From f09563169ce9f2856df692f454f5cc04ada29ed5 Mon Sep 17 00:00:00 2001 From: Marco Realacci Date: Tue, 18 Mar 2025 16:30:04 +0100 Subject: [PATCH] vault backup: 2025-03-18 16:30:04 --- .obsidian/workspace.json | 4 ++-- Concurrent Systems/notes/6 - Atomicity.md | 6 +++--- 2 files changed, 5 insertions(+), 5 deletions(-) diff --git a/.obsidian/workspace.json b/.obsidian/workspace.json index 2bcc1c5..cb59014 100644 --- a/.obsidian/workspace.json +++ b/.obsidian/workspace.json @@ -34,9 +34,9 @@ "type": "pdf", "state": { "file": "Concurrent Systems/slides/class 6.pdf", - "page": 5, + "page": 6, "left": -23, - "top": 360, + "top": 428, "zoom": 0.6627078384798101 }, "icon": "lucide-file-text", diff --git a/Concurrent Systems/notes/6 - Atomicity.md b/Concurrent Systems/notes/6 - Atomicity.md index 1552a13..0057b81 100644 --- a/Concurrent Systems/notes/6 - Atomicity.md +++ b/Concurrent Systems/notes/6 - Atomicity.md @@ -51,12 +51,12 @@ We now show that $\to$ is acyclic. - by contradiction, consider a shortest cycle - adjacent edges cannot belong to the same order (e.g. not both $\to_X$), otw. the cycle would be shortable, because of transitivity of the total order! - adjacent edges cannot belong to orders on different objects - - this would mean that an operation is involved in both $\to_X$ and $\to_Y$ but it is not possible of course - - Hence, at least one $\to_X$ exists and it must be between two $\to_H$ i.e.: $$op1 \to_H op2 \to_X op3 \to_H op4$$, with op1 = op4 + - this would mean that an operation is involved in both $\to_X$ and $\to_Y$ but it is not possible of course, so the cycle can only happen edges in $\to_X$ and $\to_H$. + - Hence, at least one $\to_X$ exists and it must be between two $\to_H$ i.e.: $$op1 \to_H op2 \to_X op3 \to_H op4$$, likely with op1 = op4 - can this be a cycle? - $op1 \to_H op2$ means that $res(op1) <_H inv(op2)$ - $op2 \to_X op3$ entails that $inv(op2) <_H res(op3)$ - - if not, as is a total order, we would have that $res(op3) <_H inv(op2)$, but we then would have a cycle of lenght 2... + - if not, as is a total order, we would have that $res(op3) <_H inv(op2)$, but we then would have $op3 \to_H op2$ a cycle of lenght 2... - $op2 \to_H op3$ entails that $inv(op2) <_H res(op3)$