From 177ff99d001f5fbeb30a237c7e8a538b469a6a93 Mon Sep 17 00:00:00 2001
From: dgelessus <dgelessus@users.noreply.github.com>
Date: Tue, 4 Apr 2023 16:13:20 +0200
Subject: [PATCH] Minor updates to FormaleSprachen.ipynb
---
info4/kapitel-1/FormaleSprachen.ipynb | 680 ++++++++++++++------------
1 file changed, 356 insertions(+), 324 deletions(-)
diff --git a/info4/kapitel-1/FormaleSprachen.ipynb b/info4/kapitel-1/FormaleSprachen.ipynb
index 44a9b80..9ae3942 100644
--- a/info4/kapitel-1/FormaleSprachen.ipynb
+++ b/info4/kapitel-1/FormaleSprachen.ipynb
@@ -38,7 +38,7 @@
"source": [
"Ein Wort ist eine endliche Folge von Elementen aus dem Alphabet.\n",
"Im Skript schreiben wir $w \\in \\Sigma^{*}$; hier im Notebook schreiben wir\n",
- "$w \\in seq(\\Sigma)$.\n",
+ "$w \\in \\mathrm{seq}(\\Sigma)$.\n",
"\n",
"Im Skript schreiben wir Wörter in dem wir die Symbole aus $\\Sigma$ hintereinander setzten: $abc$. Im Notebook schreiben wir $[a,b,c]$ für die Folge mit den drei Symbolen $a$, $b$ und $c$.\n",
"\n",
@@ -95,7 +95,7 @@
"cell_type": "markdown",
"metadata": {},
"source": [
- "Die Länge eines Wortes $w$ ist die Anzahl der Symbole in $w$. Im Skript schreiben wir $|w|$, im Notebook benutzen wir ```size(w)```:"
+ "Die Länge eines Wortes $w$ ist die Anzahl der Symbole in $w$. Im Skript schreiben wir $|w|$, im Notebook benutzen wir $\\mathrm{size}(w)$:"
]
},
{
@@ -162,7 +162,7 @@
"$\\renewcommand{\\emptyset}{\\mathord\\varnothing}\\mathit{TRUE}$\n",
"\n",
"**Solution:**\n",
- "* $ε = \\emptyset$"
+ "* $\\mathit{ε} = \\emptyset$"
],
"text/plain": [
"TRUE\n",
@@ -184,7 +184,7 @@
"cell_type": "markdown",
"metadata": {},
"source": [
- "Leider ist in der B Sprache das Symbol λ schon vergeben (zur Definition von Funktionen). Deshalb müssen wir in einer weiteren Abweichung vom Skript Epsilon verwenden.\n",
+ "Leider ist in der B-Sprache das Symbol $\\lambda$ schon vergeben (zur Definition von Funktionen). Deshalb müssen wir in einer weiteren Abweichung vom Skript Epsilon verwenden.\n",
"\n",
"Eine formale Sprache ist eine jede Teilmenge von $\\Sigma^*$."
]
@@ -200,7 +200,7 @@
"$\\mathit{TRUE}$\n",
"\n",
"**Solution:**\n",
- "* $\\mathit{Sprachen} = \\{\\mathit{L}\\mid \\mathit{L} \\subseteq \\mathit{seq}(Σ)\\}$"
+ "* $\\mathit{Sprachen} = \\{\\mathit{L}\\mid\\mathit{L} \\subseteq \\mathit{seq}(\\mathit{Σ})\\}$"
],
"text/plain": [
"TRUE\n",
@@ -222,7 +222,7 @@
"cell_type": "markdown",
"metadata": {},
"source": [
- "In folgender B Maschine werden wir ein paar Sprachen und Wörter definieren:"
+ "In folgender B-Maschine werden wir ein paar Sprachen und Wörter definieren:"
]
},
{
@@ -247,12 +247,12 @@
"SETS Σ = {a,b,c}\n",
"CONSTANTS ε, Sprachen, w₁, w₂, L₁, L₂\n",
"PROPERTIES \n",
- " ε ∈ seq(Σ) ∧ size(ε) = 0 ∧\n",
- " Sprachen = {L | L ⊆ seq(Σ)}∧\n",
- " w₁ = [a,a,b,c] ∧\n",
- " w₂ = [c,c] ∧\n",
- " L₁ = {w₁,w₂} ∧\n",
- " L₂ = {w₂, [a,a,a] }\n",
+ " ε ∈ seq(Σ) ∧ size(ε) = 0 ∧\n",
+ " Sprachen = {L | L ⊆ seq(Σ)} ∧\n",
+ " w₁ = [a,a,b,c] ∧\n",
+ " w₂ = [c,c] ∧\n",
+ " L₁ = {w₁,w₂} ∧\n",
+ " L₂ = {w₂, [a,a,a]}\n",
"END"
]
},
@@ -264,7 +264,7 @@
{
"data": {
"text/plain": [
- "Machine constants set up using operation 0: $setup_constants()"
+ "Executed operation: SETUP_CONSTANTS()"
]
},
"execution_count": 9,
@@ -284,7 +284,7 @@
{
"data": {
"text/markdown": [
- "$\\{(1\\mapsto \\mathit{a}),(2\\mapsto \\mathit{a}),(3\\mapsto \\mathit{b}),(4\\mapsto \\mathit{c})\\}$"
+ "$\\{(1\\mapsto\\mathit{a}),(2\\mapsto\\mathit{a}),(3\\mapsto\\mathit{b}),(4\\mapsto\\mathit{c})\\}$"
],
"text/plain": [
"{(1↦a),(2↦a),(3↦b),(4↦c)}"
@@ -334,7 +334,7 @@
{
"data": {
"text/markdown": [
- "$[a,\\mathit{a},\\mathit{b},c]$"
+ "$[\\mathit{a},\\mathit{a},\\mathit{b},\\mathit{c}]$"
],
"text/plain": [
