master-degree-notes/Concurrent Systems/notes/3.md

Hardware primitives

Atomic R/W registers provide quite a basic computational model.

We can strenghten the model by adding specialized HW primitives, that essentially perform in an atomic way the combination of some atomic instructions. Usually, every operating system provides at least one specilized HW primitive.

Most common ones:

• Test&set: atomic read+write of a boolean register
• Swap: atomic read+write of a general register (generalization of the above)
• Fetch&add: atomic read+increase of an integer register
• Compare&swap: tests the value of a general register, returns a boolean (result of the comparison, true if it is the same).

Test&Set

Let X be a boolean register

X.test&set() :=
	tmp <- X
	X <- 1
	return tmp

(the function is implemented in an atomic way by the hardware by suspending the interruptions!)

How do we use it for MUTEX?

lock() :=
	wait X.test&set() = 0
	return


unlock() :=
	X <- 0
	return

Swap

X general register

X.swap(v) :=
	tmp <- X
	X <- v
	return tmp

How do we use it for MUTEX?

lock() :=
	wait X.swap(1) = 0
	return

unlock() :=
	X <- 0
	return

Compare&swap

X boolean register

X.compare&swap(old, new) :=
	if X = old then
		X <- new
		return true
	return false

How do we use it for MUTEX?

Initialize X at 0

lock() :=
	wait X.compare&swap(0, 1) = true
	return

unlock() :=
	X <- 0
	return

Fetch&add

Up to now, all solutions enjoy deadlock freedom, but allow for starvation. So let's use Round Robin to promote the liveness property!

Let X be an integer register; the Fetch&add primitive is implemented as follows:

X.fetch&add(v) :=
	tmp <- X
	X <- X+v
	return tmp

How do we use it for MUTEX?

Initialize TICKET and NEXT at 0

lock() :=
	my_ticket <- TICKET.fetch&add(1)
	wait my_ticket = NEXT
	return

unlock() :=
	NEXT <- NEXT + 1
	return

[!note] a>It is bounded bypass with bound n-1>

Safe registers

Atomic R/W and specialized HW primitives provide some form of atomicity. But is it possible to enforce MUTEX without atomicity?

A MRSW Safe register is a register that provides READ and WRITE such that:

1. every READ that does not overlap with a WRITE returns the value stored in the register
2. a READ that overlaps with a WRITE can return any possible value (of the register domain).

A MRMW Safe register behaves like a MRSW safe register, when WRITE operations do not overlap. Otherwise, in case of overlapping WRITEs, the register can contain any value (of the register domain). (Ci sta comunque lo stesso problema)

Si chiama safe nel senso che se ci sono letture overlapping siamo safe. Ma è letteralmente l'unica cosa safe che abbiamo!

This is the weakest type of register that is useful in concurrency.

Bakery Algorithm

The idea is that

• every process gets a ticket
• because we don't have atomicity, tickets may be not unique
• tickets can be made unique by pairing them with the process ID
• the smallest ticket (seen as a pair) grants the access to the CS

Initialize FLAG[i] to down and MY_TURN[i] to 0, for all i

lock(i) :=
	FLAG[i] <- up
	MY_TURN[i] <- max{MY_TURN[1],...,MY_TURN[n]}+1
	FLAG[i] <- down
	forall j != i
		wait FLAG[j] = down
		wait (MY_TURN[j] = 0 OR ⟨MY_TURN[i],i⟩ < ⟨MY_TURN[j],j⟩)

unlock(i) :=
	MY_TURN[i] <- 0

Se il ticket number è minore si ottiene l'accesso, se il ticket number è uguale, allora si vede il process ID minore.

It is possible that while reading, other processes are writing their turn! So it's possible that I read something unpredictable! We need to consider it in the proofs.

Why is the flag needed?

Without it, if two process i and j sets MY_TURN at the same time, then i goes to sleep, j enters the CS, but when i wakes up, it will enter too!

MUTEX proof

Lemma 1: if p_i enters the bakery before p_j, then MY_TURN[i] < MY_TURN[j].

Proof:

• let t be the value of MY_TURN[i] after p_{i} exits the doorway (after flag <- down)
• when p_j computes its ticket, it reads t from MY_TURN[i], it reads t from MY_TURN[i] (no write overlapping with this read)
• Hence, MY_TURN[j] is at least t+1. (legge correttamente almeno t, gli altri boh ma non ci interessa ora)

Lemma 2: Let p_i be in the CS and p_j is in the doorway or in the bakery; then, $$⟨MY_TURN[i] , i⟩ < ⟨MY_TURN[j] , j⟩$$Proof: If p_i is in the CS, it has terminated its first wait for j let's consider the read of FLAG[j] done by p_i that terminates such wait

W.r.t. the execution of p_j, it can be that

• this read overlaps with FLAG[j] <- up by Lemma 1, MY_TURN[i] < MY_TURN[j] and ok
• this read is contained within the computation of MY_TURN[j], but it's not possible, since MY_TURN is computed with the FLAG up
• this read overlaps with FLAG[j] <- down or this read happens when p_j is in the bakery:
  • MY_TURN[j] has been decided and no write will change it until p_j is in the bakery, and MY_TURN[j] > 0 of course
  • When p_i evaluated the second wait for j, it found out that ⟨MY_TURN[i],i⟩ < ⟨MY_TURN[j],j⟩, good!

