10 December 2002ENS Cachan1 Generalized dining philosophers Catuscia Palamidessi, INRIA in collaboration with Mihaela Oltea Herescu, IBM Michael Pilquist, PSU

10 December 2002ENS Cachan2 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).

10 December 2002ENS Cachan3 Dining Philosophers: classic case Each philosopher needs exactly two forks Each fork is shared by exactly two philosophers

10 December 2002ENS Cachan4 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

10 December 2002ENS Cachan5 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

10 December 2002ENS Cachan6 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

10 December 2002ENS Cachan7 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  - 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  -calculus (no choice, no output guards) as an intermediate language. The asynchronous  - calculus can be implemented in a distributed, modular way.

10 December 2002ENS Cachan8 Encoding  into  pa 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 f 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. S R’i f Si

10 December 2002ENS Cachan9 Encoding  into asynchonous  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  for the implementation. Note that the DP problem can be solved in  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  into an asynchronous language.

10 December 2002ENS Cachan10 The algorithm of Lehmann and Rabin 1. Think 2. choose first_fork in {left,right} %commit 3. if taken(first_fork) then goto 3 4. take(first_fork) 5. if taken(first_fork) then goto 2 6. take(second_fork) 7. eat 8. release(second_fork) 9. release(first_fork) 10. goto 1

10 December 2002ENS Cachan11 Problems Wrt to our encoding goal, the algorithm of Lehmann and Rabin has two problems: 1. It only works for certain kinds of graphs 2. It works only for fair schedulers 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

10 December 2002ENS Cachan12 The algorithm of Lehmann and Rabin Modified so to avoid the need for fairness 1. Think 2. choose first_fork in {left,right} %commit 3. if taken(first_fork) then wait 4. take(first_fork) 5. if taken(first_fork) then goto 2 6. take(second_fork) 7. eat 8. release(second_fork) 9. release(first_fork) 10. goto 1

10 December 2002ENS Cachan13 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:

10 December 2002ENS Cachan14 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 taken committed

10 December 2002ENS Cachan15 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

10 December 2002ENS Cachan16 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.