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z4t;-$<^^zh+dtWKpoTyXz8U&?!hR=0Dj&)&p`X0n|Nnm3*w!u`*vzC553UHh95rRQ K5>D}s-~RyY!#{EW literal 0 HcmV?d00001 diff --git a/Writerside/in.tree b/Writerside/in.tree index a1110c6..a7a32cf 100644 --- a/Writerside/in.tree +++ b/Writerside/in.tree @@ -93,6 +93,8 @@ + + @@ -101,6 +103,7 @@ + @@ -115,6 +118,7 @@ + @@ -146,5 +150,4 @@ - \ No newline at end of file diff --git a/Writerside/topics/04/Rechnernetze/03_GrundlagenNetzwerke_Ethernet.md b/Writerside/topics/04/Rechnernetze/03_GrundlagenNetzwerke_Ethernet.md new file mode 100644 index 0000000..1bfaa56 --- /dev/null +++ b/Writerside/topics/04/Rechnernetze/03_GrundlagenNetzwerke_Ethernet.md @@ -0,0 +1,25 @@ +# Ethernet(IEEE 802.3) mit CSMA/CD +## Vollduplex vs Halbduplex +![image_787.png](image_787.png) + +### Vollduplex +- von einem Switch zu einem ausgewählten Host +- von einem Switch zu einem anderen Switch +- von einem Host zu einem anderen Host (über ein Crossover Kabel) + + +## Ethernet Vergleich + +| | Klassisches Ethernet | Fast Ethernet | Gigabit-Ethernet | +|-----------------------------------------|------------------------------------------------------------------------------------------------------------------------------|-----------------------------------------------------------------------------------------------------------------------------------|-----------------------------------------------------------------------------------------------------------------------------------------------| +| Übertragungsgeschwindigkeit | 10 Mbit/s | 100 Mbit/s | 1000 Mbit/s (1Gbit/s) | +| Übertragungsmedium | Koaxialkabel, (später) Twisted-Pair, Glasfaser | Twisted-Pair, Glasfaser | Twisted-Pair(IEEE 802.3ab, längere Distanzen), Glasfaser (kurze Distanzen), 1000BASE-CX(sehr kurze Distanzen) | +| Topologie | Bus-Topologie (Koaxialkabel), Stern-Topologie (Geräte über Kabel mit zentralem Hub/Switch verbunden, heute Managed Switches) | | | +| CSMA/CD | wird genutzt | Unterstützung, Veränderungen: Notwendigkeit reduziert(Einführung Switches, Full-Duplex-Verbindungen) | | +| Adressierung und Rahmenstruktur (Frame) | MAC-Adressen (Identifizierung im LAN-Netzwerk) | | | +| Rahmenstruktur (Frame) | Präambel und StartFrame-Delimiter, Ziel-/Quelladressen, Typ-/Längenfeld, Nutzdaten, Prüfsumme | | standardmäßiges Ethernet-Rahmenformat → Rückwärtskompatibilität | +| Codierung | | | PAM5 (Pulse Amplitude Modulation mit 5 Levels) für Twisted-Pair | +| Maximale Kabellänge | ~100 Meter, wegen Signalabschwächung und Anforderungen von CSMA/CD | | | +| Benutzerfreundlichkeit | Einfachheit in der Implementierung und Wartung | | | +| Übertragungsmodi | | Voll-Duplex (gleichzeitig, bidirektional, ohne Kollisionen), Halb-Duplex (mit CSMA/CD), Auto-Negotiation zwischen den beiden Modi | Full-Duplex (CSMA/CD überflüssig, wird meistens genutzt), Auto-Negotiation (ermöglicht beste verfügbare Geschwindigkeit, bester Duplex-Modus) | +| Anwendungsbereiche | | Unternehmensnetzwerke (Backbone, Bereiche mit hohen Bandbreiten), Heimnetzwerke (verbreitet, aber wird durch Gigabit ersetzt) | Unternehmens-/Heimnetzwerke/Serverfarmen (dort, wo hohe Übertragungsraten benötigt werden), Rechenzentren (Standard) | \ No newline at end of file diff --git a/Writerside/topics/04/Theoretische Informatik/Hausaufgaben/ti_hausufgabe3.md b/Writerside/topics/04/Theoretische Informatik/Hausaufgaben/ti_hausufgabe3.md new file mode 100644 index 0000000..ef0094e --- /dev/null +++ b/Writerside/topics/04/Theoretische