From e1051991a23fb441b95c03395dcf3c961e3358e3 Mon Sep 17 00:00:00 2001
From: Jan Meyer <git@lolmerkat.cc>
Date: Thu, 23 Apr 2026 22:39:23 +0200
Subject: [PATCH] vault backup: 2026-04-23 22:39:23

---
 00 Inbox/29622985 - 2026-04-28.md | 27 +++++++++++++++++++++++++++
 40 Extras/OOP/die_einen_da        |  2 +-
 2 files changed, 28 insertions(+), 1 deletion(-)

diff --git a/00 Inbox/29622985 - 2026-04-28.md b/00 Inbox/29622985 - 2026-04-28.md
index 3ba28c7..0615289 100644
--- a/00 Inbox/29622985 - 2026-04-28.md	
+++ b/00 Inbox/29622985 - 2026-04-28.md	
@@ -16,3 +16,30 @@ tags:
 ---
 
 ## 📝 Content
+
+A containment hierarchy of classes of formal grammars. Grammars are classified into four types with different limitations.
+
+### Type 0
+no restrictions
+
+### Type 1
+Each rule $sans(w)_1 -> sans(w_2)$ satisfies $sans(abs(w_1) <= abs(w_2))$ with the exception that $sans(S) -> epsilon$ is allowed if $sans(S)$ does not occur on any right hand side of rules.
+$sans(abs(w_1) <= abs(w_2))$ implies that no shortening rules are allowed.
+
+### Type 2 (context free)
+Same restriction as [[#Type 1|type 1]] and additionally for each rule $sans(w_1 -> w_2)$ the string $sans(w_1)$ contains only a single variable (i.e. $sans(w_1 in V)$)
+
+### Type 3 (regular)
+Same restriction as [[#Type 2|type 2]] and additionally $sans(w_2 in Sigma union Sigma V)$), i.e. the right hand side is either a single terminal symbol or a terminal symbol followed by a variable.
+
+> [!REMARK]
+> The order of the variable and the terminal symbol is chosen arbitrarily.
+> Important is just that within a regular grammar only a single order occurs. If for all rules $sans(w_2 in Sigma union V Sigma)$ it would also be a regular grammar.
+
+## Definition
+A language $sans(L subset Sigma^*)$ is said to be of type 0 (1,2,3) if there exist a type 0 (1,2,3) grammar $sans(G)$ for which $sans(L(G) = L)$.
+
+> [!REMARK]
+> In order to show that a language is op type 0 (1,2,3) one thus just has to find a corresponding grammar.
+> In order to show that a grammar is not of type 0 (1,2,3) one has to prove that no type 0 (1,2,3) grammar can exist generating the corresponding language.
+
diff --git a/40 Extras/OOP/die_einen_da b/40 Extras/OOP/die_einen_da
index d7b00e7..3af3ede 160000
--- a/40 Extras/OOP/die_einen_da	
+++ b/40 Extras/OOP/die_einen_da	
@@ -1 +1 @@
-Subproject commit d7b00e7700733ca4605a9597f69871b98f1adaa9
+Subproject commit 3af3edea6ffcfbe6aa68892523a07765d8538fab