"[a,a,b,c]"
@@ -364,7 +364,7 @@
{
"data": {
"text/markdown": [
- "$[c,c]$"
+ "$[\\mathit{c},\\mathit{c}]$"
],
"text/plain": [
"[c,c]"
@@ -387,7 +387,7 @@
{
"data": {
"text/markdown": [
- "$\\{[c,c],[a,\\mathit{a},\\mathit{b},c]\\}$"
+ "$\\{[\\mathit{c},\\mathit{c}],[\\mathit{a},\\mathit{a},\\mathit{b},\\mathit{c}]\\}$"
],
"text/plain": [
"{[c,c],[a,a,b,c]}"
@@ -435,7 +435,7 @@
}
],
"source": [
- "LeereSprache = ∅ ∧ LeereSprache ≠ {ε}"
+ "LeereSprache = ∅ ∧ LeereSprache ≠ {ε}"
]
},
{
@@ -488,7 +488,7 @@
}
],
"source": [
- "card( {ε} )"
+ "card({ε})"
]
},
{
@@ -511,7 +511,7 @@
}
],
"source": [
- "card( ∅ )"
+ "card(∅)"
]
},
{
@@ -529,7 +529,7 @@
{
"data": {
"text/markdown": [
- "$\\{[c,c],[a,\\mathit{a},\\mathit{b},c]\\}$"
+ "$\\{[\\mathit{c},\\mathit{c}],[\\mathit{a},\\mathit{a},\\mathit{b},\\mathit{c}]\\}$"
],
"text/plain": [
"{[c,c],[a,a,b,c]}"
@@ -552,7 +552,7 @@
{
"data": {
"text/markdown": [
- "$\\{[c,c],[a,\\mathit{a},a]\\}$"
+ "$\\{[\\mathit{c},\\mathit{c}],[\\mathit{a},\\mathit{a},\\mathit{a}]\\}$"
],
"text/plain": [
"{[c,c],[a,a,a]}"
@@ -575,7 +575,7 @@
{
"data": {
"text/markdown": [
- "$\\{[c,c],[a,\\mathit{a},a],[a,\\mathit{a},\\mathit{b},c]\\}$"
+ "$\\{[\\mathit{c},\\mathit{c}],[\\mathit{a},\\mathit{a},\\mathit{a}],[\\mathit{a},\\mathit{a},\\mathit{b},\\mathit{c}]\\}$"
],
"text/plain": [
"{[c,c],[a,a,a],[a,a,b,c]}"
@@ -598,7 +598,7 @@
{
"data": {
"text/markdown": [
- "$\\{[c,c],[a,\\mathit{a},a],[a,\\mathit{a},\\mathit{b},c]\\}$"
+ "$\\{[\\mathit{c},\\mathit{c}],[\\mathit{a},\\mathit{a},\\mathit{a}],[\\mathit{a},\\mathit{a},\\mathit{b},\\mathit{c}]\\}$"
],
"text/plain": [
"{[c,c],[a,a,a],[a,a,b,c]}"
@@ -610,7 +610,7 @@
}
],
"source": [
- "{x| x∈seq(Σ) ∧ (x∈L₁ ∨ x∈L₂)}"
+ "{x | x∈seq(Σ) ∧ (x∈L₁ ∨ x∈L₂)}"
]
},
{
@@ -621,7 +621,7 @@
{
"data": {
"text/markdown": [
- "$\\{[c,c]\\}$"
+ "$\\{[\\mathit{c},\\mathit{c}]\\}$"
],
"text/plain": [
"{[c,c]}"
@@ -644,7 +644,7 @@
{
"data": {
"text/markdown": [
- "$\\{[c,c]\\}$"
+ "$\\{[\\mathit{c},\\mathit{c}]\\}$"
],
"text/plain": [
"{[c,c]}"
@@ -656,7 +656,7 @@
}
],
"source": [
- "{x| x∈seq(Σ) ∧ (x∈L₁ ∧ x∈L₂)}"
+ "{x | x∈seq(Σ) ∧ (x∈L₁ ∧ x∈L₂)}"
]
},
{
@@ -667,7 +667,7 @@
{
"data": {
"text/markdown": [
- "$\\{[a,\\mathit{a},\\mathit{b},c]\\}$"
+ "$\\{[\\mathit{a},\\mathit{a},\\mathit{b},\\mathit{c}]\\}$"
],
"text/plain": [
"{[a,a,b,c]}"
@@ -690,7 +690,7 @@
{
"data": {
"text/markdown": [
- "$\\{[a,\\mathit{a},a]\\}$"
+ "$\\{[\\mathit{a},\\mathit{a},\\mathit{a}]\\}$"
],
"text/plain": [
"{[a,a,a]}"
@@ -716,7 +716,7 @@
"$\\mathit{TRUE}$\n",
"\n",
"**Solution:**\n",
- "* $\\mathit{Komplement} = (\\mathit{seq}(Σ) - \\{[c,c],[a,\\mathit{a},\\mathit{b},c]\\})$"
+ "* $\\mathit{Komplement} = (\\mathit{seq}(\\mathit{Σ}) - \\{[\\mathit{c},\\mathit{c}],[\\mathit{a},\\mathit{a},\\mathit{b},\\mathit{c}]\\})$"
],
"text/plain": [
"TRUE\n",
@@ -751,7 +751,7 @@
{
"data": {
"text/markdown": [
- "$[a,\\mathit{a},\\mathit{b},c]$"
+ "$[\\mathit{a},\\mathit{a},\\mathit{b},\\mathit{c}]$"
],
"text/plain": [
"[a,a,b,c]"
@@ -774,7 +774,7 @@
{
"data": {
"text/markdown": [
- "$[a,\\mathit{a},\\mathit{b},\\mathit{c},\\mathit{a},\\mathit{a},\\mathit{b},c]$"
+ "$[\\mathit{a},\\mathit{a},\\mathit{b},\\mathit{c},\\mathit{a},\\mathit{a},\\mathit{b},\\mathit{c}]$"
],
"text/plain": [
"[a,a,b,c,a,a,b,c]"
@@ -797,7 +797,7 @@
{
"data": {
"text/markdown": [
- "$[a,\\mathit{a},\\mathit{b},c]$"
+ "$[\\mathit{a},\\mathit{a},\\mathit{b},\\mathit{c}]$"
],
"text/plain": [
"[a,a,b,c]"
@@ -820,7 +820,7 @@
{
"data": {
"text/markdown": [
- "$[a,\\mathit{a},\\mathit{b},c]$"
+ "$[\\mathit{a},\\mathit{a},\\mathit{b},\\mathit{c}]$"
],
"text/plain": [
"[a,a,b,c]"
@@ -850,7 +850,7 @@
{
"data": {
"text/markdown": [
