--- /dev/null	Thu Jan 01 00:00:00 1970 +0000
+++ b/videos/02-prelims.srt	Tue Sep 29 12:52:07 2020 +0100
@@ -0,0 +1,1217 @@
+1
+00:00:09,990 --> 00:00:13,465
+Welcome back to this
+week's lecture.
+
+2
+00:00:13,465 --> 00:00:16,450
+The task this week is to actually
+
+3
+00:00:16,450 --> 00:00:20,320
+implement a regular
+expression matcher.
+
+4
+00:00:20,320 --> 00:00:22,510
+And we want to be a bit
+
+5
+00:00:22,510 --> 00:00:25,315
+faster than the regular
+expression matcher
+
+6
+00:00:25,315 --> 00:00:29,380
+in Python, Ruby,
+Javascript, and Java.
+
+7
+00:00:29,380 --> 00:00:31,960
+Remember this regular expression
+
+8
+00:00:31,960 --> 00:00:35,785
+and strings which are
+just a number of a's,
+
+9
+00:00:35,785 --> 00:00:39,670
+these regular expression matchers
+where abysmally slow.
+
+10
+00:00:39,670 --> 00:00:43,170
+They could only match
+approximately 30 a's in
+
+11
+00:00:43,170 --> 00:00:45,665
+something like half a minute.
+
+12
+00:00:45,665 --> 00:00:49,460
+What we like to do with
+our regular expression matcher.
+
+13
+00:00:49,460 --> 00:00:51,890
+We will try a few times,
+
+14
+00:00:51,890 --> 00:00:55,505
+but our second trial will already
+be much, much better.
+
+15
+00:00:55,505 --> 00:00:58,400
+It will probably
+match around maybe
+
+16
+00:00:58,400 --> 00:01:02,030
+thousand a's in something
+in half a minute.
+
+17
+00:01:02,030 --> 00:01:03,920
+But our third version in
+
+18
+00:01:03,920 --> 00:01:06,230
+comparison will be
+blindingly fast.
+
+19
+00:01:06,230 --> 00:01:08,180
+And we'll be able to match
+
+20
+00:01:08,180 --> 00:01:10,145
+strings of length 10,000 a's
+
+21
+00:01:10,145 --> 00:01:12,230
+and will not require
+
+22
+00:01:12,230 --> 00:01:14,975
+more than five seconds.
+So let's go ahead with that.
+
+23
+00:01:14,975 --> 00:01:18,095
+We will first implement
+
+24
+00:01:18,095 --> 00:01:19,880
+our regular expression
+matcher only
+
+25
+00:01:19,880 --> 00:01:22,069
+for the basic
+regular expressions.
+
+26
+00:01:22,069 --> 00:01:25,430
+So we will have only six
+cases to think about.
+
+27
+00:01:25,430 --> 00:01:27,620
+This will simplify matters at first.
+
+28
+00:01:27,620 --> 00:01:30,350
+Later we can
+think about how to
+
+29
+00:01:30,350 --> 00:01:34,100
+extend that to the extended
+regular expressions.
+
+30
+00:01:34,100 --> 00:01:37,625
+Unfortunately, before we can
+start our implementation,
+
+31
+00:01:37,625 --> 00:01:39,290
+we have to discuss
+
+32
+00:01:39,290 --> 00:01:42,470
+when two regular
+expressions are equivalent.
+
+33
+00:01:42,470 --> 00:01:46,595
+And we use here this notation
+of the triple equals.
+
+34
+00:01:46,595 --> 00:01:48,215
+We say a regular expression
+
+35
+00:01:48,215 --> 00:01:50,180
+r1 and r2 are
+
+36
+00:01:50,180 --> 00:01:56,059
+equivalent if and only
+if the language of r1 is
+
+37
+00:01:56,059 --> 00:01:59,360
+equal to the language of r2.
+
+38
+00:01:59,360 --> 00:02:02,915
+Sounds complicated,
+but essentially means
+
+39
+00:02:02,915 --> 00:02:04,700
+if the two regular expressions can
+
+40
+00:02:04,700 --> 00:02:06,950
+match exactly the same strings,
+
+41
+00:02:06,950 --> 00:02:09,800
+then we want to regard
+them as equivalent.
+
+42
+00:02:09,800 --> 00:02:14,300
+This equivalence justifies
+why we can be sloppy
+
+43
+00:02:14,300 --> 00:02:16,040
+with parentheses.