MUTEX: p_i and p_j cannot simultaneously be in the CS: Proof: By contradiction, by Lemma 2 applied twice, we would have:

⟨MY\_TURN[i] , i⟩ < ⟨MY\_TURN[j] , j⟩ \land ⟨MY\_TURN[j] , j⟩ < ⟨MY\_TURN[i] , i⟩

Which is of course not possible, how in the world is it possible that x < y and y < x at the same time? √

Deadlock freedom proof

By contradiction, assume that there is a lock but nobody enters its CS

• All processes in the bakery (we will call this set Q) are blocked in their wait
• The first wait cannot block forever
  • All p_i \in Q have their FLAG down
  • All p_i \not \in Q have their FLAG down (if not in the doorway) or will eventually put their FLAG down (cannot remain in the doorway forever)
• The second wait cannot block all of them forever
  • Tickets can be totally ordered (lexicographically)
  • Let <MY_TURN[i], j> be the minimun
  • The second wait evaluated by p_i eventually succeeds for all j
    • if p_j is before the doorway, then MY_TURN[j] = 0
    • if p_{j} is in the doorway, then MY_TURN[i] < MY_TURN[j] (bc of Lemma 1)
    • if p_j is in the bakery, by assumption ⟨MY_TURN[i] , i⟩ < ⟨MY_TURN[j] , j⟩ since it is the minimum.

Bounded bypass proof (bound n-1)

Let p_i and p_j competing for the CS and p_j wins

Then, p_j enters its CS, completes it, unlocks and then invokes lock again

• If p_i has entered the CS, √
• Otherwise, by Lemma1, MY\_TURN[i] < MY\_TURN[j], then p_j cannot bypass p_i again!
• At worse, pi has to wait all other proceeses before entering its CS
  • (indeed, since there is no deadlock, when pi is waiting somebody enters the CS)

Aravind’s algorithm

Problem with Lamport's algorithm: registers must be unbounded (every invocation of lock potentially increases the counter by 1 -> domain of the registers is all natural numbers!)

For all processes, we have a FLAG and a STAGE (both binary MRSW) and a DATE (MRMW) register that ranges from 1 to 2n.

For all i, initialize
	FLAG[i] to down
	STAGE[i] to 0
	DATE[i] to i

lock(i) :=
	FLAG[i] <- up
	repeat
		STAGE[i] <- 0
		wait (foreach j != i, FLAG[j] = down OR DATE[i] < DATE[j])
		STAGE[i] <- 1
	until foreach j != i, STAGE[j] = 0

unlock(i) :=
	tmp <- max_j{DATE[j]}+1
	if tmp >= 2n
		then foreach j, DATE[j] <- j
		else DATE[i] <- tmp
	STAGE[i] <- 0
	FLAG[i] <- down

MUTEX proof

Theorem: if p_i is in the CS, then p_j cannot simultaneously be in the CS. Proof: by contradiction.

Let's consider the execution of p_i leading to its CS: !

Corollary (of the MUTEX proof): DATE is never written concurrently.

Bounded bypass proof

Lemma 1: exactly after n CSs there is a reset of DATE. Proof:

• the first CS leads max_j{DATE[j]} to n+1
• the seconds CS leads ... to n+2
• ...
• the n-th read leads ... to n+n = 2n -> RESET

Lemma 2: there can be at most one reset of DATE during an invocation of a lock Proof:

• let p_i invoke lock, if no reset occurs, ok
• otherwise, let us consider the moment in which a reset occurs
  • if pi is the next process that enters the CS, ok
  • Otherwise let p_j be the process that enters; its next date is n+1 > DATE[i]
    • p_{j} cannot surpass p_i again (before a RESET)
  • The worst case is then all processes perform lock together and i = n (i am process n)
    • all p_{1}\dots p_{n} surpass p_{n}
    • then p_n enters and it resets the DATE in its unlock
      • only 1 reset and it is the worst case!

Theorem: the algorithm satisfies bounded bypass with bound 2n-2. Proof: ! so by this, the very worst possible case is that my lock experiences that.

It looks like I can experience at most 2n-1 other critical sections, but it is even better, let's see: ...

Improvement of Aravind’s algorithm

unlock(i) :=
	∀j≠i.if DATE[j] > DATE[i] then
		DATE[j] <- DATE[j]-1
		DATE[i] <- n STAGE[i] <- 0
		FLAG[i] <- down

Since the LOCK is like before, the revised protocol satisfies MUTEX. Furthermore, you can prove that it satisfies bounded bypass with bound n-1 -> EXERCISE!