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0b83fe9bfb
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d4f41fbd44
| Author | SHA1 | Date | |
|---|---|---|---|
| d4f41fbd44 | |||
| 1e91ba5ba6 |
@ -1,46 +1,5 @@
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"cell_type": "code",
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"outputs": [
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"data": {
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"text/plain": "<AGraph <Swig Object of type 'Agraph_t *' at 0x7f4d007b8a20>>"
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"execution_count": 1,
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"metadata": {},
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"output_type": "execute_result"
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}
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],
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"source": [
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"from automata.fa.dfa import DFA\n",
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"\n",
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"# DFA which matches all binary strings ending in an odd number of '1's\n",
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"my_dfa = DFA(\n",
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"\tstates={'q0', 'q1', 'q2'},\n",
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"\tinput_symbols={'0', '1'},\n",
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"\ttransitions={\n",
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"\t\t'q0': {'0': 'q0', '1': 'q1'},\n",
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"\t\t'q1': {'0': 'q0', '1': 'q2'},\n",
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"\t\t'q2': {'0': 'q2', '1': 'q1'}\n",
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"\t},\n",
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"\tinitial_state='q0',\n",
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"\tfinal_states={'q1'}\n",
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")\n",
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"\n",
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"my_dfa.show_diagram()\n"
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],
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"metadata": {
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"collapsed": false,
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"ExecuteTime": {
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"end_time": "2024-01-26T19:01:38.983537200Z",
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"start_time": "2024-01-26T19:01:38.652214500Z"
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}
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},
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"id": "ad7a031539dffdd7",
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"execution_count": 1
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},
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"cell_type": "markdown",
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"source": [
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@ -61,6 +20,16 @@
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},
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"id": "b67364155fcb1072"
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},
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{
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"cell_type": "markdown",
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"source": [
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"<div style=\"page-break-after: always;\"></div>"
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],
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"metadata": {
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"collapsed": false
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"cell_type": "markdown",
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@ -82,6 +51,16 @@
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},
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"id": "ead9052998c5edf6"
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},
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||||
{
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||||
"cell_type": "markdown",
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||||
"source": [
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"<div style=\"page-break-after: always;\"></div>"
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],
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"metadata": {
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"collapsed": false
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},
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"id": "f27520443479cce9"
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},
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{
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"cell_type": "markdown",
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"source": [
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@ -109,12 +88,22 @@
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},
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"id": "874c8dbcc345edad"
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},
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{
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"cell_type": "markdown",
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"source": [
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"<div style=\"page-break-after: always;\"></div>"