+
+44
+00:02:16,040 --> 00:02:23,870
+For example, if we have
+(r1 + r2) + r3,
+
+45
+00:02:23,870 --> 00:02:32,270
+then this will be equivalent
+to r1 + (r2 + r3).
+
+46
+00:02:32,270 --> 00:02:34,910
+Remember, regular
+expressions are trees,
+
+47
+00:02:34,910 --> 00:02:37,940
+so these are two different
+regular expressions,
+
+48
+00:02:37,940 --> 00:02:41,540
+but they can match
+the same strings.
+
+49
+00:02:41,540 --> 00:02:45,695
+Because, well, if we take
+here the meaning of that,
+
+50
+00:02:45,695 --> 00:02:54,350
+that would be just L(r1 + r2 + r3),
+
+
+51
+00:02:54,350 --> 00:03:04,100
+which is equal to L(r1 + r2) u L(r3).
+
+
+52
+00:03:04,100 --> 00:03:10,595
+And that is equal to of 
+
+53
+00:03:10,595 --> 00:03:15,710
+L(r1) u L(r2) u L(r3).
+
+
+54
+00:03:15,710 --> 00:03:17,930
+And if you do the
+same calculation
+
+55
+00:03:17,930 --> 00:03:19,445
+for that regular expression,
+
+56
+00:03:19,445 --> 00:03:22,940
+you will find out the
+two sets are the same.
+
+57
+00:03:22,940 --> 00:03:26,195
+So both regular expressions
+can match the same strings.
+
+58
+00:03:26,195 --> 00:03:28,805
+So even if they're different
+regular expressions,
+
+59
+00:03:28,805 --> 00:03:31,220
+we can regard them as eqivalent.
+
+60
+00:03:31,220 --> 00:03:33,290
+And so that's why we can forget
+
+61
+00:03:33,290 --> 00:03:35,270
+about writing these parentheses.
+
+62
+00:03:35,270 --> 00:03:40,205
+Let's have a look
+at some more examples.
+
+63
+00:03:40,205 --> 00:03:41,870
+So the first one here,
+
+64
+00:03:41,870 --> 00:03:43,205
+that is now clear.
+
+65
+00:03:43,205 --> 00:03:45,200
+We did this calculation already
+
+66
+00:03:45,200 --> 00:03:47,480
+for arbitrary regular expressions.
+
+67
+00:03:47,480 --> 00:03:49,520
+Here it is for
+concrete ones a, b and c.
+
+68
+00:03:49,520 --> 00:03:52,690
+Here on the left-hand side,
+
+69
+00:03:52,690 --> 00:03:54,895
+the regular expression
+has the same meaning
+
+70
+00:03:54,895 --> 00:03:56,620
+as on the right-hand side.
+
+71
+00:03:56,620 --> 00:03:58,420
+So they are equivalent.
+
+72
+00:03:58,420 --> 00:04:01,375
+We can ignore the parentheses.
+
+73
+00:04:01,375 --> 00:04:03,220
+If I have a choice,
+
+74
+00:04:03,220 --> 00:04:06,610
+but the choice is
+the same a or a,
+
+75
+00:04:06,610 --> 00:04:09,265
+then this is
+equivalent to just a.
+
+76
+00:04:09,265 --> 00:04:13,075
+So the same choice doesn't
+introduce anything more.
+
+77
+00:04:13,075 --> 00:04:15,370
+In the next case, if I have
+
+78
+00:04:15,370 --> 00:04:19,450
+a regular expression
+which can match a or b,
+
+79
+00:04:19,450 --> 00:04:23,860
+that can match the same
+strings as b or a.
+
+80
+00:04:23,860 --> 00:04:27,325
+So you have a kind of
+commutativity for the plus,
+
+81
+00:04:27,325 --> 00:04:29,485
+which is like on natural numbers.
+
+82
+00:04:29,485 --> 00:04:32,880
+But you would not have a
+communitivity for the sequence:
+
+83
+00:04:32,880 --> 00:04:37,070
+if you have
+first a and then b,
+
+84
+00:04:37,070 --> 00:04:40,850
+that's not the same as
+matching first b and then a.
+
+85
+00:04:40,850 --> 00:04:42,800
+So for the sequence ...
+
+86
+00:04:42,800 --> 00:04:44,615
+they are not equivalent.
