From ac829b7ee82de4cf57448fb7d218a62d343f451d Mon Sep 17 00:00:00 2001 From: Janis Hutz Date: Fri, 12 Dec 2025 13:18:26 +0100 Subject: [PATCH] [TI] Compact: Fix --- .../ti-compact/parts/04_computability.tex | 5 ++++- semester3/ti-compact/ti-compact.pdf | Bin 398395 -> 398639 bytes 2 files changed, 4 insertions(+), 1 deletion(-) diff --git a/semester3/ti-compact/parts/04_computability.tex b/semester3/ti-compact/parts/04_computability.tex index d54a66c..22880f7 100644 --- a/semester3/ti-compact/parts/04_computability.tex +++ b/semester3/ti-compact/parts/04_computability.tex @@ -84,12 +84,15 @@ First off, a list of important languages for this and the next section: \item $L_H = \{ \text{Kod}(M)\# x \divides x \in \wordbool \text{ and TM $M$ halts on } x \}$ ($\in \cL_{RE}$, but $\notin \cL_R$) \item $L_{\text{diag}} = \{ w \in \wordbool \divides w = w_i \text{ for an } i \in \N - \{ 0 \} \text{ and $M_i$ does not accept } w_i \}$ ($\notin \cL_{RE}$ and thus $\notin \cL_R$) \item $(L_{\text{diag}})^C$ ($\in \cL_{RE}$, but $\notin \cL_R$) - \item $L_{EQ} = \{ \text{Kod}(M)\# \text{Kod}(\overline{M}) \divides L(M) = L(\overline{M}) \}$ ($\in \cL_{RE}$, but $\notin \cL_R$) + \item $L_{EQ} = \{ \text{Kod}(M)\# \text{Kod}(\overline{M}) \divides L(M) = L(\overline{M}) \}$ ($\notin \cL_{RE}$, and thus $\notin \cL_R$) + \item $(L_{EQ})^C = \{ \text{Kod}(M)\# \text{Kod}(\overline{M}) \divides L(M) \neq L(\overline{M}) \}$ ($\notin \cL_{RE}$, and thus $\notin \cL_R$) \item $\lempty = \{ \text{Kod}(M) \divides L(M) = \emptyset \}$ ($\in \cL_{RE}$, but $\notin \cL_R$) \item $(\lempty)^C = \{ x \in \wordbool \divides x \notin \text{Kod}(\overline{M}) \forall \text{ TM } \overline{M} \text{ or } x = \text{Kod}(M) \text{ and } L(M) \neq \emptyset \}$ ($\in \cL_{RE}$, but $\notin \cL_R$) \item $L_{H, \lambda} = \{ \text{Kod}(M) \divides M \text{ halts on } \lambda \}$ ($\in \cL_{RE}$, but $\notin \cL_R$) \end{itemize} +An important consequence of the fact that both $L_{EQ}$ and its complement are $\notin \cL_{RE}$ is that it is not guaranteed +for a language's complement to \textit{necessarily} be in $\cL_{RE}$, if the language is not. \setLabelNumber{definition}{3} diff --git a/semester3/ti-compact/ti-compact.pdf b/semester3/ti-compact/ti-compact.pdf index 1a539c55146bc97961c7e73a8599a06100501c4e..b531aafa279e02736cb05eb9bcaea70a06f2d02c 100644 GIT binary patch delta 19963 zcmV)MK)AoVs2H!R7_i9-0r<1Z3Lydn1$Xanv+oSF2^-s`%#5c=msAmhh2?V<^Z22y zE-(H6FPM?NS+l+tMi(EU)=;MQQq>ukRSrEs>-K&CUGEc9i|{f1{|El|G}yBvE@>qJ zS(A-Z8Uw;||C7-J6SKWjwih2|?n)`<68{shn9u7;;UDS6KQ!W$E2kM!1Ato%E4Z=7e}3$D zi^)SC{sV2SA#an>0~3?AmJ|atHaM3NXay;M?H${a+qU(6ze1fpR67$PxC2b5Gj-E6 zZ4!5C$C*CFeV`@oE*squ>z`?nng98{(?k9Ni<;|a7 z-F*5NswWxBB$bm_cN3l=iRDCP9A!#OUM(iyPhUM0n_W?t^L%+bV~kBd->kO#y4ab2 zUwr!Qye!u9!abU~Hq#@Y*S9k&GfAdjEE%7D{D1D?!(h=zFs0vz;g8dgz1`1MwGzN_8W&0X(Vz1fy? z|6=`+*Y0t?^N)-7+hsW~>*WXcd+SHyek9h8RIC?(?iDhSTG$lI79Pe5nJ_uK#^eM?-Qm2bnGc)pzCngMF<9%3xl9dW>11 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