- "$\\{[c,c],[a,\\mathit{a},\\mathit{b},c]\\}$"
+ "$\\{[\\mathit{c},\\mathit{c}],[\\mathit{a},\\mathit{a},\\mathit{b},\\mathit{c}]\\}$"
],
"text/plain": [
"{[c,c],[a,a,b,c]}"
@@ -873,7 +873,7 @@
{
"data": {
"text/markdown": [
- "$\\{[c,c],[a,\\mathit{a},a]\\}$"
+ "$\\{[\\mathit{c},\\mathit{c}],[\\mathit{a},\\mathit{a},\\mathit{a}]\\}$"
],
"text/plain": [
"{[c,c],[a,a,a]}"
@@ -896,7 +896,7 @@
{
"data": {
"text/markdown": [
- "$\\{[c,\\mathit{c},\\mathit{c},c],[c,\\mathit{c},\\mathit{a},\\mathit{a},a],[a,\\mathit{a},\\mathit{b},\\mathit{c},\\mathit{c},c],[a,\\mathit{a},\\mathit{b},\\mathit{c},\\mathit{a},\\mathit{a},a]\\}$"
+ "$\\{[\\mathit{c},\\mathit{c},\\mathit{c},\\mathit{c}],[\\mathit{c},\\mathit{c},\\mathit{a},\\mathit{a},\\mathit{a}],[\\mathit{a},\\mathit{a},\\mathit{b},\\mathit{c},\\mathit{c},\\mathit{c}],[\\mathit{a},\\mathit{a},\\mathit{b},\\mathit{c},\\mathit{a},\\mathit{a},\\mathit{a}]\\}$"
],
"text/plain": [
"{[c,c,c,c],[c,c,a,a,a],[a,a,b,c,c,c],[a,a,b,c,a,a,a]}"
@@ -908,7 +908,7 @@
}
],
"source": [
- " {w | ∃(a,b).(a∈L₁ ∧ b∈L₂ ∧ w = a^b)}"
+ "{w | ∃(a,b).(a∈L₁ ∧ b∈L₂ ∧ w = a^b)}"
]
},
{
@@ -927,7 +927,7 @@
{
"data": {
"text/markdown": [
- "$\\{[c,\\mathit{c},\\mathit{c},c],[c,\\mathit{c},\\mathit{a},\\mathit{a},\\mathit{b},c],[a,\\mathit{a},\\mathit{b},\\mathit{c},\\mathit{c},c],[a,\\mathit{a},\\mathit{b},\\mathit{c},\\mathit{a},\\mathit{a},\\mathit{b},c]\\}$"
+ "$\\{[\\mathit{c},\\mathit{c},\\mathit{c},\\mathit{c}],[\\mathit{c},\\mathit{c},\\mathit{a},\\mathit{a},\\mathit{b},\\mathit{c}],[\\mathit{a},\\mathit{a},\\mathit{b},\\mathit{c},\\mathit{c},\\mathit{c}],[\\mathit{a},\\mathit{a},\\mathit{b},\\mathit{c},\\mathit{a},\\mathit{a},\\mathit{b},\\mathit{c}]\\}$"
],
"text/plain": [
"{[c,c,c,c],[c,c,a,a,b,c],[a,a,b,c,c,c],[a,a,b,c,a,a,b,c]}"
@@ -958,7 +958,7 @@
{
"data": {
"text/markdown": [
- "$\\{[c,\\mathit{c},\\mathit{c},\\mathit{c},\\mathit{c},c],[c,\\mathit{c},\\mathit{a},\\mathit{a},\\mathit{b},\\mathit{c},\\mathit{c},c],[a,\\mathit{a},\\mathit{b},\\mathit{c},\\mathit{c},\\mathit{c},\\mathit{c},c],[c,\\mathit{c},\\mathit{c},\\mathit{c},\\mathit{a},\\mathit{a},\\mathit{b},c],[a,\\mathit{a},\\mathit{b},\\mathit{c},\\mathit{a},\\mathit{a},\\mathit{b},\\mathit{c},\\mathit{c},c],[c,\\mathit{c},\\mathit{a},\\mathit{a},\\mathit{b},\\mathit{c},\\mathit{a},\\mathit{a},\\mathit{b},c],[a,\\mathit{a},\\mathit{b},\\mathit{c},\\mathit{c},\\mathit{c},\\mathit{a},\\mathit{a},\\mathit{b},c],[a,\\mathit{a},\\mathit{b},\\mathit{c},\\mathit{a},\\mathit{a},\\mathit{b},\\mathit{c},\\mathit{a},\\mathit{a},\\mathit{b},c]\\}$"
+ "$\\{[\\mathit{c},\\mathit{c},\\mathit{c},\\mathit{c},\\mathit{c},\\mathit{c}],[\\mathit{c},\\mathit{c},\\mathit{a},\\mathit{a},\\mathit{b},\\mathit{c},\\mathit{c},\\mathit{c}],[\\mathit{a},\\mathit{a},\\mathit{b},\\mathit{c},\\mathit{c},\\mathit{c},\\mathit{c},\\mathit{c}],[\\mathit{c},\\mathit{c},\\mathit{c},\\mathit{c},\\mathit{a},\\mathit{a},\\mathit{b},\\mathit{c}],[\\mathit{a},\\mathit{a},\\mathit{b},\\mathit{c},\\mathit{a},\\mathit{a},\\mathit{b},\\mathit{c},\\mathit{c},\\mathit{c}],[\\mathit{c},\\mathit{c},\\mathit{a},\\mathit{a},\\mathit{b},\\mathit{c},\\mathit{a},\\mathit{a},\\mathit{b},\\mathit{c}],[\\mathit{a},\\mathit{a},\\mathit{b},\\mathit{c},\\mathit{c},\\mathit{c},\\mathit{a},\\mathit{a},\\mathit{b},\\mathit{c}],[\\mathit{a},\\mathit{a},\\mathit{b},\\mathit{c},\\mathit{a},\\mathit{a},\\mathit{b},\\mathit{c},\\mathit{a},\\mathit{a},\\mathit{b},\\mathit{c}]\\}$"
],
"text/plain": [
"{[c,c,c,c,c,c],[c,c,a,a,b,c,c,c],[a,a,b,c,c,c,c,c],[c,c,c,c,a,a,b,c],[a,a,b,c,a,a,b,c,c,c],[c,c,a,a,b,c,a,a,b,c],[a,a,b,c,c,c,a,a,b,c],[a,a,b,c,a,a,b,c,a,a,b,c]}"
@@ -981,7 +981,7 @@
{
"data": {
"text/markdown": [
