From 5f6d97ad434743ec6221984fa2b71d1481bdd86d Mon Sep 17 00:00:00 2001 From: Janis Hutz Date: Fri, 12 Dec 2025 10:45:06 +0100 Subject: [PATCH] [TI] Compact: Add intuition for reduction ordering --- .../ti-compact/parts/04_computability.tex | 4 ++++ semester3/ti-compact/ti-compact.pdf | Bin 397980 -> 398395 bytes 2 files changed, 4 insertions(+) diff --git a/semester3/ti-compact/parts/04_computability.tex b/semester3/ti-compact/parts/04_computability.tex index df03a2a..d54a66c 100644 --- a/semester3/ti-compact/parts/04_computability.tex +++ b/semester3/ti-compact/parts/04_computability.tex @@ -70,6 +70,10 @@ we only need to show that there exists no TM $M$, for which $L(M) \in \cL_{RE}$. \label{sec:reductions} This is the start of the topics that are explicitly part of the endterm. +For the reductions, it is important to get the order right.\\ +To show that a language $L_1$ is not part of e.g. $\cL_R$, show that there exists a reduction into a language $L_2 \notin \cL_R$, i.e. e.g. show $L_2 \leq_R L_1$.\\ +To show that a language $L_1$ is part of e.g. $\cL_R$, show that there exists a reduction into a language $L_2 \in \cL_R$, i.e. e.g. show $L_1 \leq_R L_2$. + For a language to be in $\cL_R$, in contrast to $L \in \cL_{RE}$, the TM has to halt also for \texttt{no} instances, i.e. it has to be an algorithm. 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