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],
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"metadata": {
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"collapsed": false
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},
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"id": "ca477f12a96f397a"
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},
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{
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||||
"cell_type": "markdown",
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"source": [
|
||||
"# Question 11\n",
|
||||
"Consider the relation $R$ defined on the set $\\mathbb{Z}$ as follows:\n",
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||||
"$$\\forall m,n \\in \\mathbb{Z}, (m,n \\in R \\text{ if and only if } m + n = 2k \\text{ for some integer } k$$\n",
|
||||
"$$\\forall m,n \\in \\mathbb{Z}, (m,n) \\in R \\text{ if and only if } m + n = 2k \\text{ for some integer } k$$\n",
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"\n",
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||||
"(a) Is this relation reflexive? Yes\n",
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||||
"For every integer $x$, $x + x = 2x$, therefore it is reflexive\n",
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||||
@ -126,12 +115,147 @@
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||||
"\n",
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||||
"(c) Is this relation transitive? Yes\n",
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||||
"Let $(a,b)$ and $(b,c)$ be valid pairs of integers of the relation $R$\n",
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||||
"$a + b = 2n$, and $b + c = 2p$"
|
||||
"$a + b = 2n$, and $b + c = 2p$, then $b = 2n - a$ and $b = 2p - c$\n",
|
||||
"$-2n + a = 2p - c$\n",
|
||||
"$a + c = 2p + 2n$\n",
|
||||
"Since the sum of a and c is the sum of two even numbers (numbers multiplied by 2 must be even), then the result must be even, so the result is divisible by 2\n",
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||||
"$\\therefore$ The relationship is transitive"
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||||
],
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"metadata": {
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"collapsed": false
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},
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"id": "5fe823d05dd08b12"
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},
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{
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"cell_type": "markdown",
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"source": [
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"<div style=\"page-break-after: always;\"></div>"
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],
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"metadata": {
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"collapsed": false
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},
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"id": "889a2a7359618ee7"
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},
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{
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"cell_type": "markdown",
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"source": [
|
||||
"# Question 14\n",
|
||||
"There are 7 equivalence classes as the results are grouped by the integer returned by $n_0(x) - n_1(x)$/$n_0(y) - n_1(y)$ for any relation pair, the results are that being set of differences $\\{-3, -2, -1, 0, 1, 2, 3\\}$"
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],
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"metadata": {
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"collapsed": false
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},
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"id": "fd69c73a15e8e6a0"
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},
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{
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"cell_type": "code",
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"outputs": [
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{
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"name": "stdout",
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"output_type": "stream",
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"text": [
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"{-3: [('111', '111')],\n",
|
||||
" -2: [('11', '11')],\n",
|
||||
" -1: [('1', '1'),\n",
|
||||
" ('1', '011'),\n",
|
||||
" ('1', '101'),\n",
|
||||
" ('1', '110'),\n",
|
||||
" ('011', '1'),\n",
|
||||
" ('011', '011'),\n",
|
||||
" ('011', '101'),\n",
|
||||
" ('011', '110'),\n",
|
||||
" ('101', '1'),\n",
|
||||
" ('101', '011'),\n",
|
||||
" ('101', '101'),\n",
|
||||
" ('101', '110'),\n",
|
||||
" ('110', '1'),\n",
|
||||
" ('110', '011'),\n",
|
||||
" ('110', '101'),\n",
|
||||
" ('110', '110')],\n",
|
||||
" 0: [('', ''),\n",
|
||||
" ('', '01'),\n",
|
||||
" ('', '10'),\n",
|
||||
" ('01', ''),\n",
|
||||
" ('01', '01'),\n",
|
||||
" ('01', '10'),\n",
|
||||
" ('10', ''),\n",
|
||||
" ('10', '01'),\n",
|
||||
" ('10', '10')],\n",
|
||||
" 1: [('0', '0'),\n",
|
||||
" ('0', '001'),\n",
|
||||
" ('0', '010'),\n",
|
||||
" ('0', '100'),\n",
|
||||
" ('001', '0'),\n",
|
||||
" ('001', '001'),\n",
|
||||
" ('001', '010'),\n",
|
||||
" ('001', '100'),\n",
|
||||
" ('010', '0'),\n",
|
||||
" ('010', '001'),\n",
|
||||
" ('010', '010'),\n",
|
||||
" ('010', '100'),\n",
|
||||
" ('100', '0'),\n",
|
||||
" ('100', '001'),\n",
|
||||
" ('100', '010'),\n",
