From c656bfa2e7c4a5c499a7030ecd6cd79e54d2780e Mon Sep 17 00:00:00 2001 From: Janis Hutz Date: Sat, 6 Dec 2025 10:31:33 +0100 Subject: [PATCH] [TI] Compact: Almost finish reductions --- .../ti-compact/parts/04_computability.tex | 81 +++++++++++++++++- semester3/ti-compact/parts/05_complexity.tex | 3 + semester3/ti-compact/ti-compact.pdf | Bin 335815 -> 346398 bytes semester3/ti-compact/ti-compact.tex | 1 + 4 files changed, 84 insertions(+), 1 deletion(-) diff --git a/semester3/ti-compact/parts/04_computability.tex b/semester3/ti-compact/parts/04_computability.tex index 462f4ce..9d8e458 100644 --- a/semester3/ti-compact/parts/04_computability.tex +++ b/semester3/ti-compact/parts/04_computability.tex @@ -65,11 +65,90 @@ First off, a list of important languages for this and the next section: \item $L_{H, \lambda} = \{ \text{Kod}(M) \divides M \text{ halts on } \lambda \}$ ($\in \cL_{RE}$, but $\notin \cL_R$) \end{itemize} +\newpage +\setLabelNumber{definition}{3} +\fancydef{Recursively reducible languages} $L_1 \leq_R L_2$ ($L_1$ reducible into $L_2$), if $L_2 \in \cL_R \Rightarrow L_1 \in \cL_R$ + +\fancydef{$EE$-Reductions} $L_1 \leq_{EE} L_2$ if there exists a TM $M$ that implements image $f_M : \word_1 \rightarrow \word_2$, +for which we have $x \in L_1 \Leftrightarrow f_M(x) \in L_2$ for all $x \in \wordbool_1$ + +\setLabelNumber{lemma}{3} +\inlinelemma If $L_1 \leq_{EE} L_2$ then also $L_1 \leq_R L_2$ + +\inlinelemma For each language $L \subseteq \word$ we have: $L \leq_R L^C$ and $L^C \leq_R L$ + \setLabelNumber{theorem}{6} \fancytheorem{Universal TM} A TM $U$, such that $L(U) = L_U$ + +\fhlc{Cyan}{Showing reductions} First, a general guide to reductions and below what else we need to keep in mind for specific reductions: +\begin{enumerate} + \item We construct a TM $A$ that: + \begin{enumerate} + \item Checks if the input has the right form and if it does not, returns some output that is $\notin L_2$ + \item Applies the transformation to all remaining input + \end{enumerate} + \item We show $x \in L_1 \Leftrightarrow A(x) \in L_2$ by showing the implications: + \begin{enumerate} + \item For $\Rightarrow$, we show it directly, by assuming that $x \in L_1$ (obviously) and we can ignore the invalid input (as that $\notin L_1$ anyway) + \item For $\Leftarrow$, we have two options (mention what happens to invalid input here): + \begin{itemize} + \item We show $A(x) \in L_2 \Rightarrow x \in L_1$ directly (usually harder) + \item We show $x \notin L_1 \Rightarrow A(x) \notin L_2$ (contraposition) + \end{itemize} + \end{enumerate} + \item Show that the TM always halts (for $P$, $EE$ and $R$ reductions at least) +\end{enumerate} + +\shade{Cyan}{$EE$-reductions} They follow the above scheme exactly + +\shade{Cyan}{$R$-reductions} It is usually a good idea to draw the setup here. We have a TM $C$ that basically executes an $EE$-reduction and we have a TM $A$ that can check $L_2$. +Then, we have a TM $B$ that wraps the whole thing: It first executes TM $C$, which will either output an transformation of $L_1$ for $L_2$ (i.e. execute an $EE$-reduction) or +output some encoding for \textit{invalid word}. +If it outputs the encoding for \textit{invalid word}, $B$ will output $x \notin L_1$. + +If $C$ does not output an encoding for \textit{invalid word}, then $B$ will execute $A$ on the output of $C$ and then use the output of $A$ (either accepting or rejecting) +to output the same (i.e. if $A$ accepts, then $B$ will output $x \in L_1$ and if $A$ rejects, $B$ outputs $x \notin L_1$) + +\inlineintuition In $R$-reductions, we construct a full verifier for $L_1$ using the verifier for $L_2$, i.e. we can use TM $B$ directly to check if a given word is in $L_1$ +given that the transformed word is also in $L_2$. + + +\shade{Cyan}{$P$-reductions} (Used in Chapter \ref{sec:complexity}). We need to also show that $A$ terminates in polynomial time. + +\shade{orange}{Tips \& Tricks:} +\begin{itemize} + \item The TM $A$ has to terminate always + \item Check the input for the correct form first + \item For the correctness, show $x \in L_1 \Leftrightarrow A(x) \in L_2$ + \item The following tricks can be useful: + \begin{itemize} + \item Transitions into $\qacc$ and $\qrej$ can be redirected to $\qacc / \qrej$ or into an infinite loop + \item Construct TM $M'$ that ignores input and does the same, regardless of input + \end{itemize} + \item Generate encoding of a $TM$ with special properties (e.g. accepts all input, never halts, \dots) +\end{itemize} + + % ──────────────────────────────────────────────────────────────────── + +\newpage \subsection{Rice's Theorem} +\setLabelNumber{definition}{7} +\inlinedef $L$ is called a \bi{semantically non-trivial decision problem}, if these conditions apply: +\begin{enumerate}[label=\textit{(\roman*)}] + \item There exists a TM $M_1$, such that $\text{Kod}(M_1) \in L$ (i.e. $L \neq \emptyset$) + \item There exists a TM $M_2$, such that $\text{Kod}(M_2) \notin L$ (not all encodings are in $L$) + \item For two TM $A$ and $B$: $L(A) = L(B) \Rightarrow \text{Kod}(A) \in L \Rightarrow \text{Kod}(B) \in L$ +\end{enumerate} + + \setLabelNumber{theorem}{9} -\fancytheorem{Rice's Theorem} +\fancytheorem{Rice's Theorem} Every semantically non-trivial decision problem over TMs is undecidable + +\fhlc{Cyan}{Using Rice's Theorem} We only need to show that a language is semantically non-trivial, which we do by checking the above conditions. +For the third condition, intuitively, we only need to check if in the definition of $L$ only $L(M)$ appears and nowhere $M$ directly (except of course, to say that $M$ has to be a TM), +or the condition can be restated such that only $L(M)$ is described by it. + +For a more formal proof of that condition, simply show that the implication holds diff --git a/semester3/ti-compact/parts/05_complexity.tex b/semester3/ti-compact/parts/05_complexity.tex index e69de29..8df6e7a 100644 --- a/semester3/ti-compact/parts/05_complexity.tex +++ b/semester3/ti-compact/parts/05_complexity.tex @@ -0,0 +1,3 @@ +\newsection +\section{Complexity} +\label{sec:complexity} diff --git a/semester3/ti-compact/ti-compact.pdf b/semester3/ti-compact/ti-compact.pdf index 744b5b8c51319fb630fcc916c001d3cb0b2c78c2..2b8e146ed88ca4b6ac9345e62b77bab09d272705 100644 GIT binary patch delta 140311 zcmZttV~j3L(6$SYZO^Q+ZQHii*tR`$jcwbuZJTRs+xB-qPu_fcC;Lr0o&Hgk^snxt z&Z={)4=pzly1W2 zQgxCj7*5JOu{lGQrv0hzmCMa609ffc-otzI^}WYD@&-vvV&GGX@uMHrBwR_SU;<4_ z1Wiryk5qK*!V!kwPgO3Xyfns3JUkD4Wp+#FB-_TV~Ip=dhT_YE0MylOBt z@FN!vWFF{60WZf3Qi-SzN{)pY6Eigs!=ftko15XkdvbA&8m2+w#(NTSltdsgWDZg( z6mzqFc#u6d@n~e!u`RFTOoviMQUEq`115pTzSYRo87Nps0R%pD33MbD7+sIvf~x?A zy+qNidLNKnBHJ>3*)$Ty6uZ=l4A(KykRw1MZUXGl zC}|?RAsq_19BY}hbSue|`P~C*g>o#dfc;Gu>Lu=L_GhFl>5+tkKwl?Y0B{QBAW2;8 zL`%BI%AN|IABTN<`UiE6N|^)Mc=obo 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\input{parts/02_finite-automata.tex} \input{parts/03_turing-machines.tex} \input{parts/04_computability.tex} +\input{parts/05_complexity.tex}