+
+87
+00:04:44,615 --> 00:04:49,475
+However, for the sequence I
+can, like for the plus, ignore
+
+88
+00:04:49,475 --> 00:04:51,245
+prarentheses.
+
+89
+00:04:51,245 --> 00:04:55,070
+If I have the parentheses
+and this way,
+
+90
+00:04:55,070 --> 00:04:57,785
+or I have it in this way.
+
+91
+00:04:57,785 --> 00:05:00,605
+These are two different
+regular expressions,
+
+92
+00:05:00,605 --> 00:05:02,120
+but they have the same meaning.
+
+93
+00:05:02,120 --> 00:05:03,815
+So they are equivalent.
+
+94
+00:05:03,815 --> 00:05:05,780
+And so I can leave out parentheses
+
+95
+00:05:05,780 --> 00:05:09,170
+for times as well.
+
+96
+00:05:09,170 --> 00:05:12,185
+The next one is a slightly
+more interesting one.
+
+97
+00:05:12,185 --> 00:05:15,020
+If I have a regular
+expression which can match
+
+98
+00:05:15,020 --> 00:05:19,655
+c first followed by (a + b),
+
+99
+00:05:19,655 --> 00:05:25,355
+then this is the same as
+first c followed by a,
+
+100
+00:05:25,355 --> 00:05:28,640
+or c followed by b.
+
+101
+00:05:28,640 --> 00:05:31,265
+So that's the kind of
+distributivity law
+
+102
+00:05:31,265 --> 00:05:33,545
+on regular expressions.
+
+103
+00:05:33,545 --> 00:05:36,350
+But they're also regular expressions
+which are not equivalent.
+
+104
+00:05:36,350 --> 00:05:38,990
+If I have the regular
+expression which can
+
+105
+00:05:38,990 --> 00:05:42,800
+match the string containing
+exactly two a's.
+
+106
+00:05:42,800 --> 00:05:44,240
+That is not equivalent
+
+107
+00:05:44,240 --> 00:05:46,730
+to the regular
+expression which can just match
+
+108
+00:05:46,730 --> 00:05:49,250
+a single a. And similarly
+
+109
+00:05:49,250 --> 00:05:51,680
+in this case, if I can match
+
+110
+00:05:51,680 --> 00:05:56,075
+a or the string b followed by c,
+
+111
+00:05:56,075 --> 00:05:59,825
+that is not the same as a or b,
+
+112
+00:05:59,825 --> 00:06:02,000
+followed by a or c.
+
+113
+00:06:02,000 --> 00:06:05,990
+I'll let you think about
+why is that not equivalent.
+
+114
+00:06:05,990 --> 00:06:08,060
+Essentially you
+have to find out what's
+
+115
+00:06:08,060 --> 00:06:10,475
+the meaning of both
+regular expressions.
+
+116
+00:06:10,475 --> 00:06:14,090
+And are they the
+same sets or not?
+
+117
+00:06:14,090 --> 00:06:17,435
+There are again some
+interesting corner cases.
+
+118
+00:06:17,435 --> 00:06:20,540
+If I have a regular expression
+that can match a,
+
+119
+00:06:20,540 --> 00:06:23,450
+but followed by the regular
+expression which cannot match
+
+120
+00:06:23,450 --> 00:06:25,670
+anything, that is not equivalent
+
+121
+00:06:25,670 --> 00:06:29,255
+to the regular
+expression which can match a.
+
+122
+00:06:29,255 --> 00:06:31,340
+Again here you have
+to do the calculation
+
+123
+00:06:31,340 --> 00:06:32,915
+to convince you of that.
+
+124
+00:06:32,915 --> 00:06:35,840
+Similarly, if I have a regular
+expression which can
+
+125
+00:06:35,840 --> 00:06:38,750
+match a or the empty string,
+
+126
+00:06:38,750 --> 00:06:40,640
+this is not equivalent to
+
+127
+00:06:40,640 --> 00:06:43,355
+the regular expression
+which can just match a.
+
+128
+00:06:43,355 --> 00:06:46,925
+Here are some interesting
+ones with zeros and ones.
+
+129
+00:06:46,925 --> 00:06:48,860
+So if I have the regular expression
+
+130
+00:06:48,860 --> 00:06:50,975
+that can match the empty string,
+
+131
+00:06:50,975 --> 00:06:53,540
+this will be actually equivalent to
+
+132
+00:06:53,540 --> 00:06:56,435
+the regular expression
+which can match nothing,
+
+133
+00:06:56,435 --> 00:06:59,405
+but star of that.
