Probabilistic Methods in Concurrency Lecture 7 The probabilistic asynchronous p-calculus

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Probabilistic Methods in ConcurrencyLecture 7
The probabilistic asynchronous p-calculus

• Catuscia Palamidessi
• catuscia_at_lix.polytechnique.fr
• www.lix.polytechnique.fr/catuscia
• Page of the course
• www.lix.polytechnique.fr/catuscia/teaching/Pisa/

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The probabilistic asynchronous p-calculus

• Originally developed as an intermediate language
  for the fully distributed implementation of the
  p-calculus (Herescu and Palamidessi)
• The results of Lecture 4 show that a fully
  distributed implementation of p must necessarily
  be randomized
• A two-steps approach

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

probabilistic asynchronous p
ltlt gtgt
distributed machine
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ppa the Probabilistic Asynchonous p

• Syntax based on the asynchronous p of Amadio,
  Castellani, Sangiorgi
• 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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The expressive power of p

• Example of distributed agreement
• the leader election problem

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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
• The probabilistic input guarded construct is
  implemented as a while loop in which channels to
  be tried are selected according to their
  probability. The loop repeats until an input is
  successful