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

This lecture explores the probabilistic asynchronous p-calculus, its operational semantics, expressive power, and implementation in Java.

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

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  1. Probabilistic Methods in ConcurrencyLecture 7The probabilistic asynchronous p-calculus Catuscia Palamidessi catuscia@lix.polytechnique.fr www.lix.polytechnique.fr/~catuscia Page of the course: www.lix.polytechnique.fr/~catuscia/teaching/Pisa/

  2. 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 << >> distributed machine Prob methods in Concurrency

  3. ppa: the Probabilistic Asynchonous p Syntax: based on the asynchronous p of Amadio, Castellani, Sangiorgi g ::= x(y) | tprefixes P ::= Sipigi . Pi pr. inp. guard. choiceSi pi = 1 | x^y output action | P | Pparallel | (x) P new name | recA P recursion | Aprocedure name Prob methods in Concurrency

  4. 1/2 1/3 1/2 1/3 1/2 1/3 1/3 1/3 1/2 1/3 1/2 1/2 1/2 1/3 1/3 1/3 1/2 1/2 1/2 1/3 1/3 1/3 1/3 1/3 1/3 1/2 1/2 1/2 1/3 1/3 1/3 1/2 1/2 1/2 2/3 1/3 1/3 1/3 1/3 1/3 1/3 1/3 2/3 1/3 2/3 2/3 2/3 2/3 2/3 2/3 1/3 1/3 1/3 1/3 1/3 1/3 1/2 1/3 1/3 1/2 1/3 2/3 1/3 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 choosesprobabilistically the transition within the group Prob methods in Concurrency

  5. The operational semantics of ppa • Representation of a group of transition P { --gi-> piPi } i • Rules Choice Si pi gi . Pi {--gi-> piPi }i P{--gi-> piPi }i Par ____________________ Q | P {--gi-> piQ | Pi }i Prob methods in Concurrency

  6. The operational semantics of ppa • Rules (continued) P{--xi(yi)-> piPi }i Q{--x^z-> 1 Q’}i Com ____________________________________ P | Q {--t-> piPi[z/yi]|Q’ }xi=x U { --xi(yi)-> pi Pi |Q }xi=/=x P{--xi(yi)-> piPi }i Res ___________________ qi renormalized (x) P { --xi(yi)-> qi (x) Pi }xi =/= x Prob methods in Concurrency

  7. y P Q x The expressive power of p • Example of distributed agreement: the leader election problem Prob methods in Concurrency

  8. Implementation of ppa • Compilation in Java << >> : ppa Java • Distributed << P | Q >> = << P >>.start(); << Q >>.start(); • Compositional << P op Q >> = << P >> jop << Q >> 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 Prob methods in Concurrency

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