1. Catuscia Palamidessi, INRIA Saclay, France
On the expressive power of synchronization primitives in the π-calculus
Catuscia Palamidessi, INRIA Saclay, France
14 October 2009
BASICS'09, Shanghai

2. Focus on the -calculus
Contents
The -calculus with mixed choice ()
Expressive power of the -calculus and problems with its fully distributed implementation
The asynchronous -calculus (a)
The π hierarchy
Towards a randomized fully distributed implementation of 
The probabilistic asynchronous -calculus (pa)
Encoding  into pa using a generalized dining cryptographers algorithm
14 October 2009
BASICS'09, Shanghai

3. The p-calculus Q z z x y R P z
Proposed by [Milner, Parrow, Walker ‘92] as a formal language to reason about concurrent systems
Concurrent: several processes running in parallel
Asynchronous cooperation: every process proceeds at its own speed
Synchronous communication: handshaking, input and output prefix
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
14 October 2009
BASICS'09, Shanghai

4. p: the p-calculus with mixed choice
Syntax
g ::= x(y) | x^y | 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
14 October 2009
BASICS'09, Shanghai

5. Operational semantics
Transition system P -a Q
Rules
Choice Si gi . Pi –gi Pi
P -x^y P’
Open ___________________
(y) P -x^(y) P’
14 October 2009
BASICS'09, Shanghai

6. Operational semantics
Rules (continued)
P -x(y) P’ Q -x^z 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 _________________ f(Q), b(g) disjoint
Q | P -g Q | P
14 October 2009
BASICS'09, Shanghai

7. 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
14 October 2009
BASICS'09, Shanghai

8. 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
x.Pwins + y^.Ploses | y.Qwins + x^.Qloses P Q y x
 Ploses | Qwins
 Pwins | Qloses
14 October 2009
BASICS'09, Shanghai

9. 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
14 October 2009
BASICS'09, Shanghai

10. A solution to the leader election problem in p
looser
winner
winner
winner
looser
looser
14 October 2009
BASICS'09, Shanghai

11. 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
In this talk we propose an implementation fully distributed, symmetric, and using no synchronous hypotheses.
14 October 2009
BASICS'09, Shanghai

12. 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-Tokoro’91, Boudol, ’92] (Asynchronous = Asynchronous communication)
14 October 2009
BASICS'09, Shanghai

13. pa : the Asynchonous p Version of [Amadio, Castellani, Sangiorgi ’97]
Syntax
g ::= x(y) | t prefixes
P ::= Si gi . Pi input guarded choice
| x^y output action
| P | P parallel
| (x) P new name
| recA P recursion
| A procedure name
14 October 2009
BASICS'09, Shanghai

14. Characteristics of pa Asynchronous communication:
we can’t write a continuation after an output,
i.e. no x^y.P, but only x^y | 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 Odersky’s group’s project PiLib)
14 October 2009
BASICS'09, Shanghai

15. The π hierarchy
We can relate the various sublanguages of π by using encodings
Preserving certain observable properties of runs. Here we will consider as observable properties the presence/absence of certain actions.
Existence of such encoding represented by
14 October 2009
BASICS'09, Shanghai

16. The π hierarchy mixed choice Value-passing CCS Internal mobility
Separate choice
Input guarded choice
output prefix
asynchronous
14 October 2009
BASICS'09, Shanghai

17. The π hierarchy mixed choice Palamidessi Value-passing CCS
Internal mobility
Separate choice
Input guarded choice
output prefix
Nestmann
asynchronous
14 October 2009
BASICS'09, Shanghai

18. Separation result 1
It is not possible to encode mixed-choice π into separate-choice π
Homomorphically wrt |:
Preserving 2 distinct observable actions
This result is based on a sort of confluence property, which holds for the separate-choice π and not for the separate-choice π
The proof proceeds by showing that the separate-choice π cannot solve the leader election problem for 2 nodes
14 October 2009
BASICS'09, Shanghai

