master-degree-notes/Concurrent Systems/notes/8 - Enhancing Liveness Properties.md

Can we take the most basic protocol that satisfies the most basic liveness property (obstruction freedom) and "upgrade" it to bounded wait freedom?

Contention manager: is an object that allows progress of processes by providing contention-free periods for completing their invocations. It provides 2 operations:

• need_help(i): invoked by p_i when it discovers that there is contention
• stop_help(i): invoked by p_{i} when it terminates its current invocation

Enriched implementation: when a process realizes that there is contention, it invokes need_help; when it completes its current operation, it invokes stop_help.

Why is it different from lock/unlock? Because this allows failures, and they can also happen in the contention-free period.

PROBLEM: to distinguish a failure from a long delay, we need objects called failure detectors, that provide processes information on the failed processes of the system. According to the type/quality of the info, several F.D.s can be defined.

Eventually restricted leadership: given a non-empty set of process IDs X, the failure detector \Omega_{X} provides each process a local variable ev_leader(X) such that:

1. (Validity) ev_leader(x) always contains a process ID
2. (Eventual leadership) eventually, all ev_leader(X) of all non-crashed processes of X for ever contain the same process ID, that is one of them

REMARK: the moment in which all variables contain the same leader is unknown

NEED_HELP[1..n] : SWMR atomic R/W boolean registers init at false

need_help(i) :=
	NEED_HELP[i] <- true
	repeat
		X <- {j : NEED_HELP[j]}
	until ev_leader(X) = i

stop_help(i) :=
	NEED_HELP[i] <- false

Theorem: the contention manager just seen transforms an obstr.-free implementation into a non-blocking enriched implementation.

Proof: By contr., \exists \tau s.t. \exists many (> 0) op.'s invoked concurrently that never terminate. Let Q be the set of proc.'s that performed these invocations.

• by enrichment, eventually NEED_HELP[i]=T (\forall i\in Q) forever
• since crashed are fail-stop, eventually NEED_HELP[j] is no longer modified (\forall j \not \in Q)
  • \exists \tau' \geq \tau where all proc.'s in Q compute the same X

Observation: Q \subseteq X (it is possible that p_j sets NEED_HELP[j] and then fails)

By definition of \Omega_{X}, \exists \tau'' \geq t' s.t. all proc.'s in Q have the same ev_leader(X)

• the leader belongs to Q, since it cannot be failed
• this is the only process allowed to proceed
• because run in isolation, it eventually terminates (because of obstruction freedom)

On implementing \Omega

It can be proved that there exists no wait-free implementation of \Omega in an asynchronous system with atomic R/W registers and any number of crashes

• crashes are indistinguishable from long delays
• need of timing constraints

1. \exists time \tau_{1}, time interval \nabla and correct process p_{L} s.t. after \tau_{1} every two consecutive writes to a specific SWMR atomic R/W by p_{L} are at most \nabla time units apart one from the other

2. let t be an upper bound on the number of possible failing processes and f the real number of process failed (hence, 0\leq f\leq t\leq n-1, with f unknown and t known in advance). Then, there are at least t-f correct processes different from p_L with a timer s.t. \exists time \tau_{2} \forall time interval \delta, if their timer is set to \delta after \tau_{2}, it expires at least after \delta.

REMARK: \tau_{1}, \tau_{2}, \nabla and p_L are all unknown.

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