From f2e975ade7011de883582ec7f4fc98aaa43d1b30 Mon Sep 17 00:00:00 2001 From: Janis Hutz Date: Tue, 11 Nov 2025 16:31:43 +0100 Subject: [PATCH] [TI] Summarized reduction chapter --- .../parts/04_computability/01_reduction.tex | 73 ++++++++++++++++++ semester3/ti/ti-summary.pdf | Bin 967696 -> 975671 bytes semester3/ti/ti-summary.tex | 2 + 3 files changed, 75 insertions(+) diff --git a/semester3/ti/parts/04_computability/01_reduction.tex b/semester3/ti/parts/04_computability/01_reduction.tex index c498c02..99c9959 100644 --- a/semester3/ti/parts/04_computability/01_reduction.tex +++ b/semester3/ti/parts/04_computability/01_reduction.tex @@ -1 +1,74 @@ +\newpage \subsection{Die Methode der Reduktion} +\fancydef{Rekursiv reduzierbare Sprache} Eine Sprache $L_1 \subseteq \word_1$ ist auf $L_2 \subseteq \word_2$ rekursiv reduzierbar, geschrieben $L_1 \leq_R L_2$, +falls $L_2 \in \cL_R \Rightarrow L_1 \in \cL_R$. + +\shade{teal}{Intuition:} $L_2$ ist bezüglich der algorithmischen Lösbarkeit mindestens so schwer wie $L_1$. $\cL_R$ ist die Menge aller rekursiv reduzierbaren Sprachen. +Ist also $L_2$ lösbar, so muss auch $L_1$ lösbar sein. + +\fancydef{EE-reduzierbare Sprache} $L_1$ ist auf $L_2$ \bi{EE-reduzierbar}, geschrieben $L_1 \leq_{EE} L_2$, wenn eine TM $M$ existiert, +die eine Abbildung $f_M : \word_1 \rightarrow \word_2$ mit der Eigenschaft $x \in L_1 \Leftrightarrow f_M(x) \in L_2$ für alle $x \in \word_1$ berechnet. +Anders ausgedrückt: die TM $M$ reduziert die Sprache $L_1$ auf die Sprache $L_2$ + +\inlinelemma Falls $L_1 \leq_{EE} L_2$, dann auch $L_1 \leq_R L_2$. \inlineproof Im Buch auf Seite 135 (= 148 im PDF) + +Wir müssen also nur zeigen, dass $L_1 \leq_{EE} L_2$ um zu zeigen, dass $L_1 \leq_R L_2$ + +\inlinelemma Für jede Sprache $L \subseteq \word$ gilt: $L \leq_R L^C$ und $L^C \leq_R L$ + +\inlinecorollary $(L_\text{diag})^C \notin \cL_R$ +\inlineproof Folgt davon, dass $L_\text{diag} \notin \cL_{RE}$ (was heisst, dass $L_\text{diag} \notin \cL_R$) +und nach Lemma 5.4 $L_\text{diag} \leq_R (L_\text{diag})^C$ und das umgekehrte gelten muss. + +\inlinelemma $(\ldiag)^C \in \cL_{RE}$ +\inlineproof Auf Seite 137 (= 150 im PDF) wird eine Turingmaschine aufgezeigt, die $(L_\text{diag})^C$ akzeptiert. + +\inlinecorollary $(\ldiag)^C \in \cL_{RE} - \cL_R$ und daher $\cL_R \subsetneq \cL_{RE}$ + +Folgende Sprachen sind nicht rekursiv, liegen aber in $\cL_{RE}$ + +\fancydef{Universelle Sprache} $L_U = \{ \text{Kod}(M)\# w \divides w \in \wordbool \text{ und TM } M \text{ akzeptiert } w \}$ + +\fancytheorem{Universelle TM} Eine TM $U$, so dass $L(U) = L_U$, also gilt $L_U \in \cL_{RE}$\\ +% +\inlineproof