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Welcome back.

Remember this slide.

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This slide said, What is

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all wreck expression Metro

actually supposed to do?

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It will take two arguments and

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reg expression R and a string

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S. And it's supposed to say yes,

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the wreck expression matches

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the string if and only

if the string is in

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the language of R.

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And if the string is not

in the language of our,

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then our algorithm has to say no.

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And we can't use

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this specification

directly because remember,

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this l Sometimes

produces infinite sets.

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And so we can test whether a

string is an infinite set,

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at least not easily.

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And so what we have to do

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instead is we have

to be a bit more

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clever and implement

some operations

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on Rekha expressions instead.

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Because Weka expressions

are always finite trees.

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I should say the

person who has been

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clever is called brush-off ski.

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It's his, I've written,

I'm introducing here.

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And his algorithm consists

of two functions.

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One is called

nullable and it takes

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a regular expression as argument.

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And the idea is that

this function nullable.

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Testing whether

the reg expression

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can match the empty string.

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So 0 cannot match any string.

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So it cannot match the

empty string either.

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So that's defined as false.

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This reg expression,

the whole purpose

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is that it can match

the empty string.

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So it's defined as true.

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If a reg expression

can match a character,

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then it cannot match

the empty string.

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So that is defined as false.

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If an alternative can

match the empty string,

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then either or one can

match the empty string.

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Or R2 can match the empty string.

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So either nullable

of R1 has to hold,

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or nullable of R2 has to hold.

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In this sequence, it's

the other way around.

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If this reg expression can

match the empty string,

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then R1 has to be able to

match the empty string,

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and R2 has to be able to

match the empty string.

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So here it's just the opposite.

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In one case it is o and

the other case it's end.

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In the store record

expression can match

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always the empty string.

So that is true.

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So this is a simple

recursive function

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and should not be too

difficult to implement.

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Okay, this nullable function

that is easy-peasy.

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The second function, however,

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is a bit more involved and

that's just to be expected.

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Remember people working in

this area already for decades.

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If they have some problems

with runtime, for example,

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we can't expect that as

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simple fix will solve all

the problems in the world.

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So I admit the second

function is a bit more

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complicated and make sure

that you understand it.

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And it also just

chose this preserves

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the is a very clever guy.

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Actually, I have to say he was

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a clever guy because

I think he either

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died last year or

the year before.

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And he came up with this

algorithm already in 1964.

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It somehow got lost,

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but has been rediscovered

in the last ten years.

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So the idea of the second

function is the following.

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Imagine you have a

reexpression and it can

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match a string of the

form C followed by as.

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So the C is the first

character of that string.

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So I mentioned that can

match this kind of string.

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The question is now,

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what would a wrecker

expression look

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like that can match chest

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s. You probably remember

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that from the semantic

that every relative,

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there was also

something of chopping

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off the first character.

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Here it's the same.

If a reg expression

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can match a string

starting with a C,

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we're looking for a wreck

expression which can match

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the rest of the string where

the c has been chopped off.

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And this operation will be called

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the derivative of a

wreck expression.

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And it will also take

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a character as argument

and the rec expression.

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And in contrast to

the semantic records,

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semantic derivative, which works

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over languages or

sets of strings.

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This derivative works

over regular expressions.

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Okay, here's this function.

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It's defined recursively over

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the structure of rec expressions.

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The first argument

is the character,

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and the second one is

a wreck expression.

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And remember the idea

is we're looking for

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a wreck expression that

can match everything.

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This reg expression

as argument can match

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except for the C. So now let's

look at this first case.

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So this reg expression

cannot match any string.

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So it certainly cannot

match any string starting

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a C. So we have to look

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for and reg expression which

can not much anything.

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Well, that's the purpose

of this record expression,

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so we define it as 0.

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In the next case,

this reg expression

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can match the empty string,

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but it cannot match

any string that starts

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with C. So also in this case,