- "$\\{[c,\\mathit{c},\\mathit{c},\\mathit{c},\\mathit{c},\\mathit{c},\\mathit{c},c],[c,\\mathit{c},\\mathit{c},\\mathit{c},\\mathit{a},\\mathit{a},\\mathit{b},\\mathit{c},\\mathit{c},c],[c,\\mathit{c},\\mathit{a},\\mathit{a},\\mathit{b},\\mathit{c},\\mathit{c},\\mathit{c},\\mathit{c},c],[a,\\mathit{a},\\mathit{b},\\mathit{c},\\mathit{c},\\mathit{c},\\mathit{c},\\mathit{c},\\mathit{c},c],[c,\\mathit{c},\\mathit{c},\\mathit{c},\\mathit{c},\\mathit{c},\\mathit{a},\\mathit{a},\\mathit{b},c],[a,\\mathit{a},\\mathit{b},\\mathit{c},\\mathit{a},\\mathit{a},\\mathit{b},\\mathit{c},\\mathit{c},\\mathit{c},\\mathit{c},c],[c,\\mathit{c},\\mathit{c},\\mathit{c},\\mathit{a},\\mathit{a},\\mathit{b},\\mathit{c},\\mathit{a},\\mathit{a},\\mathit{b},c],[c,\\mathit{c},\\mathit{a},\\mathit{a},\\mathit{b},\\mathit{c},\\mathit{c},\\mathit{c},\\mathit{a},\\mathit{a},\\mathit{b},c],[c,\\mathit{c},\\mathit{a},\\mathit{a},\\mathit{b},\\mathit{c},\\mathit{a},\\mathit{a},\\mathit{b},\\mathit{c},\\mathit{c},c],[a,\\mathit{a},\\mathit{b},\\mathit{c},\\mathit{c},\\mathit{c},\\mathit{c},\\mathit{c},\\mathit{a},\\mathit{a},\\mathit{b},c],[a,\\mathit{a},\\mathit{b},\\mathit{c},\\mathit{c},\\mathit{c},\\mathit{a},\\mathit{a},\\mathit{b},\\mathit{c},\\mathit{c},c],[c,\\mathit{c},\\mathit{a},\\mathit{a},\\mathit{b},\\mathit{c},\\mathit{a},\\mathit{a},\\mathit{b},\\mathit{c},\\mathit{a},\\mathit{a},\\mathit{b},c],[a,\\mathit{a},\\mathit{b},\\mathit{c},\\mathit{c},\\mathit{c},\\mathit{a},\\mathit{a},\\mathit{b},\\mathit{c},\\mathit{a},\\mathit{a},\\mathit{b},c],[a,\\mathit{a},\\mathit{b},\\mathit{c},\\mathit{a},\\mathit{a},\\mathit{b},\\mathit{c},\\mathit{c},\\mathit{c},\\mathit{a},\\mathit{a},\\mathit{b},c],[a,\\mathit{a},\\mathit{b},\\mathit{c},\\mathit{a},\\mathit{a},\\mathit{b},\\mathit{c},\\mathit{a},\\mathit{a},\\mathit{b},\\mathit{c},\\mathit{c},c],[a,\\mathit{a},\\mathit{b},\\mathit{c},\\mathit{a},\\mathit{a},\\mathit{b},\\mathit{c},\\mathit{a},\\mathit{a},\\mathit{b},\\mathit{c},\\mathit{a},\\mathit{a},\\mathit{b},c]\\}$"
+ "$\\{[\\mathit{c},\\mathit{c},\\mathit{c},\\mathit{c},\\mathit{c},\\mathit{c},\\mathit{c},\\mathit{c}],[\\mathit{c},\\mathit{c},\\mathit{c},\\mathit{c},\\mathit{a},\\mathit{a},\\mathit{b},\\mathit{c},\\mathit{c},\\mathit{c}],[\\mathit{c},\\mathit{c},\\mathit{a},\\mathit{a},\\mathit{b},\\mathit{c},\\mathit{c},\\mathit{c},\\mathit{c},\\mathit{c}],[\\mathit{a},\\mathit{a},\\mathit{b},\\mathit{c},\\mathit{c},\\mathit{c},\\mathit{c},\\mathit{c},\\mathit{c},\\mathit{c}],[\\mathit{c},\\mathit{c},\\mathit{c},\\mathit{c},\\mathit{c},\\mathit{c},\\mathit{a},\\mathit{a},\\mathit{b},\\mathit{c}],[\\mathit{a},\\mathit{a},\\mathit{b},\\mathit{c},\\mathit{a},\\mathit{a},\\mathit{b},\\mathit{c},\\mathit{c},\\mathit{c},\\mathit{c},\\mathit{c}],[\\mathit{c},\\mathit{c},\\mathit{c},\\mathit{c},\\mathit{a},\\mathit{a},\\mathit{b},\\mathit{c},\\mathit{a},\\mathit{a},\\mathit{b},\\mathit{c}],[\\mathit{c},\\mathit{c},\\mathit{a},\\mathit{a},\\mathit{b},\\mathit{c},\\mathit{c},\\mathit{c},\\mathit{a},\\mathit{a},\\mathit{b},\\mathit{c}],[\\mathit{c},\\mathit{c},\\mathit{a},\\mathit{a},\\mathit{b},\\mathit{c},\\mathit{a},\\mathit{a},\\mathit{b},\\mathit{c},\\mathit{c},\\mathit{c}],[\\mathit{a},\\mathit{a},\\mathit{b},\\mathit{c},\\mathit{c},\\mathit{c},\\mathit{c},\\mathit{c},\\mathit{a},\\mathit{a},\\mathit{b},\\mathit{c}],[\\mathit{a},\\mathit{a},\\mathit{b},\\mathit{c},\\mathit{c},\\mathit{c},\\mathit{a},\\mathit{a},\\mathit{b},\\mathit{c},\\mathit{c},\\mathit{c}],[\\mathit{c},\\mathit{c},\\mathit{a},\\mathit{a},\\mathit{b},\\mathit{c},\\mathit{a},\\mathit{a},\\mathit{b},\\mathit{c},\\mathit{a},\\mathit{a},\\mathit{b},\\mathit{c}],[\\mathit{a},\\mathit{a},\\mathit{b},\\mathit{c},\\mathit{c},\\mathit{