vault backup: 2025-04-05 00:07:51
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11 changed files with 30 additions and 30 deletions
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@ -62,7 +62,7 @@ function withdraw() {
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While `read()` and `write()` may be considered as atomic, their sequential composition **is not**.
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#### Mutual Exclusion (MUTEX)
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Ensure that some parts of the code are executed as *atomic*.
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@ -100,7 +100,7 @@ Every solution to a problem should satisfy at least:
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**Both inclusions are strict:**
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$$\text{Deadlock freedom} \not \implies \text{Starvation freedom}$$
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*p1 is starving!*
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$$\text{Starvation freedom} \not \implies \text{Bounded bypass}$$
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Assume a $f$ and consider the scheduling above, where p2 wins $f(3)$ times and so does p3, then p1 looses (at least) $2f(3)$ times before winning.
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@ -30,7 +30,7 @@ A configuration C obtained during the execution of all A is called:
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If A wait-free implements binary consensus for n processes, then there exists a bivalent initial configuration.
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*Proof:*
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### CN(Atomic R/W registers) = 1
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**Thm:** There exists no wait-free implementation of binary consensus for 2 processes that uses atomic R/W registers.
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@ -135,7 +135,7 @@ propose(i, v) :=
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Let us consider a verison of the compare&swap where, instead of returning a boolean, it always returns the previous value of the object, i.e.:
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```
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CS a compare&swap object init at ⊥
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@ -56,7 +56,7 @@ a) `FLAG[1] = down`, this is possible only with the following interleaving:
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b) `AFTER_YOU = 1`, this is possible only with the following interleaving:
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##### Bounded Bypass proof (with bound = 1)
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- If the wait condition is true, then it wins (and waits 0).
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@ -49,9 +49,9 @@ unlock(i) :=
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##### MUTEX proof
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How can pi enter its CS?
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(*must finished before nel senso che $p_i$ deve aspettare $p_j$*)
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##### Deadlock freedom
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Let $p_i$ invoke lock
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@ -72,6 +72,6 @@ Let $p_i$ invoke lock
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- In the second wait Y = ⊥: but then there exists a $p_h$ that eventually enters its CS -> good
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- In the ∀j.wait FLAG[j]=down: this wait cannot block a process forever
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esercizio: prova che NON soddisfa starvation freedom
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@ -50,5 +50,5 @@ By Deadlock freedom of RR, at least one process eventually unlocks
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The worst case is when TURN = *i+1* mod n when FLAG[i] is set.
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@ -31,7 +31,7 @@ unlock(i) :=
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*Proof:* by contradiction.
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Let's consider the execution of $p_i$ leading to its CS:
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**Corollary** (of the MUTEX proof)**:** DATE is never written concurrently.
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@ -57,7 +57,7 @@ Let's consider the execution of $p_i$ leading to its CS:
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**Theorem:** the algorithm satisfies bounded bypass with bound $2n-2$.
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*Proof:*
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so by this, the very worst possible case is that my lock experiences that.
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It looks like I can experience at most $2n-1$ other critical sections, but it is even better, let's see:
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@ -105,7 +105,7 @@ The **casual past** of a transaction T is the set of all T' and T'' such that
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VWC allows more transactions to commit -> it is a more liberal property than opacity.
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#### A Vector clock based STM system
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We have m shared MRMW registers; register X is represented by a pair XX, with:
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@ -21,9 +21,9 @@ A complete history $\hat{H}$ is **linearizable** if there exists a sequential hi
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Given an history $\hat{K}$, we can define a binary relation on events $⟶_{K}$ s.t. (op, op’) ∈ ⟶K if and only if res[op] <K inv[op’]. We write op ⟶K op’ for denoting (op, op’) ∈ ⟶K. Hence, condition 3 of the previous Def. requires that ⟶H ⊆ ⟶S.
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But there is another linearization possible! I can also push a before if I pull it before c!
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But there is another linearization possible! I can also push a before if I pull it before c!
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Of course I have to respect the semantics of a Queue (if I push "a" first, I have to pop "a" first because it's a fucking FIFO)
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#### Compositionality theorem
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@ -2,9 +2,9 @@ Let us define $op ->_{proc} op'$ to hold whenever there exists a process p that
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### Sequential consistency
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**Def:** a complete history is sequentially consistent if there exists a sequential history $𝑆$ s.t.
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>[!warning]
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>The problem with sequential consistency is that it is NOT compositional.
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@ -64,7 +64,7 @@ this implementation satisfies the three requirements for the splitter
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- let us consider the last process that writes into LAST (this is an atomic register, so this is meaningful)
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- if the door is closed, it receives R and √
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3. let $p_i$ be the first process that receives $S \to LAST=i$ in its second if
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### An Obstruction-free Timestamp Generator
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A **timestamp generator** is a concurrent object that provides a single operation get_ts such that:
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@ -98,7 +98,7 @@ this implementation satisfies the three properties of the timestamp generator
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- every process that starts after its termination will find NEXT to a greater value (NEXT never decreases!)
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3. Obstruction freedom is trivial
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**REMARK:** this implementation doesn’t satisfy the non-blocking property:
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**REMARK:** this implementation doesn’t satisfy the non-blocking property:
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### A Wait-free Stack
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REG is an unbounded array of atomic registers (the stack)
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@ -149,7 +149,7 @@ This is needed for the so called ABA problem with compare&set:
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- with the compare&set you mainly test that the sequence_number has not changed
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TOP : a register that can be read or compare&setted
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```
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push(w) :=
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