  delivery of the message
• Input-guarded choice only input prefixes are
  allowed in a choice.
• Note the original asynchronous p-calculus did
  not contain a choice construct. However the
  version presented here was shown by Nestmann and
  Pierce, 96 to be equivalent to the original
  asynchronous p-calculus
• It can be implemented in a fully distributed
  fashion (see for instance Oderskys groups
  project PiLib)

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Towards a fully distributed implementation of p

• The results of previous pages show that a fully
  distributed implementation of p must necessarily
  be randomized
• A two-steps approach

Advantages the correctness proof is easier
since (which is the difficult part of the
implementation) is between two similar languages
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ppa the Probabilistic Asynchonous p

• Syntax
• g x(y) t prefixes
• P Si pi gi . Pi pr. inp. guard. choice
  Si pi 1
• xy output action
• P P parallel
• (x) P new name
• recA P recursion
• A procedure name

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The operational semantics of ppa

• Based on the Probabilistic Automata of Segala and
  Lynch
• Distinction between
• nondeterministic behavior (choice of the
  scheduler) and
• probabilistic behavior (choice of the process)

Scheduling Policy The scheduler chooses the
group of transitions
Execution The process chooses probabilistically
the transition within the group
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The operational semantics of ppa

• Representation of a group of transition
• P --gi-gt pi Pi i
• Rules
• Choice Si pi gi . Pi --gi-gt pi Pi i
• P --gi-gt piPi i
• Par ____________________
• Q P --gi-gt piQ Pi i

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The operational semantics of ppa

• Rules (continued)
• P --xi(yi)-gt piPi i Q --xz-gt 1 Q
  i
• Com ____________________________________
• P Q --t-gt piPiz/yi Q xix U
  --xi(yi)-gt pi Pi Q xi/x
• P --xi(yi)-gt piPi i
• Res ___________________ qi renormalized
• (x) P --xi(yi)-gt qi (x) Pi xi / x

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Implementation of ppa

• Compilation in Java ltlt gtgt ppa ? Java
• Distributed
• ltlt P Q gtgt ltlt P gtgt.start() ltlt Q gtgt.start()
  
• Compositional
• ltlt P op Q gtgt ltlt P gtgt jop ltlt Q gtgt for all
  op
• Channels are one-position buffers with
  test-and-set (synchronized) methods for input and
  output

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Encoding p into ppa

• p ? ppa
• Fully distributed
• P Q P Q
• Preserves the communication structure
• P s P s
• Compositional
• P op Q Cop P , Q
• Correct wrt a notion of probabilistic testing
  semantics
• P must O iff P must O with
  prob 1

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Encoding p into ppa

• Idea (from an idea in Nestmann97)
• Every mixed choice is translated into a parallel
  comp. of processes corresponding to the branches,
  plus a lock f
• The input processes compete for acquiring both
  its own lock and the lock of the partner
• The input process which succeeds first,
  establishes the communication. The other
  alternatives are discarded

The problem is reduced to a generalized dining
philosophers problem where each fork (lock) can
be adjacent to more than two philosophers Further,
we can reduce the generalized DP to the classic
case, and then apply the randomized algorithm of
Lehmann and Rabin for the dining philosophers
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Dining Philosophers classic case

• Each fork is shared by exactly two philosophers

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The algorithm of Lehmann and Rabin

1. think
2. choose probabilistically first_fork in
   left,right
3. if not taken(first_fork) then take(first_fork)
   else goto 3
4. if not taken(second_fork) then take(second_fork)
   else release(first_fork) goto 2
5. eat
6. release(second_fork)
7. release(first_fork)
8. goto 1

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Dining Philosophers generalized case

• Each fork can be shared by more than two
  philosophers
• The classical algorithm of Lehmann and Rabin, as
  it is, does not work in the generalized case.
  However, we can transform it into the classic case

Transformation into the classic case each fork
is initially associated with a token. Each phil
needs to acquire a token in order to participate
to the competition. The competing phils determine
a set of subgraphs in which each subgraph
contains at most one cycle
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Generalized philosophers

• Another problem we had to face the solution of
  Lehmann and Rabin works only for fair
  schedulers, while ppa does not provide any
  guarantee of fairness
• Fortunately, it turns out that the fairness is
  required only in order to avoid a busy-waiting
  livelock at instruction 3. If we replace
  busy-waiting with suspension, then the algorithm
  works for any scheduler
• This result was achieved independently also by
  Duflot, Fribourg, Picarronny 02.

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The algorithm of Lehmann and RabinModified so to
avoid the need for fairness
The algorithm of Lehmann and Rabin

1. think
2. choose probabilistically first_fork in
   left,right
3. if not taken(first_fork) then take(first_fork)
   else wait
4. if not taken(second_fork) then take(second_fork)
   else release(first_fork) goto 2
5. eat
6. release(second_fork)
7. release(first_fork)
8. goto 1

1. think
2. choose probabilistically first_fork in
   left,right
3. if not taken(first_fork) then take(first_fork)
   else goto 3
4. if not taken(second_fork) then take(second_fork)
   else release(first_fork) goto 2
5. Eat
6. release(second_fork)
7. release(first_fork)
8. goto 1

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Conclusion

• We have provided an encoding of the
• p calculus into a probabilistic version of its
  asynchronous fragment
• fully distributed
• compositional
• correct wrt a notion of testing semantics
• Advantages
• high-level solutions to distributed algorithms
• Easier to prove correct (no reasoning about
  randomization required)