c},\\mathit{a},\\mathit{a},\\mathit{b},\\mathit{c},\\mathit{a},\\mathit{a},\\mathit{b},\\mathit{c}],[\\mathit{a},\\mathit{a},\\mathit{b},\\mathit{c},\\mathit{a},\\mathit{a},\\mathit{b},\\mathit{c},\\mathit{c},\\mathit{c},\\mathit{a},\\mathit{a},\\mathit{b},\\mathit{c}],[\\mathit{a},\\mathit{a},\\mathit{b},\\mathit{c},\\mathit{a},\\mathit{a},\\mathit{b},\\mathit{c},\\mathit{a},\\mathit{a},\\mathit{b},\\mathit{c},\\mathit{c},\\mathit{c}],[\\mathit{a},\\mathit{a},\\mathit{b},\\mathit{c},\\mathit{a},\\mathit{a},\\mathit{b},\\mathit{c},\\mathit{a},\\mathit{a},\\mathit{b},\\mathit{c},\\mathit{a},\\mathit{a},\\mathit{b},\\mathit{c}]\\}$"
],
"text/plain": [
"{[c,c,c,c,c,c,c,c],[c,c,c,c,a,a,b,c,c,c],[c,c,a,a,b,c,c,c,c,c],[a,a,b,c,c,c,c,c,c,c],[c,c,c,c,c,c,a,a,b,c],[a,a,b,c,a,a,b,c,c,c,c,c],[c,c,c,c,a,a,b,c,a,a,b,c],[c,c,a,a,b,c,c,c,a,a,b,c],[c,c,a,a,b,c,a,a,b,c,c,c],[a,a,b,c,c,c,c,c,a,a,b,c],[a,a,b,c,c,c,a,a,b,c,c,c],[c,c,a,a,b,c,a,a,b,c,a,a,b,c],[a,a,b,c,c,c,a,a,b,c,a,a,b,c],[a,a,b,c,a,a,b,c,c,c,a,a,b,c],[a,a,b,c,a,a,b,c,a,a,b,c,c,c],[a,a,b,c,a,a,b,c,a,a,b,c,a,a,b,c]}"
@@ -1027,7 +1027,7 @@
"cell_type": "markdown",
"metadata": {},
"source": [
- "Die Spiegelung eines Wortes kann im Notebook mit dem ```rev``` Operator verwirklicht werden:"
+ "Die Spiegelung eines Wortes kann im Notebook mit dem Operator `rev` verwirklicht werden:"
]
},
{
@@ -1038,7 +1038,7 @@
{
"data": {
"text/markdown": [
- "$[c,\\mathit{b},\\mathit{a},a]$"
+ "$[\\mathit{c},\\mathit{b},\\mathit{a},\\mathit{a}]$"
],
"text/plain": [
"[c,b,a,a]"
@@ -1068,7 +1068,7 @@
{
"data": {
"text/markdown": [
- "$\\{[c,c],[c,\\mathit{b},\\mathit{a},a]\\}$"
+ "$\\{[\\mathit{c},\\mathit{c}],[\\mathit{c},\\mathit{b},\\mathit{a},\\mathit{a}]\\}$"
],
"text/plain": [
"{[c,c],[c,b,a,a]}"
@@ -1087,7 +1087,7 @@
"cell_type": "markdown",
"metadata": {},
"source": [
- "Die Teilwort und Präfix Relationen definieren wir in folgender B Maschine:"
+ "Die Teilwort- und Präfix-Relationen definieren wir in folgender B-Maschine:"
]
},
{
@@ -1112,16 +1112,16 @@
"SETS Sigma = {a,b,c}\n",
"CONSTANTS ε, Sprachen, w₁, w₂, L₁, L₂\n",
"ABSTRACT_CONSTANTS\n",
- " teilwort, präfix\n",
+ " teilwort, präfix\n",
"PROPERTIES \n",
- " ε ∈ seq(Sigma) ∧ size(ε) = 0 ∧\n",
- " Sprachen = {L | L ⊆ seq(Sigma)}∧\n",
- " w₁ = [a,a,b,c] ∧\n",
- " w₂ = [c,a,b,a] ∧\n",
- " L₁ = {w₁,w₂} ∧\n",
- " L₂ = {w₂, [a,a,a] } ∧\n",
- " teilwort = {ut,vt | ut∈seq(Sigma) & vt∈seq(Sigma) ∧ ∃(vt1,vt2).(vt1^ut^vt2 = vt)} ∧\n",
- " präfix = {up,vp | up∈seq(Sigma) & vp∈seq(Sigma) ∧ ∃(wp).(up^wp = vp)}\n",
+ " ε ∈ seq(Sigma) ∧ size(ε) = 0 ∧\n",
+ " Sprachen = {L | L ⊆ seq(Sigma)}∧\n",
+ " w₁ = [a,a,b,c] ∧\n",
+ " w₂ = [c,a,b,a] ∧\n",
+ " L₁ = {w₁,w₂} ∧\n",
+ " L₂ = {w₂, [a,a,a]} ∧\n",
+ " teilwort = {ut,vt | ut∈seq(Sigma) & vt∈seq(Sigma) ∧ ∃(vt1,vt2).(vt1^ut^vt2 = vt)} ∧\n",
+ " präfix = {up,vp | up∈seq(Sigma) & vp∈seq(Sigma) ∧ ∃(wp).(up^wp = vp)}\n",
"DEFINITIONS SET_PREF_PP_SEQUENCES == TRUE\n",
"END"
]
@@ -1134,7 +1134,7 @@
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- "Machine constants set up using operation 0: $setup_constants()"
+ "Executed operation: SETUP_CONSTANTS()"
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"</g>\n",
"</g>\n",
- "</svg>"
+ "</svg>\n"
],
"text/plain": [
- "<Dot visualization: expr_as_graph [TWTW={[],[a],[a,b],[b],[b,c],[c],[a,b,c]}(\"tw\",{x,y|(x,y):teilwort & y:TW})]>"
+ "<Dot visualization: expr_as_graph [LET TW BE TW={[],[a],[a,b],[b],[b,c],[c],[a,b,c]} IN(\n",
+ "(\"tw\",{x,y | x |-> y : teilwort & y : TW})\n",
+ ")END]>"
]
},
"execution_count": 52,
@@ -1539,7 +1541,7 @@
}
],
"source": [
- ":dot expr_as_graph (\"tw\",{x,y | x |-> y : teilwort & y : TW })"
+ ":dot expr_as_graph (\"tw\",{x,y | x |-> y : teilwort & y : TW})"