Informatik/Hausaufgaben/ti_hausufgabe3.md @@ -0,0 +1,250 @@ +# Übungsblatt 3 +## Übung 1 +Betrachten Sie den nichtdeterministischen Automaten $N$ aus Abbildung 1 über dem +Alphabet $Σ = \{ 0, 1 \}^*$. + +![image_784.png](image_784.png) +Automat $N$ für Worte aus $\{ 0, 1 \}^*$, deren drittletztes Zeichen eine `0` ist. + +### 1(a) +Wie viele Berechnungspfade gibt $e$, die das Wort `1111` lesen? Wie viele Berechnungsschritte (Verarbeitungen eines Zeichens) sind insgesamt notwendig, um all diese +Berechnungspfade (deterministisch) zu simulieren? Begründen Sie nachvollziehbar, +wie Sie auf Ihre Antworten gekommen sind. + + + +### 1(b) +Konstruieren Sie den Potenzautomaten $A_N$ zu $N$. Geben Sie bei der Konstruktion +die Übergangsfunktion $δ$ des Automaten $A_N$ in tabellarischer Form an. Machen Sie +nach dem Abarbeiten einer Zeile deutlich, welche neuen Metazustände Sie gefunden +haben. + +| Zustand | δ(.,0) | δ(.,1) | +|-----------------------|-----------------|---------------| +| a | **{a,b}** | a | +| {a,b} | **{a,b,c}** | **{a,c}** | +| {a,b,c} | **{a,b,c,d}** | **{a,c,d}** | +| {a,c} | **{a,b,d}** | **{a,d}** | +| {a,b,c,d} [$∈ Q_A$] | **{a,b,c,d,ε}** | **{a,c,d,ε}** | +| {a,c,d} [$∈ Q_A$] | **{a,b,d,ε}** | **{a,d,ε}** | +| {a,b,d} [$∈ Q_A$] | **{a,b,c,ε}** | **{a,c,ε}** | +| {a,d} [$∈ Q_A$] | **{a,b,ε}** | **{a,ε}** | +| {a,b,c,d,ε} [$∈ Q_A$] | {a,b,c,d,ε} | {a,c,d,ε} | +| {a,c,d,ε} [$∈ Q_A$] | {a,b,d,ε} | {a,d,ε} | +| {a,b,d,ε} [$∈ Q_A$] | {a,b,c,ε} | {a,c,ε} | +| {a,d,ε} [$∈ Q_A$] | {a,b,ε} | {a,ε} | +| {a,b,c,ε} | {a,b,c,d,ε} | {a,c,d,ε} | +| {a,c,ε} | {a,b,d,ε} | {a,d,ε} | +| {a,b,ε} | {a,b,c,ε} | {a,c,ε} | +| {a,ε} | {a,b,ε} | {a,ε} | +| b | c | c | +| c | d | d | +| d [$∈ Q_A$] | ε | **ε** | +| ε | ε | ε | + + +```plantuml +@startuml +scale 0.75 + +top to bottom direction +skinparam dpi 150 + +skinparam state { + BackgroundColor #FFFACD + BorderColor black + FontName Helvetica + RoundCorner 30 + Shadowing false + LineThickness 0 +} + +state "d" as d##[bold] +state "a" as a +state "{a,b}" as ab +state "{a,b,c}" as abc +state "{a,c}" as ac +state "{a,b,c,d}" as abcd##[bold] +state "{a,c,d}" as acd##[bold] +state "{a,b,d}" as abd##[bold] +state "{a,d}" as ad##[bold] +state "{a,b,c,d,ε}" as abcde##[bold] +state "{a,c,d,ε}" as acde##[bold] +state "{a,b,d,ε}" as abde##[bold] +state "{a,d,ε}" as ade##[bold] +state "{a,b,c,ε}" as abce +state "{a,c,ε}" as ace +state "{a,b,ε}" as abe +state "{a,ε}" as ae +state "b" as b +state "c" as c +state "ε" as e + + + +[*] --> a + +' Übergänge von a +a --> a : 1 +a --> ab : 0 + +' Übergänge von {a,b} +ab --> abc : 0 +ab --> ac : 1 + +' Übergänge von {a,b,c} +abc --> abcd : 0 +abc --> acd : 1 + +' Übergänge von {a,c} +ac --> abd : 0 +ac --> ad : 1 + +' Übergänge von {a,b,c,d} +abcd --> abcde : 0 +abcd --> acde : 1 + +' Übergänge von {a,c,d} +acd --> abde : 0 +acd --> ade : 1 + +' Übergänge von {a,b,d} +abd --> abce : 0 +abd --> ace : 1 + +' Übergänge von {a,d} +ad --> abe : 0 +ad --> ae : 1 + +' Übergänge von {a,b,c,d,ε} +abcde --> abcde : 0 +abcde --> acde : 1 + +' Übergänge von {a,c,d,ε} +acde --> abde : 0 +acde --> ade : 1 + +' Übergänge von {a,b,d,ε} +abde --> abce : 0 +abde --> ace : 1 + +' Übergänge von {a,d,ε} +ade --> abe : 0 +ade --> ae : 1 + +' Übergänge von {a,b,c,ε} +abce --> abcde : 0 +abce --> acde : 1 + +' Übergänge von {a,c,ε} +ace --> abde : 0 +ace --> ade : 1 + +' Übergänge von {a,b,ε} +abe --> abce : 0 +abe --> ace : 1 + +' Übergänge von {a,ε} +ae --> abe : 0 +ae --> ae : 1 + +' Übergänge von b +b --> c : 0 +b --> c : 1 + +' Übergänge von c +c --> d : 0 +c --> d : 1 + +' Übergänge von d (Endzustand Q_A) +d --> e : 0 +d --> e : 1 + +' Schleife auf ε +e --> e : 0 +e --> e : 1 +@enduml +``` + +### 1(c) +Gibt es einen deterministischen endlichen Automaten mit weniger Zuständen als +$A_N$, der $L(N)$ akzeptiert? Begründen Sie Ihre Antwort. + + +Um $A_N$ zu minimieren, könnte man nicht erreichbare Zustände, in diesem Fall +{{b},{c},{d},{ε}}, entfernen. +Ansonsten ist der Automat bereits in minimaler Form, da die Zustände sich alle in ihrer +Erreichbarkeit und/oder ihrem Umgang mit Eingaben unterscheiden. + + +## Übung 2 +Betrachten Sie den nichtdeterministischen Automaten N aus Abbildung 2 über dem +Alphabet $Σ = \{ x, y, z \}^*$. Weiterhin seien die Zeichenketten $s_1 = zzx$, $s_2 = xxyz$, +$s_3 = yyy$, $s_4 = xxz$ und $s_5 = xxzxxzxxzxxz$ definiert. + +![image_785.png](image_785.png) +Automat $N$ für Worte aus $\{x,y,z\}^*$, die kein $y$ enthalten und die Zeichenkette $xxy$ beinhalten + +### 2(a) +Geben Sie für jedes $s_i (i ∈ \{ 1, 2, . . . , 5 \})$ an, ob es eine Berechnung (Bearbeitungspfad) für den Automaten N gibt, welche die Zeichenkette si vollständig liest (also +alle Zeichen abarbeitet bevor entschieden wird, ob $s_i$ akzeptiert wird oder nicht.) + + +### 2(b) +Geben Sie für jedes $s_i (i ∈ \{ 1, 2, . . . , 5 \})$ die Menge aller Zustände an, die $N$ durch +die Zeichenkette si erreichen kann + + +### 2(c) +Wie viele Berechnungspfade gibt es, die das Wort $s_5$ vollständig lesen? Begründen +Sie nachvollziehbar, wie Sie auf Ihre Antwort gekommen sind. + +### 2(d) +Beschreiben Sie die Sprache $L(N)$ aller Worte, die der Automat $N$ akzeptiert +(formal oder informal, Ihre Wahl). Welche der Zeichenketten $s_i (i ∈ \{ 1, 2, . . . , 5 \})$ +gehören zu $L(N )$, welche nicht? + +### 2(e) +Beschreiben Sie die Sprache aller Worte, welche der Automat $N$ nicht vollständig +bearbeiten kann, unabhängig davon ob $N$ die Worte akzeptiert oder nicht (formal +oder informal, Ihre Wahl). Also die Worte, für die kein Berechnungspfad existiert +der alle Zeichen liest + + +## Übung 3 +### 3(a) +Gegeben sei ein beliebiger deterministischer endlicher Automat $A = (Σ, Q, q_s, Q_a, δ)$. +Konstruieren Sie einen deterministischen endlichen Automaten $A$ in formaler Tupel- +Darstellung, der genau die Worte akzeptiert die $A$ nicht akzeptiert. Beweisen Sie +die Korrektheit Ihrer Konstruktion. + + +### 3(b) +Gegeben sei ein beliebiger nichtdeterministischer endlicher Automat $N = (Σ, Q, q_s, Q_a, δ)$. +Können Sie hier auf ähnliche Weise wie in [Punkt (a)](#3-a) einen nichtdeterministischen +endlichen Automaten $N$ konstruieren, der genau die Worte akzeptiert, die $N$ nicht +akzeptiert? Falls ja, beweisen Sie die Korrektheit der Konstruktion. Falls nein, +geben Sie ein Beispiel für das die Konstruktion scheitert. + + + +## Übung 4 +Betrachten Sie die beiden deterministischen endlichen Automaten $A_1$ und $A_2$ aus Abbildung 3 +![image_786.png](image_786.png) + +### 4(a) +Beschreiben Sie den Aufbau von Worten $w$ aus der Sprache $L(A_1) ∪ L(A_2)$ (formal +oder informal, Ihre Wahl) + + +### 4(b) +Konstruieren Sie einen nichtdeterministischen endlichen Automaten $N_∪$, der die +Sprache $L(A_1) ∪ L(A_2)$ akzeptiert + +### 4(c) +Beschreiben Sie den Aufbau von Worten $w$ aus der Sprache $L(A_1) ∩ L(A_2)$ (formal +oder informal, Ihre Wahl) + +### 4(d) +Konstruieren Sie einen nichtdeterministischen endlichen Automaten $N_∩$, der die +Sprache $L(A_1) ∩ L(A_2)$ akzeptiert \ No newline at end of file