Generalized dining philosophers

1. Generalized dining philosophers Catuscia Palamidessi, INRIA in collaboration with Mihaela Oltea Herescu, IBM Michael Pilquist, PSU ENS Cachan

2. Plan of the talk • What we mean by “Generalized dining philosophers” • two levels of generalization • Motivations • The classic deadlock-free solution by Lehmann and Rabin • Problems with the classic solution • fairness assumption adversaries • restrictions on the connection structure • A solution for the first generalization (w/o the fairness assumption) • A solution for the second generalization (w/ fairness assumption). ENS Cachan

3. Dining Philosophers: classic case • Each philosopher needs exactly two forks • Each fork is shared by exactly two philosophers ENS Cachan

4. Dining Phils: generalization 1 • Each phil still needs two forks, but • Each fork can now be shared by more than two philosophers • We will consider arbitrary graphs • arcs  philosophers • vertices forks • Of course, the problem is interesting when there is at least one cycle ENS Cachan

5. Dining Phils: generalization 2 • Each phil needs an arbitrary number of forks • Each fork can be shared by an arbitrary number of phils • We will consider arbitrary bipartite graphs • vertices 1 philosophers • vertices 2 forks • Of course, the problem is interesting when there is at least one cycle ENS Cachan

6. Intended properties of soln • 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 adversaries: Adversaries decide who does the next move (schedulers) • 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 ENS Cachan

7. Motivations • To model real systems, in which the relation between resources and users may be more complicate than a ring • Our main motivation: Distributed and modular implementation of the mixed choice mechanism of the p-calculus (and CCS, CSP etc.) • mixed choice : both input and output guards are present (out(a).P1 + in(b).P2) | (in(a).Q1 + out(b).Q2)  P1 | Q1  P2 | Q2 • Our approach: use the asynchronous p-calculus (no choice, no output guards) as an intermediate language. The asynchronous p-calculus can be implemented in a distributed, modular way. ENS Cachan

8. Encoding p into ppa • 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 f Pi P R f Ri Qi R’i Q S f Si f The problem is reduced to a generalized DP problem where each fork (lock) can be adjacent to more than two philosophers (generalization 1). We are interested in progress, i.e. deadlock freedom. ENS Cachan

9. Encoding p into asynchonous p • The requirements on the encoding imply symmetry and full distribution • There are many solution to the DP problem, but only randomized solutions can be symmetric and fully distributed. [Lehmann and Rabin 81] - They also provided a randomized algorithm (for the classic case) Hence we have actually considered a probabilistic extension of the asynchronous p for the implementation. • Note that the DP problem can be solved in p in a fully distributed, symmetric way [Francez and Rodeh ’80] – they actually use CSP Hence the need for randomization is not a characteristic of our approach: it would arise in any encoding of pinto an asynchronous language. ENS Cachan

10. The algorithm of Lehmann and Rabin • Think • choose first_fork in {left,right} %commit • if taken(first_fork) then goto 3 • take(first_fork) • if taken(first_fork) then goto 2 • take(second_fork) • eat • release(second_fork) • release(first_fork) • goto 1 ENS Cachan

11. Problems • Wrt to our encoding goal, the algorithm of Lehmann and Rabin has two problems: • It only works for certain kinds of graphs • It works only for fairschedulers • Problem 2 however can be solved by replacing the busy waiting in step 3 with suspension. [Duflot, Friburg, Picaronny 2002] – see also Herescu’s PhD thesis ENS Cachan

12. The algorithm of Lehmann and RabinModified so to avoid the need for fairness • Think • choose first_fork in {left,right} %commit • if taken(first_fork) then wait • take(first_fork) • if taken(first_fork) then goto 2 • take(second_fork) • eat • release(second_fork) • release(first_fork) • goto 1 ENS Cachan

13. Restriction 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: ENS Cachan

14. Proof of the theorem • 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 ENS Cachan

15. 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 ENS Cachan

16. Solution for the Generalization 1 • 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 phil needs to acquire a token in order to participate to the competition. After this initial phase, the algorithm is the same as the Lemahn & Rabin’s Theorem: The competing phils 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. ENS Cachan