Auf Seite 138 (= 151 im PDF) + +Was dies bedeutet, es existiert eine TM ohne Haltegarantie, die eine beliebige Turingmaschine auf einer gegebenen Eingabe simulieren kann. +Untenstehendes Resultat bedeutet, dass man das Resultat der Berechnung einer TM $M$ auf einer Eingabe $x$ anders berechnen kann, als die Berechnung von $M$ auf $x$ zu simulieren. + +\inlinetheorem $L_U \notin \cL_R$ + +Wenn jetzt aber $M$ unendlich lange auf $x$ arbeitet, so wissen wir nicht, ob wir die Simulation beenden können. Dies führt zum Halteproblem + +\begin{definition}[]{Halteproblem} + Das Halteproblem ist das Entscheidungsproblem $(\{ 0, 1, \# \}, L_H)$ mit + \begin{align*} + L_H = \{ \text{Kod}(M)\# x \divides x \in \{ 0, 1 \}^* \text{ und } M \text{ hält auf } x \} + \end{align*} +\end{definition} + +Dies scheint vorerst nicht ein allzu grosses Problem zu sein, jedoch besagt das nächste Resultat, dass es keinen Algorithmus gibt, +der testen kann, ob ein gegebenes Programm immer terminiert. + +\inlinetheorem $L_H \in \cL_R$ +\inlineproof Auf Seiten 140 - 142 (153 - 155 im PDF) + +Betrachten wir die Sprache $\lempty = \{ \text{Kod}(M) \divides L(M) = \emptyset \}$, die die Kodierungen aller Turingmaschinen enthält, +die die leere Menge (kein Wort) akzeptieren. Es gilt +\rmvspace +\begin{align*} + (\lempty)^C = \{ x \in \wordbool \divides x \notin \text{Kod}(\overline{M}) \forall \text{ TM } \overline{M} \text{ oder } x = \text{Kod}(M) \text{ und } L(M) \neq \emptyset \} +\end{align*} + +\drmvspace +\inlinelemma $(\lempty)^C \in \cL_{RE}$ +\inlineproof Auf Seiten 142 - 143 (155 - 156 im PDF) + +\inlinelemma $(\lempty)^C \notin \cL_R$ + +Wir haben als wiederum die Nichtexistenz eines Algorithmus zur Überprüfung, ob ein gegebenes Programm die leere Menge akzeptiert. +Ein Beweis dazu findet sich auf Seiten 143 und 144 im Buch (156 - 157 im PDF) + +\inlinecorollary $\lempty \notin \cL_R$ + +\inlinecorollary $L_{EQ} = \{ \text{Kod}(M)\# \text{Kod}(\overline{M}) \divides L(M) = L(\overline{M}) \}$ ist nicht entscheidbar (also $L_{EQ} \notin \cL_R$) diff --git a/semester3/ti/ti-summary.pdf b/semester3/ti/ti-summary.pdf index 8fd64f8c88043813100f9200be83aac0fa2fd7ad..56b3921da8acf0cb198a385c17950c0dff5549f5 100644 GIT binary patch delta 86049 zcmZtsV{|V((?5>Zwr$&XyK9@fw(Wjv+qP}nwr$(kZU5Kre$LDD;xlVyWhR*pvZKlzYl;hM* zhrn>y@rD^$vaK`_BNI_P-2ZOn^dlruZ>;OntIvQ+2VF8mYbmO}-7eu3M6>nRakW=f zBnVVKZGV9IxC+ohDyTQ1RCw-^E?}C5E$=6v?T>?w%#tPVTPifJFs*K)fZPPrRt!k}`w4>s9=4skA#Tqcz!i$k-Z)%)Pd%vrR8nZ$h zItE^d7F)T__HmH9O8tX)K>zPTMQ2T@?&=7e6c0x9YB`IJFJE(-?j3j*Sp!=}!RY5g zgF;?