]
},
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"text/plain": [
- "<Dot visualization: expr_as_graph [TWTW={[],[a],[a,b],[b],[b,c],[c],[a,b,c]}(\"präfix\",{x,y|(x,y):präfix & y:TW})]>"
+ "<Dot visualization: expr_as_graph [LET TW BE TW={[],[a],[a,b],[b],[b,c],[c],[a,b,c]} IN(\n",
+ "(\"präfix\",{x,y | x |-> y : präfix & y : TW})\n",
+ ")END]>"
]
},
"execution_count": 53,
@@ -1735,7 +1739,7 @@
}
],
"source": [
- ":dot expr_as_graph (\"präfix\",{x,y | x |-> y : präfix & y : TW })"
+ ":dot expr_as_graph (\"präfix\",{x,y | x |-> y : präfix & y : TW})"
]
},
{
@@ -1749,7 +1753,7 @@
},
{
"cell_type": "code",
- "execution_count": 63,
+ "execution_count": 55,
"metadata": {},
"outputs": [
{
@@ -1758,7 +1762,7 @@
"Loaded machine: FormaleSprachen"
]
},
- "execution_count": 63,
+ "execution_count": 55,
"metadata": {},
"output_type": "execute_result"
}
@@ -1768,115 +1772,138 @@
"/* Formulierung in B der Grundbegriffe aus dem Foliensatz 1 von Informatik 4 */\n",
"/* erstellt von Michael Leuschel, 2014 */\n",
"SETS\n",
- " Alphabet /* Ein Alphabet ist eine endliche, nichtleere Menge von Buchstaben (oder Symbolen). */\n",
- " = {a,b,c} /* Hier als Beispiel {a,b,c} */\n",
+ "Alphabet /* Ein Alphabet ist eine endliche, nichtleere Menge von Buchstaben (oder Symbolen). */\n",
+ " = {a,b,c} /* Hier als Beispiel {a,b,c} */\n",
"ABSTRACT_CONSTANTS\n",
- " kat, Worte, Sprachen, kleene, sp, teilwort, praefix\n",
+ " kat, Worte, Sprachen, kleene, sp, teilwort, praefix\n",
"CONSTANTS\n",
- " w1, w2, lambda,\n",
- " L1, L2, LeereSprache, KomplementL1, L1L2\n",
+ " w1, w2, lambda,\n",
+ " L1, L2, LeereSprache, KomplementL1, L1L2\n",
"PROPERTIES\n",
- " Worte = seq(Alphabet) /* Ein Wort ist eine endliche Folge von Elementen aus dem Alphabet */\n",
- " &\n",
- " /* Beispiele */\n",
- " w1 = [a,a,b,c] & /* in den Folien schreiben wir aabc anstatt [a,a,b,c] */\n",
- " w1 : Worte & /* w1 ist ein Wort */\n",
- " w2 = [c,a,b,a] &\n",
- " w2 : Worte &\n",
- " /* Die Laenge eines Wortes ist die Anzahl der Symbole in w */\n",
- " /* Beispiel: */\n",
- " size(w1) = 4 &\n",
- " \n",
- "/* Das leere Wort ist das eindeutig bestimmte Wort der Laenge 0 */\n",
- " lambda: Worte & size(lambda) = 0 &\n",
- "\n",
- "/* Die Menge aller Worte bezeichnen wir mit \\Sigma^*. Hier verwenden wir seq(Alphabet) */\n",
- "\n",
- " /* Eine formale Sprache ist eine jede Teilmenge von \\Sigma^*. */\n",
- " Sprachen = {L | L <: Worte} &\n",
- " /* Beispiel: */\n",
- " L1 = {w1,w2} & L1 : Sprachen &\n",
- " L2 = {w2, [a,a,a] } & L2 : Sprachen &\n",
- "\n",
- " /* Die Leere Sprache ist die Sprache die keine Wörter enthält */\n",
- " LeereSprache = {} &\n",
- " /* Beachte: */\n",
- " LeereSprache /= {lambda} &\n",
- "\n",
- " /* Die Kardinalität einer Sprache L ist die Anzahl der Wörter von L. \n",
- " Beispiel: */\n",
- " card(L1) = 2 & card({lambda}) = 1 & card(LeereSprache) = 0 &\n",
- "\n",
- " /* Beispiele für die Vereinigung von Sprachen: */\n",
- " L1 \\/ L2 = {x| x:Worte & (x:L1 or x:L2)} &\n",
- "/* Beispiele für die Durchchnitt von Sprachen: */\n",
- " L1 /\\ L2 = {x| x:Worte & x:L1 & x:L2} &\n",
- "/* Beispiele für die Differenz von Sprachen: */\n",
- " L1 - L2 = {x| x:Worte & x:L1 & x/: L2} &\n",
- "/* Beispiele für das Komplement von Sprachen: */\n",
- " KomplementL1 = Worte - L1 /* kann man auch so schreiben {x| x:Worte & x/: L1} */ &\n",
- " w1 /: KomplementL1 & [a,a,a] : KomplementL1 & lambda : KomplementL1 &\n",
- "\n",
- " /* Konkatenation von Wörtern wird in B mit ^ geschrieben: */\n",
- " w1^w2 = [a,a,b,c,c,a,b,a] &\n",
- " w1^lambda = w1 & lambda^w1 = w1 &\n",
- "\n",
- " /* Verkettung von Sprachen */\n",
- " L1L2 = {w | #(a,b).(a:L1 & b:L2 & w = a^b)}\n",