19. Separation result 2
It is not possible to encode mixed-choice π into value-passing ccs or π with internal mobil.
Homomorphically wrt |:
Without introducing extra channels
Preserving 2 distinct observable actions
The proof proceeds by showing that the separate-choice π cannot solve the leader election problem for certain kinds of graphs
14 October 2009
BASICS'09, Shanghai

20. 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: p
probabilistic asynchronous p
distributed machine
[[ ]]
<< >>
Advantages:
the correctness proof is easier since [[ ]] (which is the difficult part of the implementation) is between two similar languages
14 October 2009
BASICS'09, Shanghai

21. ppa: the Probabilistic Asynchonous p
Syntax
g ::= x(y) | t prefixes
P ::= Si pi gi . Pi pr. inp. guard. choice Si pi = 1
| x^y output action
| P | P parallel
| (x) P new name
| recA P recursion
| A procedure name
14 October 2009
BASICS'09, Shanghai

22. 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
1/2
1/3
2/3
1/2
1/3
2/3
1/2
1/3
2/3
Execution:
The process chooses probabilistically the transition within the group
14 October 2009
BASICS'09, Shanghai

23. The operational semantics of ppa
Representation of a group of transition
P { --gi-> pi Pi } i
Rules
Choice Si pi gi . Pi {--gi-> pi Pi }i
P {--gi-> piPi }i
Par ____________________
Q | P {--gi-> piQ | Pi }i
14 October 2009
BASICS'09, Shanghai

24. The operational semantics of ppa
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
14 October 2009
BASICS'09, Shanghai

25. Implementation of ppa Compilation in a DM << >> : ppa  DM
Distributed
<< P | Q >> = << P >>.start() | << Q >>.start();
Compositional
<< P op Q >> = << P >> jop << Q >> for all op
Channels are buffers with test-and-set (synchronized) methods for input and output. The input-guarded choice selects probabilistically one of the channels with available data
14 October 2009
BASICS'09, Shanghai

26. 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
14 October 2009
BASICS'09, Shanghai

27. Encoding p into ppa Idea (from an idea of Uwe Nestmann):
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 P Q R Pi Qi Ri f S
R’i Si
The problem is reduced to a dining philosophers problem: each lock is a fork, each input process is a philosopher, and enters a competition to get his adjacent forks. The winners of the competition can synchronize, which corresponds to eating in the DP. There can be more than one winner
Generalized DP: each fork can be adjacent to more than two Philosophers
14 October 2009
BASICS'09, Shanghai

28. Dining Philosophers: classic case
Each fork is shared by exactly two philosophers
14 October 2009
BASICS'09, Shanghai

29. Dining Philosophers: generalized case
Each fork can be shared by more than two philosophers
14 October 2009
BASICS'09, Shanghai

30. Intended properties of solution
Deadlock freedom (aka progress): if there is a hungry philosopher, a philosopher will eventually eat
Starvation freedom: every hungry philosopher will eventually eat (but we won’t consider this property here)
Robustness wrt a large class of schedulers: A scheduler decides who does the next move , not necessarily in cooperation with the program, maybe even against it
Fully distributed: no centralized control or memory
Symmetric:
All philosophers run the same code and are in the same initial state
The same holds for the forks
14 October 2009
BASICS'09, Shanghai

31. The Dining Philosophers - a brief history
Problem proposed by Edsger Dijkstra in 1965 (actually the popular formulation is due to Tony Hoare)
Many solutions had been proposed for the DP, but none of them satisfied all requirements
In 1981, Lehmann and Rabin proved that
There was no “deterministic” solution satisfying all requirements
They proposed a randomized solution and proved that it satisfies all requirement. Progress is satisfied in the probabilistic sense, I.e. there is probability 1 that a philosopherwill eventually eat.
Meanwhile, Francez and Rodeh had come out in 1980 with solution to the DC written in CSP
The controversy was solved by Lehmann and Rabin who proved that CSP (with guarded choice) is not implementable in a distributed fashion (deterministically).
14 October 2009
BASICS'09, Shanghai