+
+134
+00:06:59,405 --> 00:07:01,970
+Remember the star
+always introduces,
+
+135
+00:07:01,970 --> 00:07:04,370
+no matter what, the empty string.
+
+136
+00:07:04,370 --> 00:07:06,170
+So this regular expression can match
+
+137
+00:07:06,170 --> 00:07:08,930
+the empty string,
+just like the 1.
+
+138
+00:07:08,930 --> 00:07:12,125
+So these two expressions
+are not the same,
+
+139
+00:07:12,125 --> 00:07:14,210
+but they are equivalent.
+
+140
+00:07:14,210 --> 00:07:16,700
+And it doesn't matter
+whether you take
+
+141
+00:07:16,700 --> 00:07:20,090
+the empty string to  the star,
+
+142
+00:07:20,090 --> 00:07:23,855
+because if I can match 0 or
+more times the empty string,
+
+143
+00:07:23,855 --> 00:07:27,450
+that will be equivalent to 
+just the empty string.
+
+144
+00:07:27,520 --> 00:07:32,510
+Slightly similar to the
+third case is this one.
+
+145
+00:07:32,510 --> 00:07:35,720
+Zero star is not equivalent to 0
+
+146
+00:07:35,720 --> 00:07:40,025
+because that can match
+exactly the empty string.
+
+147
+00:07:40,025 --> 00:07:42,740
+This cannot match anything.
+
+148
+00:07:42,740 --> 00:07:44,839
+While the previous
+
+149
+00:07:44,839 --> 00:07:47,540
+equivalences are all very
+instructive to look at,
+
+150
+00:07:47,540 --> 00:07:49,910
+these are the
+equivalences we need
+
+151
+00:07:49,910 --> 00:07:52,685
+in our regular expression matchers.
+
+152
+00:07:52,685 --> 00:07:54,350
+And they are defined for
+
+153
+00:07:54,350 --> 00:07:58,160
+all regular expressions r.
+So r plus 0,
+
+154
+00:07:58,160 --> 00:08:00,470
+no matter what r looks
+like is equivalent
+
+155
+00:08:00,470 --> 00:08:02,945
+to just r. Similarly
+
+156
+00:08:02,945 --> 00:08:05,930
+0 plus r is also
+equivalent to r.
+
+157
+00:08:05,930 --> 00:08:08,525
+The order of this
+choice doesn't matter;
+
+158
+00:08:08,525 --> 00:08:11,495
+r followed by 1,
+
+159
+00:08:11,495 --> 00:08:14,030
+that's the sequence
+regular expression, is
+
+160
+00:08:14,030 --> 00:08:16,760
+equivalent to r. The
+sam, r followed by
+
+161
+00:08:16,760 --> 00:08:19,010
+r is equivalent to r.
+
+162
+00:08:19,010 --> 00:08:20,990
+And r followed by
+
+163
+00:08:20,990 --> 00:08:23,390
+the regular expression which
+cannot match anything,
+
+164
+00:08:23,390 --> 00:08:27,455
+is equivalent to just 0.
+
+165
+00:08:27,455 --> 00:08:30,740
+And 0 followed by r is also equivalent to 0.
+
+166
+00:08:30,740 --> 00:08:33,615
+And if you have the
+choice of r plus r,
+
+167
+00:08:33,615 --> 00:08:37,210
+that is also
+equivalent to just r.
+
+168
+00:08:37,210 --> 00:08:40,270
+What we're going to do
+with these equivalences is
+
+169
+00:08:40,270 --> 00:08:42,820
+that we regard them as
+simplification rules.
+
+170
+00:08:42,820 --> 00:08:43,930
+So whenever we see
+
+171
+00:08:43,930 --> 00:08:46,465
+this regular expression
+in our algorithm,
+
+172
+00:08:46,465 --> 00:08:48,580
+we will replace it by
+the right-hand side.
+
+173
+00:08:48,580 --> 00:08:51,715
+So we will orient
+these equivalences.
+
+174
+00:08:51,715 --> 00:08:53,650
+If we see, this regular expression,
+
+175
+00:08:53,650 --> 00:08:55,225
+we will replace it by that one.
