vault backup: 2025-04-04 23:39:27
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13 changed files with 28 additions and 28 deletions
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.obsidian/workspace.json
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.obsidian/workspace.json
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@ -13,12 +13,12 @@
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"state": {
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"type": "markdown",
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"state": {
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"file": "Concurrent Systems/notes/2 - Fast mutex by Lamport.md",
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"file": "Concurrent Systems/notes/4c - Dining Philosophers.md",
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"mode": "source",
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"source": false
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},
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"icon": "lucide-file",
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"title": "2 - Fast mutex by Lamport"
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"title": "4c - Dining Philosophers"
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}
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},
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{
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@ -65,7 +65,7 @@
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"state": {
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"type": "search",
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"state": {
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"query": "",
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"query": "images/",
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"matchingCase": true,
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"explainSearch": false,
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"collapseAll": false,
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@ -203,9 +203,10 @@
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},
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"active": "7c5b0ca6f7687800",
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"lastOpenFiles": [
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"Concurrent Systems/notes/4b - Monitors.md",
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"Concurrent Systems/notes/3a - Hardware primitives & Lamport Bakery algorithm.md",
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"Concurrent Systems/notes/2 - Fast mutex by Lamport.md",
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"Concurrent Systems/notes/1 - CS Basics.md",
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"Concurrent Systems/notes/2 - Fast mutex by Lamport.md",
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"Concurrent Systems/ignore this folder/Untitled.md",
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"Concurrent Systems/notes/4 - Semaphores.md",
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"Foundation of data science/notes/9 XGBoost.md",
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@ -228,7 +229,6 @@
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"Concurrent Systems/notes/6 - Atomicity.md",
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"Concurrent Systems/slides/class 5.pdf",
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"Concurrent Systems/notes/6a - Alternatives to Atomicity.md",
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"Concurrent Systems/notes/4b - Monitors.md",
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"Concurrent Systems/notes/4c - Dining Philosophers.md",
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"Concurrent Systems/slides/class 4.pdf",
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"Concurrent Systems/slides/class 9.pdf",
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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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![[images/Pasted image 20250303090135.png]]
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![[/Concurrent Systems/notes/images/Pasted image 20250303090135.png]]
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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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![[images/Pasted image 20250303093116.png]]
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![[/Concurrent Systems/notes/images/Pasted image 20250303093116.png]]
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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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![[images/Pasted image 20250401083747.png]]
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![[/Concurrent Systems/notes/images/Pasted image 20250401083747.png]]
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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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![[images/Pasted image 20250401092557.png]]
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![[/Concurrent Systems/notes/images/Pasted image 20250401092557.png]]
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```
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CS a compare&swap object init at ⊥
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@ -53,10 +53,10 @@ unlock(i) :=
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**How has p0 entered its CS?**
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a) `FLAG[1] = down`, this is possible only with the following interleaving:
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![[images/Pasted image 20250303100721.png]]
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![[/Concurrent Systems/notes/images/Pasted image 20250303100721.png]]
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b) `AFTER_YOU = 1`, this is possible only with the following interleaving:
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![[images/Pasted image 20250303100953.png]]
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![[/Concurrent Systems/notes/images/Pasted image 20250303100953.png]]
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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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@ -140,7 +140,7 @@ Easy to generalize to k-MUTEX.
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Peterson's algorithm cost $O(n^2)$
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A first way to reduce this cost is by using a tournament of MUTEX between pairs of processes:
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![[images/Pasted image 20250304082459.png|350]]
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![[/Concurrent Systems/notes/images/Pasted image 20250304082459.png|350]]
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Of course this is a binary tree, and the height of a binary tree is logaritmic to the number of leaves. A process then wins after $\lceil \log_{2}n \rceil$ competitions $\to O(\log n)$ cost.
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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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![[images/Pasted image 20250304084537.png]]
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![[/Concurrent Systems/notes/images/Pasted image 20250304084537.png]]
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![[images/Pasted image 20250304084901.png]]
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![[/Concurrent Systems/notes/images/Pasted image 20250304084901.png]]
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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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![[images/Pasted image 20250304090219.png]]
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![[/Concurrent Systems/notes/images/Pasted image 20250304090219.png]]
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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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![[images/Pasted image 20250304093223.png]]
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![[/Concurrent Systems/notes/images/Pasted image 20250304093223.png]]
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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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![[images/Pasted image 20250310172134.png]]
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![[/Concurrent Systems/notes/images/Pasted image 20250310172134.png]]
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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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![[images/Pasted image 20250310103703.png]]
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![[/Concurrent Systems/notes/images/Pasted image 20250310103703.png]]
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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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@ -180,7 +180,7 @@ producer A:
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- so producer A will write at `BUF[0]`
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- but wait! Consumer B is still reading there
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- **Producer A doesn't give a fuck.**
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![[images/Pasted image 20250312121828.png|200]]
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![[/Concurrent Systems/notes/images/Pasted image 20250312121828.png|200]]
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*don't be like Producer A, be more like Bob, who always scans EMPTY before!*
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So the issue here is that producers just assume that IN is the first available slot. But it its not necessarily the case.
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@ -3,7 +3,7 @@ The first real practical example of a concurrent system.
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- one chopstick between each pair of philosophers
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- a philosophers must pick up its two nearest chopsticks in order to eat
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- a philosopher must pick up first one chopstick, then the second one, not both at once
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![[images/Pasted image 20250317100456.png|100]]
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![[/Concurrent Systems/notes/images/Pasted image 20250317100456.png|100]]
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**PROBLEM:** *Devise a deadlock-free algorithm for allocating these limited resources (chopsticks) among several processes (philosophers).*
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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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![[images/Pasted image 20250317105355.png]]
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![[/Concurrent Systems/notes/images/Pasted image 20250317105355.png]]
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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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![[images/Pasted image 20250318090733.png]]
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![[/Concurrent Systems/notes/images/Pasted image 20250318090733.png]]
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![[images/Pasted image 20250318090909.png]]But there is another linearization possible! I can also push a before if I pull it before c!
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![[/Concurrent Systems/notes/images/Pasted image 20250318090909.png]]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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![[images/Pasted image 20250324082534.png]]
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![[/Concurrent Systems/notes/images/Pasted image 20250324082534.png]]
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![[images/Pasted image 20250324082545.png]]
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![[/Concurrent Systems/notes/images/Pasted image 20250324082545.png]]
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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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![[images/Pasted image 20250324091452.png]]
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![[/Concurrent Systems/notes/images/Pasted image 20250324091452.png]]
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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:![[images/Pasted image 20250324092633.png]]
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**REMARK:** this implementation doesn’t satisfy the non-blocking property:![[/Concurrent Systems/notes/images/Pasted image 20250324092633.png]]
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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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![[images/Pasted image 20250324100652.png]]
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![[/Concurrent Systems/notes/images/Pasted image 20250324100652.png]]
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```
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push(w) :=
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