|
||||
" ('100', '100')],\n",
|
||||
" 2: [('00', '00')],\n",
|
||||
" 3: [('000', '000')]}\n"
|
||||
]
|
||||
}
|
||||
],
|
||||
"source": [
|
||||
"# Set of all strings containing 0 and 1 up to length 3\n",
|
||||
"def get_all_strings_with_a_given_alphabet_and_length(_alphabet, _length):\n",
|
||||
"\t_strings = ['']\n",
|
||||
"\tfor i in range(_length):\n",
|
||||
"\t\ts = [s + c for s in _strings for c in _alphabet]\n",
|
||||
"\t\t_strings += s\n",
|
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"\n",
|
||||
"\t# Remove duplicates (set() isn't as nice as it doesn't preserve order, at least for verification purposes)\n",
|
||||
"\t_strings = list(dict.fromkeys(_strings))\n",
|
||||
"\treturn _strings\n",
|
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"\n",
|
||||
"alphabet = ['0', '1']\n",
|
||||
"strings = get_all_strings_with_a_given_alphabet_and_length(alphabet, 3)\n",
|
||||
"\n",
|
||||
"# The relation is valid if the number of zeros from string x minus the number of ones from string x is equal to the number of zeros from string y minus the number of ones from string y\n",
|
||||
"def is_valid_relation(_x, _y):\n",
|
||||
"\t# Count the number of zeros and ones in each string\n",
|
||||
"\tx_zeros = _x.count('0')\n",
|
||||
"\tx_ones = _x.count('1')\n",
|
||||
"\ty_zeros = _y.count('0')\n",
|
||||
"\ty_ones = _y.count('1')\n",
|
||||
"\n",
|
||||
"\t# Return true if the difference between the number of zeros and ones is equal\n",
|
||||
"\treturn (x_zeros - x_ones) == (y_zeros - y_ones)\n",
|
||||
"\n",
|
||||
"from collections import defaultdict\n",
|
||||
"\n",
|
||||
"# Print all valid relations\n",
|
||||
"equivalence_classes = defaultdict(list)\n",
|
||||
"for x in strings:\n",
|
||||
"\tfor y in strings:\n",
|
||||
"\t\tif is_valid_relation(x, y):\n",
|
||||
"\t\t\t# create a tuple of the two strings\n",
|
||||
"\t\t\tt = (x, y)\n",
|
||||
"\t\t\t# calculate the difference between the number of zeros and ones\n",
|
||||
"\t\t\td = x.count('0') - x.count('1')\n",
|
||||
"\t\t\t# add the tuple a list under the key of the difference\n",
|
||||
"\t\t\tequivalence_classes[d].append(t)\n",
|
||||
"\t\t\t\t\n",
|
||||
"# Print the dict\n",
|
||||
"equivalence_classes = dict(equivalence_classes)\n",
|
||||
"from pprint import pprint\n",
|
||||
"pprint(equivalence_classes)"
|
||||
],
|
||||
"metadata": {
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"collapsed": false,
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"ExecuteTime": {
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"end_time": "2024-01-26T22:03:01.411977200Z",
|
||||
"start_time": "2024-01-26T22:03:01.364977700Z"
|
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}
|
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},
|
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"id": "94712ae197e8639c",
|
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"execution_count": 5
|
||||
}
|
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],
|
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"metadata": {
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|
||||
51
Assignment 3.ipynb
Normal file
51
Assignment 3.ipynb
Normal file
@ -0,0 +1,51 @@
|
||||
{
|
||||
"cells": [
|
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{
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"cell_type": "code",
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"execution_count": null,
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"id": "initial_id",
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"metadata": {
|
||||
"collapsed": true
|
||||
},
|
||||
"outputs": [],
|
||||
"source": [
|
||||
"from automata.fa.dfa import DFA\n",
|
||||
"\n",
|
||||
"# DFA which matches all binary strings ending in an odd number of '1's\n",
|
||||
"my_dfa = DFA(\n",
|
||||
"\tstates={'q0', 'q1', 'q2'},\n",
|
||||
"\tinput_symbols={'0', '1'},\n",
|
||||
"\ttransitions={\n",
|
||||
"\t\t'q0': {'0': 'q0', '1': 'q1'},\n",
|
||||
"\t\t'q1': {'0': 'q0', '1': 'q2'},\n",
|
||||
"\t\t'q2': {'0': 'q2', '1': 'q1'}\n",
|
||||
"\t},\n",
|
||||
"\tinitial_state='q0',\n",
|
||||
"\tfinal_states={'q1'}\n",
|
||||
")\n",
|
||||
"my_dfa.show_diagram()"
|
||||
]
|
||||
}
|
||||
],
|
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"metadata": {
|
||||
"kernelspec": {
|
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"display_name": "Python 3",
|
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"language": "python",
|
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"name": "python3"
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},
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"language_info": {
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"codemirror_mode": {
|
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"name": "ipython",
|
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"version": 2
|
||||
},
|
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"file_extension": ".py",
|
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"mimetype": "text/x-python",
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"name": "python",
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"nbconvert_exporter": "python",
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"pygments_lexer": "ipython2",
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"version": "2.7.6"
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}
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},
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"nbformat": 4,
|
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"nbformat_minor": 5
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}
|
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