diff --git a/.obsidian/workspace.json b/.obsidian/workspace.json
index 035da25..17dc80e 100644
--- a/.obsidian/workspace.json
+++ b/.obsidian/workspace.json
@@ -34,9 +34,9 @@
               "type": "pdf",
               "state": {
                 "file": "Concurrent Systems/slides/class 4.pdf",
-                "page": 2,
+                "page": 3,
                 "left": -23,
-                "top": 165,
+                "top": 298,
                 "zoom": 0.652019002375297
               },
               "icon": "lucide-file-text",
@@ -217,16 +217,16 @@
   },
   "active": "6edd4157a160e462",
   "lastOpenFiles": [
-    "Concurrent Systems/slides/class 4.pdf",
-    "Concurrent Systems/notes/4 - Semaphores.md",
+    "Concurrent Systems/notes/3a - Hardware primitives & Lamport Bakery algorithm.md",
     "Concurrent Systems/notes/3b - Aravind's algorithm and improvements.md",
+    "Concurrent Systems/notes/4 - Semaphores.md",
+    "Concurrent Systems/slides/class 4.pdf",
     "Concurrent Systems/notes/images/Pasted image 20250310172134.png",
     "Concurrent Systems/notes/1b - Peterson algorithm.md",
     "Concurrent Systems/slides/class 3.pdf",
     "Concurrent Systems/notes/images/Pasted image 20250310103703.png",
     "Concurrent Systems/notes/2 - Fast mutex by Lamport.md",
     "Concurrent Systems/notes/1 - CS Basics.md",
-    "Concurrent Systems/notes/3a - Hardware primitives & Lamport Bakery algorithm.md",
     "Concurrent Systems/notes/2b - Round Robin algorithm.md",
     "Concurrent Systems/notes/1 - CS Basics2.md",
     "HCIW/slides/Interface and Interaction for IoT.pdf",
diff --git a/Concurrent Systems/notes/4 - Semaphores.md b/Concurrent Systems/notes/4 - Semaphores.md
index 99f5001..b758a9b 100644
--- a/Concurrent Systems/notes/4 - Semaphores.md	
+++ b/Concurrent Systems/notes/4 - Semaphores.md	
@@ -13,7 +13,7 @@ Main use: **prevent busy waiting**: suspend processes that cannot perform *down*
 **Binary semaphore:** if it is at most 1 (also *up* are blocking).
 
 A semaphore needs two underlying objects:
-- a counter initialized at s0 that can also become negative
+- a counter initialized at s0 that can also become negative (see the implementation below to understand)
 - a data structure (typically a queue), initially empty, to store suspended processes.
 
 #### Ideal implementation
@@ -29,4 +29,30 @@ S.up() :=
 	S.counter++
 	if S.counter <= 0 then
 		activate a proc from S.queue
-	return 
\ No newline at end of file
+	return
+```
+>[!note] note
+>if S.counter ≥ 0, then this is the value of the semaphore; otherwise, S.counter tells you how many processes are suspended in S
+>
+>all operations are in MUTEX
+
+
+#### Actual implementation
+```
+Let t be a test&set register initialized at 0
+
+S.down() :=
+	Disable interrupts
+	wait S.t.test&set() = 0
+	S.counter--
+	if S.counter < 0 then
+		enter into S.queue
+		S.t <- 0
+		SUSPEND
+	return
+
+S.up() :=
+	S.counter++
+	if S.counter <= 0 then
+		activate a proc from S.queue
+	return