+
+176
+00:08:55,225 --> 00:08:58,945
+And it will not matter
+because the left-hand sides
+
+177
+00:08:58,945 --> 00:09:01,255
+can match exactly
+the same strings
+
+178
+00:09:01,255 --> 00:09:03,475
+as the right-hand sides.
+
+179
+00:09:03,475 --> 00:09:06,370
+Here two quick examples.
+
+180
+00:09:06,370 --> 00:09:08,680
+The first one, let's
+assume you have
+
+181
+00:09:08,680 --> 00:09:10,270
+a regular expression r and
+
+182
+00:09:10,270 --> 00:09:11,905
+there is something
+in front of it.
+
+183
+00:09:11,905 --> 00:09:13,720
+This "something in front of it"
+
+184
+00:09:13,720 --> 00:09:15,870
+can be simplified as follows.
+
+185
+00:09:15,870 --> 00:09:20,000
+This 1 times b
+can be simplified to b.
+
+186
+00:09:20,000 --> 00:09:23,555
+Then this b plus 0 can
+be simplified to b.
+
+187
+00:09:23,555 --> 00:09:25,490
+And assuming that nothing can
+
+188
+00:09:25,490 --> 00:09:27,470
+be simplified inside this r,
+
+189
+00:09:27,470 --> 00:09:28,700
+then this would be
+
+190
+00:09:28,700 --> 00:09:33,050
+the simplified version
+of this regular expression.
+
+191
+00:09:33,050 --> 00:09:34,820
+And because the simplification
+
+192
+00:09:34,820 --> 00:09:36,965
+rules preserve what can be matched,
+
+193
+00:09:36,965 --> 00:09:39,080
+we can be sure that
+this regular expression
+
+194
+00:09:39,080 --> 00:09:41,255
+can match exactly the strings
+
+195
+00:09:41,255 --> 00:09:43,940
+this regular expression can match.
+
+196
+00:09:43,940 --> 00:09:45,740
+Here is the other example.
+
+197
+00:09:45,740 --> 00:09:49,550
+This has just a tiny change
+that this becomes here as 0.
+
+198
+00:09:49,550 --> 00:09:54,485
+Then 0 times b can be
+replaced by just 0.
+
+199
+00:09:54,485 --> 00:09:56,705
+Then they are actually two
+rules which could apply
+
+200
+00:09:56,705 --> 00:09:59,014
+either that we have
+the same choice
+
+201
+00:09:59,014 --> 00:10:01,160
+or we can just say something plus
+
+202
+00:10:01,160 --> 00:10:04,100
+0 will always go to something.
+
+203
+00:10:04,100 --> 00:10:06,245
+So we can simplify it to that.
+
+204
+00:10:06,245 --> 00:10:08,510
+And then we have
+0 times r again,
+
+205
+00:10:08,510 --> 00:10:10,460
+and that can be simplified to 0.
+
+206
+00:10:10,460 --> 00:10:12,080
+So actually what we find out with
+
+207
+00:10:12,080 --> 00:10:14,645
+this calculation is that
+this regular expression,
+
+208
+00:10:14,645 --> 00:10:16,835
+even if it looks
+quite complicated,
+
+209
+00:10:16,835 --> 00:10:19,295
+actually doesn't
+match any string.
+
+210
+00:10:19,295 --> 00:10:21,290
+Because it matches exactly
+
+211
+00:10:21,290 --> 00:10:23,420
+those string this regular
+expression can match.
+
+212
+00:10:23,420 --> 00:10:26,285
+And this reg expression
+cannot match any.
+
+213
+00:10:26,285 --> 00:10:29,930
+We need one more
+operation on languages.
+
+214
+00:10:29,930 --> 00:10:31,700
+I call this operation
+
+215
+00:10:31,700 --> 00:10:35,165
+the semantic derivative
+of a language.
+
+216
+00:10:35,165 --> 00:10:37,775
+This operation takes
+two arguments.
+
+217
+00:10:37,775 --> 00:10:40,670
+It takes a character
+which we take
+
+218
+00:10:40,670 --> 00:10:43,925
+the semantic derivative
+according to,
+
+219
+00:10:43,925 --> 00:10:45,980
+and it takes a language.
+
+220
+00:10:45,980 --> 00:10:49,579
+And what it does is it
+looks into this language
+
+221
+00:10:49,579 --> 00:10:51,365
+and looks for all the strings
+
+222
+00:10:51,365 --> 00:10:53,735
+that start with this character,
+
+223
+00:10:53,735 --> 00:10:56,405
+let's say c, and then
+
+224
+00:10:56,405 --> 00:11:00,920
+just strips off that c.