32. The algorithm of Lehmann and Rabin
think;
choose probabilistically first_fork in {left,right};
if not taken(first_fork) then take(first_fork) else goto 3;
if not taken(second_fork) then take(second_fork); else { release(first_fork); goto 2 }
eat;
release(second_fork);
release(first_fork);
goto 1
14 October 2009
BASICS'09, Shanghai

33. Problems
Wrt to our encoding goal, the algorithm of Lehmann and Rabin has two problems:
It only works for the classical case (not for the generalized one)
It works only for fair schedulers
14 October 2009
BASICS'09, Shanghai

34. Conditions on the graph
Theorem: The algorithm of Lehmann and Rabin is deadlock-free if and only if all cycles are pairwise disconnected
There are essentially three ways in which two cycles can be connected:
14 October 2009
BASICS'09, Shanghai

35. Proof of the theorem committed taken
If part) Each cycle can be considered separately. On each of them the classic algorithm is deadlock-free. Some additional care must be taken for the arcs that are not part of the cycle.
Only if part) By analysis of the three possible cases. Actually they are all similar. We illustrate the first case
committed
taken
14 October 2009
BASICS'09, Shanghai

36. Proof of the theorem The initial situation has probability p > 0
The scheduler forces the processes to loop
Hence the system has a deadlock (livelock) with probability p
Note that this scheduler is not fair. However we can define even a fair scheduler which induces an infinite loop with probability > 0. The idea is to have a scheduler that “gives up” after n attempts when the process keep choosing the “wrong” fork, but that increases (by f) its “stubborness” at every round.
With a suitable choice of n and f we have that the probability of a loop is p/4
14 October 2009
BASICS'09, Shanghai

37. Solution for the Generalized DP
As we have seen, the algorithm of Lehmann and Rabin does not work on general graphs
However, it is easy to modify the algorithm so that it works in general
The idea is to reduce the problem to the pairwise disconnetted cycles case:
Each fork is initially associated with one token. Each philosopher needs to acquire a token in order to participate to the competition. After this initial phase, the algorithm is the same as the Lehman & Rabin’s
Theorem: The competing philosophers determine a graph in which all cycles are pairwise disconnected
Proof: By case analysis. To have a situation with two connected cycles we would need a node with two tokens.
14 October 2009
BASICS'09, Shanghai

38. Generalized philosophers
The other 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].
14 October 2009
BASICS'09, Shanghai

39. The algorithm of Lehmann and Rabin
The algorithm of Lehmann and Rabin Modified so to avoid the need for fairness
The algorithm of Lehmann and Rabin
(second_fork);
release(first_fork);
goto 1 think;
choose probabilistically first_fork in {left,right};
if not taken(first_fork) then take(first_fork) else goto 3;
if not taken(second_fork) then take(second_fork); else { release(first_fork); goto 2 }
Eat;
release
think;
choose probabilistically first_fork in {left,right};
if not taken(first_fork) then take(first_fork) else wait;
if not taken(second_fork) then take(second_fork); else { release(first_fork); goto 2 }
eat;
release(second_fork);
release(first_fork);
goto 1
14 October 2009
BASICS'09, Shanghai

40. The encoding [[ (x) P ]] = (x) [[ P ]] [[P | Q ]] = [[ P ]] | [[ Q ]]
[[ ∑ gi.Pi ]] = the translation we have just seen
Theorem: For every P, [[P]] and P are testing-equivalent. Namely for every test T,
inf (Prob (succ, [[ P ]] | [[T ]] ) = inf (Prob (succ , P | T))
sup (Prob (succ, [[ P ]] | [[T ]] ) = sup (Prob (succ , P | T))
14 October 2009
BASICS'09, Shanghai

41. 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)
14 October 2009
BASICS'09, Shanghai