&9MtRp0uRr5_$V00gNX#V1BdpSiKMJ3=N)j}vAG+DDc*sSNrnG`6K{^eaC)|@V5OZ7-P4Q}Ls_NY196vj`pROP#)wCxQ)cd)KmmroXZ|8Jic@s8ew&oIvw_N4%k{=5X+z6Ac#7T#ockjL zT~Dbh{Uj2myf_Q?D>1T}mUp%OVw&>?K(}>s_(W9}!ZP$qU_RN#T7(+($;7?P~RFG^)|-Rc)eoxy=GYZ(d{jU<}$H*if@sMv<2`IPg-I-YHJz zsV;p#?0?V{79(deAJ9F7)QTM&>z!S$CHw_)BCKgkS^z~&0!P3AVPQ@S0EVJY5=TG> zw@kEKbWJiqzyh3%cX*6-t;!lHLbe6n;rVa@31{n)aeX*G9=klAbE`#&^OSiydF499 zIAv2RI~SKo%AZYaJ-aBi>TPITZ6sH^D9N*qdVCgg+jn)IS4~~mmocIuutT$1S$ETA zTG>W6JQHU96j3vIy77GY8cFX5!|fo6CCi(=9N*Y;WdKBXr@%-{)gE_Ks|!qG9cK4E zGrBXna}DY~)FF zzp9yT{sO9V8Tn`$Uz> zTO=vYKoh+P6TuwM!=e|W3z-Xp6p1EXMTlv`=m4yK!f4pSm*BkNUKRL!qOwZ-wQxmq z;3Iskf~htI*dxYlK}=TYUEy95YJn!fR!G7+1tCN*7q$N&l7oyQY|A+Y2vaw)|~ z(*d=?!;wlrrDS-K)D9EERz7*6TI9ia#lg;G9YATgJXm7FkXuIap+Bj{hRZZ=hAFXD z?-zuZqZfBp@I?PDZ(-*GF{57}1SOq@B0wFAsd*JvpL+U!j9ckH`!T*r@Cx6nRzB2qFJF}hv z4YtH4`L$3BZOYG=-IR>JE6$WXizSmm_S_7lRg_0a2(u_C=XA#=R=Tsn-0|MA6u@~v zjPb&k2`}aarAZP$51}tNfk}kxJqu2vO@TdxM@>T#?}SJ2C6XAJ6_})rHNm!|S}fSy z5^z^-71?`Jxv?2C0y1bcAYw2b?Lc`v%T*6Lf&+m#)p;-c-z~yMny$>01l{#jn`0I~ z%h}Xxg$q^NAicnbinX^CZ0*~J4j_Y5F-K;aLwH{f`Yo;0m|ChIhy zVB`fkv=SN}=zXs&59(y)8!bU5lVglI+{9+}q@u^1GJg0r?XB z(E)iJ=fZ} zJut1^FzoWOs!0&^lD?%Z_|o699Qwrvy#O+T?qbHouT|H@G~{%;u&^mUS=U4VHPB?X z{E<32maJ<7`Ph_K@b&<7I4x|bsr7O~`Xc%%1F{UY&z;?II$Z}d0m!jzkoIM#-}O$X z`M2RO4^Ayqv;7OyNWvfZeEpek_SS;r@$rhVq_w^!zaB%eVzz5(IdyMlE9k7w&plK( z5d=`4otCEzZ#PFf9V}s8sp&*-YHPXlQ=K-`Lg9%O+wQfr%#$0f`U@WQ zSffT#GqE8f4GzKI08Okuj#Q#ZV?#r_S0{*avWp>C{(IWf>I4(uF47)TDmmZ?r~NgdErwO18ugxTM6FP$jGSg={d50?edOEJPd zlXWrRbFjM;$x>v@ExZXl2?6UxfoyJ$-Dlkm#82jz(6Nom?ZI&}`_M(rdAiWzq3ssK;ab#e3s}R{w{9x^cov zuF$gi>&(h#A6Rx;j`usZ8TK*VH7yMhPddzugtLu%^q2U3|Dt&7F4mLiOyTv??>|x>&QTq>x3o(Oz%9?JuSWSw-2% zW{5In4Zlm~5Y7RZ<70N_5>1t;-upeQs(<&F-CY+`Y0Kg6d87mZ4>jhF*1~kt_xJ}W zYMXCB7~ysFrb}(V{wOj7>s!Yf0}c>i|1D39s`mj35D~Ylf{)ka$xpk^you#c&Pf>Y zD2mtf&EGGlf`gZ|J3zer9S~1oqJIH|9_yQztamZ=G`fpP+bNTj>Q~k*+MkS1VPMb9 z40fvKxWF;esI6gYUgSLT;RyUq*LB+iL=wtg`#1(pY1#)hwLMGnX3f2)+j(R)5L=F; z`mvh@C@IP+{s}xeM>$=+QFLEpjY{Kpw@W1GX1MCZm=w>0-H^%ECg)x}wFY*dP&pk2_%I_t- zF}E$cAQ!@N#T@q1&VL7Qi^Khq+a8uNVXu(_*yyi|p;DMTSWvgN)+p6azCDhNDS8?fw$Acmvr>9*>FY4) z?sL1u`pIcWF+rmxw00b!R7agjB@Q^Il94rl?aj=V2X@FsIW4;lBLtR{LUUa^4dbZ; z5VTfAy`-U#*AKI)t*0F zzPHw)>uS^w^Y zW4iEMt17V&((&?od+K)cRCg-F?R(<1C;?&j1bq={4q=!t2KhH=f92%i%0>q47s%nWLJmz~vP z#^qzi4z3qhEYi_gWTPeJGFoYs46b^aoz1Qnmy2tr^a{(5V*mfi>Z1w0!vDx6y65RY z(BNfsy2B#LBD`(QM06>?ECuRlD?-$>KDmzYfrxk(AR8zX2!JHO8o;4S4YGz{^w3A8 z^Jmbr6p1^rw{7_PAQ(u{jUTPUm1^}%*0`BOh;=hGq!>1l!{CoCV&r~$HXH@1M8}lU z{h%npx^C&RllJBhZ=;@2vaIEox)4Yyyf4BOW*>2``F+a3Wcur>WriggT$GfuOw+E$ zCwBP)UsJR39Duf1UD%sQSZO^R_Zv=hl4f7n30-~gPdGk$f5P9n7*7bh;1Ok^#h=A# z;sxypk6pUx8E3_z28@CjfTp~~G_HEOY#4lb!&cd-LC+a8sigL%fmy30uev=3$18R_ zGr^T%wK

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