- "\n",
- " & kat = %(x,y).(x:Worte & y:Worte | x^y) /* Die Konkatenierung als explizite Funktion */\n",
- " & kat(w1,w2) = [a,a,b,c,c,a,b,a] \n",
- " & L1L2 = kat[L1*L2]\n",
- "\n",
- " /* Eine rekursive Definition von A^n: */\n",
- " & kleene = %(A,n).(A : Sprachen & n=0|{lambda}) \\/\n",
- " %(A,n).(A : Sprachen & n>0| kat[A*kleene(A,n-1)])\n",
- " & kleene(L1,1) = L1\n",
- " & kleene(L2,1) = L2\n",
- "/* A^* ist momentan schwieriger darstellbar, zumindest so, dass man mit ProB damit\n",
- " arbeiten könnte */\n",
- "\n",
- " & /* Die Spiegelbildoperation */\n",
- " sp = %u.(u:Worte| %i.(i:dom(u)|u(size(u)+1-i)))\n",
- " & sp(w1) = [c,b,a,a]\n",
- " & sp(sp(w1)) = w1\n",
- " & sp(lambda) = lambda\n",
- "\n",
- " /* Die Spiegelung einer Sprache bekommt man einfach durch das relational image: */\n",
- " & sp[L1] = {sp(w1),sp(w2)}\n",
- " & sp[sp[L1]] = L1\n",
- " & sp[sp[L2]] = L2\n",
- "\n",
- " /* Die Teilwortrelation */\n",
- " & teilwort = {u,v | u:Worte & v:Worte & #(v1,v2).(v1^u^v2 = v)}\n",
- " & [a,a] |-> [c,b,a,a,b] : teilwort\n",
- " & [a,a,a] |-> [c,b,a,a,b] /: teilwort\n",
- " /* Wir können alle Teilworte durch das relational image der Umkehrrelation berechnen: */\n",
- " & teilwort~[{[a,a,b]}] = { lambda, [a], [b], [a,a],[a,b], [a,a,b] }\n",
- "\n",
- " /* Die Anfangswortrelation */\n",
- " & praefix = {u,v | u:Worte & v:Worte & #(w).(u^w = v)}\n",
- " & [a,a] |-> [c,b,a,a,b] /: praefix\n",
- " & [c,b] |-> [c,b,a,a,b] : praefix\n",
- " /* Wir können alle Präfixe durch das relational image der Umkehrrelation berechnen: */\n",
- " & praefix~[{[a,a,b]}] = { lambda, [a], [a,a], [a,a,b] }\n",
+ " Worte = seq(Alphabet) /* Ein Wort ist eine endliche Folge von Elementen aus dem Alphabet */\n",
+ " &\n",
+ " /* Beispiele */\n",
+ " w1 = [a,a,b,c] & /* in den Folien schreiben wir aabc anstatt [a,a,b,c] */\n",
+ " w1 : Worte & /* w1 ist ein Wort */\n",
+ " w2 = [c,a,b,a] &\n",
+ " w2 : Worte &\n",
+ " /* Die Laenge eines Wortes ist die Anzahl der Symbole in w */\n",
+ " /* Beispiel: */\n",
+ " size(w1) = 4 &\n",
+ " \n",
+ " /* Das leere Wort ist das eindeutig bestimmte Wort der Laenge 0 */\n",
+ " lambda: Worte & size(lambda) = 0 &\n",
+ " \n",
+ " /* Die Menge aller Worte bezeichnen wir mit \\Sigma^*. Hier verwenden wir seq(Alphabet) */\n",
+ " \n",
+ " /* Eine formale Sprache ist eine jede Teilmenge von \\Sigma^*. */\n",
+ " Sprachen = {L | L <: Worte} &\n",
+ " /* Beispiel: */\n",
+ " L1 = {w1,w2} & L1 : Sprachen &\n",
+ " L2 = {w2, [a,a,a] } & L2 : Sprachen &\n",
+ " \n",
+ " /* Die Leere Sprache ist die Sprache die keine Wörter enthält */\n",
+ " LeereSprache = {} &\n",
+ " /* Beachte: */\n",
+ " LeereSprache /= {lambda} &\n",
+ " \n",
+ " /* Die Kardinalität einer Sprache L ist die Anzahl der Wörter von L. \n",
+ " Beispiel: */\n",
+ " card(L1) = 2 & card({lambda}) = 1 & card(LeereSprache) = 0 &\n",
+ " \n",
+ " /* Beispiele für die Vereinigung von Sprachen: */\n",
+ " L1 \\/ L2 = {x| x:Worte & (x:L1 or x:L2)} &\n",
+ " /* Beispiele für die Durchchnitt von Sprachen: */\n",
+ " L1 /\\ L2 = {x| x:Worte & x:L1 & x:L2} &\n",
+ " /* Beispiele für die Differenz von Sprachen: */\n",
+ " L1 - L2 = {x| x:Worte & x:L1 & x/: L2} &\n",
+ " /* Beispiele für das Komplement von Sprachen: */\n",
+ " KomplementL1 = Worte - L1 /* kann man auch so schreiben {x| x:Worte & x/: L1} */ &\n",
+ " w1 /: KomplementL1 & [a,a,a] : KomplementL1 & lambda : KomplementL1 &\n",
+ " \n",
+ " /* Konkatenation von Wörtern wird in B mit ^ geschrieben: */\n",
+ " w1^w2 = [a,a,b,c,c,a,b,a] &\n",
+ " w1^lambda = w1 & lambda^w1 = w1 &\n",
+ " \n",
+ " /* Verkettung von Sprachen */\n",
+ " L1L2 = {w | #(a,b).(a:L1 & b:L2 & w = a^b)}\n",
+ " \n",