+So here's an example.
+
+225
+00:11:00,920 --> 00:11:02,975
+Suppose you have a language A,
+
+226
+00:11:02,975 --> 00:11:04,460
+with the strings
+
+227
+00:11:04,460 --> 00:11:06,965
+foo, bar and frak.
+
+228
+00:11:06,965 --> 00:11:10,835
+And you take the semantic
+derivative according to f,
+
+229
+00:11:10,835 --> 00:11:14,450
+then we discard all the
+strings which do not
+
+230
+00:11:14,450 --> 00:11:18,320
+start with an f. So
+bar will be discarded.
+
+231
+00:11:18,320 --> 00:11:22,850
+Foo and frac start with
+an f. So we keep them
+
+232
+00:11:22,850 --> 00:11:25,265
+but we strip off the first f.
+
+233
+00:11:25,265 --> 00:11:29,045
+So here we have only oo and rak.
+
+234
+00:11:29,045 --> 00:11:31,609
+If you take the
+semantic derivative
+
+235
+00:11:31,609 --> 00:11:33,830
+of that language according to b,
+
+236
+00:11:33,830 --> 00:11:37,190
+then we will discard foo and
+frak because they don't
+
+237
+00:11:37,190 --> 00:11:40,940
+start with b, and just keep bar.
+
+238
+00:11:40,940 --> 00:11:44,585
+But again, we have to
+strip off this first b.
+
+239
+00:11:44,585 --> 00:11:49,305
+And if you take the semantic
+derivative according to a,
+
+240
+00:11:49,305 --> 00:11:52,585
+then none of these
+strings start with a.
+
+241
+00:11:52,585 --> 00:11:56,800
+So this will be defined
+as just the empty set.
+
+242
+00:11:56,800 --> 00:11:59,695
+You can slightly 
+generalized this
+
+243
+00:11:59,695 --> 00:12:02,560
+semantic derivative
+to also strings.
+
+244
+00:12:02,560 --> 00:12:05,170
+So imagine you have
+a language A and you
+
+245
+00:12:05,170 --> 00:12:08,274
+build the semantic derivative
+according to a string s.
+
+246
+00:12:08,274 --> 00:12:10,990
+Then you look in
+this language and
+
+247
+00:12:10,990 --> 00:12:13,420
+you look for all the
+strings that start with
+
+248
+00:12:13,420 --> 00:12:19,555
+this s. But you strip
+off that s. Ok?
+
+249
+00:12:19,555 --> 00:12:23,830
+So if you have a string
+starting with the s,
+
+250
+00:12:23,830 --> 00:12:26,065
+then you keep that string,
+
+251
+00:12:26,065 --> 00:12:27,490
+but you keep only the rest...
+
+252
+00:12:27,490 --> 00:12:28,810
+what's coming after this s.
+
+253
+00:12:28,810 --> 00:12:32,010
+That is here indicated
+with this s'.
+
+254
+00:12:32,010 --> 00:12:35,330
+I also have this convention,
+this personal convention
+
+255
+00:12:35,330 --> 00:12:40,055
+I have to say, that everything
+which works on lists,
+
+256
+00:12:40,055 --> 00:12:42,185
+remember strings are
+lists of characters.
+
+257
+00:12:42,185 --> 00:12:46,655
+I just put there an 's'. So
+here's the one for characters.
+
+258
+00:12:46,655 --> 00:12:48,680
+I just call it Der with a capital
+
+259
+00:12:48,680 --> 00:12:51,740
+D. And I call that Ders,
+
+260
+00:12:51,740 --> 00:12:54,350
+because that works over strings.
+
+261
+00:12:54,350 --> 00:12:55,865
+And you can see how it would
+
+262
+00:12:55,865 --> 00:12:59,750
+be defined if you only take this
+as a primitive operation.
+
+263
+00:12:59,750 --> 00:13:01,340
+We would just need to iterate
+
+264
+00:13:01,340 --> 00:13:04,050
+that essentially for this Ders.
+
+265
+00:13:04,060 --> 00:13:07,970
+Ok, we now have all
+the theory in place.
+
+266
+00:13:07,970 --> 00:13:11,345
+And we can finally start
+implementing our algorithm.
+
+267
+00:13:11,345 --> 00:13:14,580
+That's when we'll do
+in the next video.