Chapter 4
Chapter 4
Chapter 4
Language-Defining Symbol
• We now introduce the use of the Kleene star, applied not to a set, but
directly to the letter x and written as a superscript: x∗.
• This simple expression indicates some sequence of x’s (may be none
at all):
x∗ = Λ or x or x 2 or x 3 ...
= xn for some n = 0, 1, 2, 3,...
L 4 =language(x∗)
L = language(ab∗)
• From now on, for convenience, we will simply say some b’s to mean
some or no b’s. When we want to mean some positive number of
b’s, we will explicitly say so.
• Observe that
(ab)∗ ƒ= a∗b∗
because the language defined by the expression on the left contains
the word abab, whereas the language defined by the expression on
the right does not.
7
• If we want to define the language L 1 = {x, xx, xxx, ...} using the
language-defining symbol, we can write
L 1 =language(xx∗)
• Note that we can also define L 1 using the notation + (as an exponent)
introduced in Chapter 2:
L 1 =language(x+)
Example
ab∗a
is the set of all strings of a’s and b’s that have at least two letters, that
begin and end with a’s and that have only b’s in between (if any at
all).
• That is,
Plus Sign
x +y
10
Example
T = language((a+ c)b∗)
11
Example
• Consider a finite language L that contains all the strings of a’s and b’s
of length three exactly:
or for short,
L = language((a + b)3)
12
Example
language((a +b)∗)
• This is the set of all possible strings of letters from the alphabet
Σ = {a, b}, including the null string.
• This is powerful notation. For instance, we can describe all the words
that begin with first an a, followed by anything (i.e., as many choices
as we want of either a or b) as
a(a +b)∗
13
14
Definition
Example
(a + b)∗a(a + b)∗
• For example, the word abbaab can be considered to come from this
expression by 3 different choices:
16
Example (cont.)
• This language is the set of all words over the alphabet Σ = {a, b}
that have at least one a.
• The only words left out are those that have only b’s and the word Λ.
These left out words are exactly the language defined by the
expression b∗.
Example
• The language of all words that have at least two a’s can be defined by
the expression:
• Another expression that defines all the words with at least two a’s is
b∗ab∗a(a +b)∗
where by the equal sign we mean that these two expressions are
equivalent in the sense that they describe the same language.
18
Example
• If we want a language of all the words with exactly two a’s, we could
use the expression
b∗ab∗ab∗
• For example, to make the word aab, we let the first and second b∗
become Λ and the last become b.
19
Example
• The language of all words that have at least one a and at least one bis
somewhat trickier. If we write
• Note that the only words that are omitted by the first term
(a + b)∗a(a + b)∗b(a + b)∗ are the words of the form some b’s
followed by some a’s. They are defined by the expressionbb∗aa∗.
20
Example (cont.)
• We can add these specific exceptions. So, the language of all words
over the alphabet Σ = {a, b} that contain at least one a and at least
one bis defined by the expression:
Example
• In the above example, the language of all words that contain both an
a and a bis defined by the expression
• The only words that do not contain both an a and a bare the words of
all a’s, all b’s, or Λ.
defines all possible strings of a’s and b’s, including Λ (accounted for
in both a∗and b∗).
22
• Thus,
Example
– The second term, b∗a∗ describes all the words that do not contain the
substring ab (i.e., all a’s, all b’s, Λ, or some b’s followed by some a’s).
24
Example
• Let V be the language of all strings of a’s and b’s in which either the
strings are all b’s, or else an a followed by some b’s. Let V also
contain the word Λ. Hence,
b∗ + ab∗
which means that in front of the string of some b’s, we have either an
a or nothing.
25
Example (cont.)
• Hence,
(Λ + a)b∗ = b∗ + ab∗
• Since b∗ = Λb∗, we have
26
Definition: ProductSet
If S and T are sets of strings of letters (whether they are finite or infinite
sets), we define the product set of strings of letters to be
ST = {all combinations of a string from S concatenated with a string
from T in that order}
27
Example
(a + aa + aaa)(bb + bbb)
= abb + abbb + aabb + aabbb + aaabb + aaabbb
28
Examples
L{Λ} = {Λ}L = L
rΛ = Λr = r
29
Example
30
Definition
The following rules define the language associated with any regular
expression:
language(r1r2) = L 1 L 2
32
Definition (cont.)
• Rule 2 (cont.):
(ii) The regular expression r 1 + r 2 is associated with the language
formed by the union of L 1 and L 2 :
language(r1 + r2) = L 1 + L 2
(iiii) The language associated with the regular expression (r1)∗ is L∗1,
the Kleene closure of the set L 1 as a set of words:
language(r1∗) = L∗1
33
Remarks
• We shall show in the next slide that every finite language can be
defined by a regular expression.
34
Theorem 5
36
Example
• This is the set of strings of a’s and b’s that at some point contain a
double letter.
38
Example (cont.)
• The expression (ab)∗ covers all of these except those that begin with
bor end with a. Adding these choices gives us the expression:
(Λ + b)(ab)∗(Λ + a)
• Combining the two expressions gives us the one that defines the set
of all strings
Example
• Let us break up the middle plus sign into two cases: either that
middle factor contributes an a or else it contributes a Λ. Therefore,
40
Example (cont.)
• The first term clearly represents all words that have at least three a’s
in them. Let us analyze the second term.
• Observe that
(a + b)∗Λ(a + b)∗ = (a + b)∗
• This reduces the second term to
• Hence, the language associates with E is the union of all words that
have three or more a’s with all words that have two or more a’s.
41
Example (cont.)
• But all words with three or more a’s are themselves already words
with two or more a’s. So, the whole language is just the second set
alone, ie.
E = (a + b)∗a(a + b)∗a(a + b)∗
42
Examples
• Note that
(a + b∗)∗ = (a + b)∗
since the internal ∗adds nothing to the language.
• However,
(aa + ab∗)∗ ƒ= (aa + ab)∗
since the language on the left includes the word abbabb, whereas the
language on the right does not. (The language on the right cannot
contain any word with a double b.)
43
Example
• Since both the single letter a and the single letter bare words of the
form a∗b∗, this language contains all strings of a’s and b’s. That is,
(a∗b∗)∗ = (a + b)∗
• This equation gives a big doubt on the possibility of finding a set of
algebraic rules to reduce one regular expression to another equivalent
one.
44
Introducing EVEN-EVEN
• This expression represents all the words that are made up of syllables
of three types:
type1 = aa type2 = bb
type3 = (ab + ba)(aa + bb)∗(ab + ba)
• Every word of the language defined by E contains an even number
of a’s and an even number of b’s.
• All strings with an even number of a’s and an even number of b’s
belong to the language defined by E.
45
We want to determine whether a long string of a’s and b’s has the property
that the number of a’s is even and the number of b’s is even.
• Algorithm 1: Keep two binary flags, the a-flag and the b-flag. Every
time an a is read, the a-flag is reversed (0 to 1, or 1 to 0); and every
time a bis read, the b-flag is reversed. We start both flags at 0 and
check to be sure they are both 0 at the end.
46
Example
(aa)(ab)(bb)(ba)(ab)(bb)(bb)(bb)(ab)(ab)(bb)(ba)(aa)