+ " & kat = %(x,y).(x:Worte & y:Worte | x^y) /* Die Konkatenierung als explizite Funktion */\n",
+ " & kat(w1,w2) = [a,a,b,c,c,a,b,a] \n",
+ " & L1L2 = kat[L1*L2]\n",
+ " \n",
+ " /* Eine rekursive Definition von A^n: */\n",
+ " & kleene = %(A,n).(A : Sprachen & n=0|{lambda}) \\/\n",
+ " %(A,n).(A : Sprachen & n>0| kat[A*kleene(A,n-1)])\n",
+ " & kleene(L1,1) = L1\n",
+ " & kleene(L2,1) = L2\n",
+ " /* A^* ist momentan schwieriger darstellbar, zumindest so, dass man mit ProB damit\n",
+ " arbeiten könnte */\n",
+ " \n",
+ " /* Die Spiegelbildoperation */\n",
+ " & sp = %u.(u:Worte| %i.(i:dom(u)|u(size(u)+1-i)))\n",
+ " & sp(w1) = [c,b,a,a]\n",
+ " & sp(sp(w1)) = w1\n",
+ " & sp(lambda) = lambda\n",
+ " \n",
+ " /* Die Spiegelung einer Sprache bekommt man einfach durch das relational image: */\n",
+ " & sp[L1] = {sp(w1),sp(w2)}\n",
+ " & sp[sp[L1]] = L1\n",
+ " & sp[sp[L2]] = L2\n",
+ " \n",
+ " /* Die Teilwortrelation */\n",
+ " & teilwort = {u,v | u:Worte & v:Worte & #(v1,v2).(v1^u^v2 = v)}\n",
+ " & [a,a] |-> [c,b,a,a,b] : teilwort\n",
+ " & [a,a,a] |-> [c,b,a,a,b] /: teilwort\n",
+ " /* Wir können alle Teilworte durch das relational image der Umkehrrelation berechnen: */\n",
+ " & teilwort~[{[a,a,b]}] = { lambda, [a], [b], [a,a],[a,b], [a,a,b] }\n",
+ " \n",
+ " /* Die Anfangswortrelation */\n",
+ " & praefix = {u,v | u:Worte & v:Worte & #(w).(u^w = v)}\n",
+ " & [a,a] |-> [c,b,a,a,b] /: praefix\n",
+ " & [c,b] |-> [c,b,a,a,b] : praefix\n",
+ " /* Wir können alle Präfixe durch das relational image der Umkehrrelation berechnen: */\n",
+ " & praefix~[{[a,a,b]}] = { lambda, [a], [a,a], [a,a,b] }\n",
"ASSERTIONS\n",
- " card(L1L2) = 4\n",
+ " card(L1L2) = 4\n",
+ "DEFINITIONS SET_PREF_PP_SEQUENCES == TRUE\n",
"END"
]
},
{
"cell_type": "code",
- "execution_count": 66,
+ "execution_count": 56,
"metadata": {},
"outputs": [
{
- "ename": "IllegalArgumentException",
- "evalue": "Executing operation $initialise_machine with predicate 1=1 produced errors: Could not execute operation INITIALISATION in state root",
- "output_type": "error",
- "traceback": [
- "\u001b[1m\u001b[31mjava.lang.IllegalArgumentException: Executing operation $initialise_machine with predicate 1=1 produced errors: Could not execute operation INITIALISATION in state root\u001b[0m"
- ]
+ "data": {
+ "text/plain": [
+ "Executed operation: SETUP_CONSTANTS()"
+ ]
+ },
+ "execution_count": 56,
+ "metadata": {},
+ "output_type": "execute_result"
+ }
+ ],
+ "source": [
+ ":constants"
+ ]
+ },
+ {
+ "cell_type": "code",
+ "execution_count": 57,
+ "metadata": {},
+ "outputs": [
+ {
+ "data": {
+ "text/plain": [
+ "Executed operation: INITIALISATION()"
+ ]
+ },
+ "execution_count": 57,
+ "metadata": {},
+ "output_type": "execute_result"
}
],
"source": [
@@ -1885,16 +1912,21 @@
},
{
"cell_type": "code",
- "execution_count": 67,
+ "execution_count": 58,
"metadata": {},
"outputs": [
{
- "ename": "CommandExecutionException",
- "evalue": ":eval: NOT-INITIALISED",
- "output_type": "error",
- "traceback": [
- "\u001b[1m\u001b[31m:eval: NOT-INITIALISED\u001b[0m"
- ]
+ "data": {
+ "text/markdown": [
+ "$\\{[\\mathit{c},\\mathit{a},\\mathit{b},\\mathit{a},\\mathit{a},\\mathit{a},\\mathit{b},\\mathit{c}],[\\mathit{c},\\mathit{a},\\mathit{b},\\mathit{a},\\mathit{c},\\mathit{a},\\mathit{b},\\mathit{a}],[\\mathit{a},\\mathit{a},\\mathit{b},\\mathit{c},\\mathit{a},\\mathit{a},\\mathit{b},\\mathit{c}],[\\mathit{a},\\mathit{a},\\mathit{b},\\mathit{c},\\mathit{c},\\mathit{a},\\mathit{b},\\mathit{a}]\\}$"
+ ],
+ "text/plain": [
+ "{[c,a,b,a,a,a,b,c],[c,a,b,a,c,a,b,a],[a,a,b,c,a,a,b,c],[a,a,b,c,c,a,b,a]}"
+ ]
+ },
+ "execution_count": 58,
+ "metadata": {},
+ "output_type": "execute_result"
}
],
"source": [
--
GitLab