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```									Regular Languages, Regular
Operations
September 11, 2001

1
Agenda
Today
   Regular languages
 Finite languages are regular
   Regular operations on languages
 Union ()
 Concatenation ()
 Kleene star (*)
For next time:
   Read 1.3 and handout on minimization
Thursday, 9/20 (revised ): HW1 collected

2
Definition of Regular
Language
Recall the definition of a regular language:
DEF: The language accepted by an FA M is
the set of all strings which are accepted by M
and is denoted by L (M).
Would like to understand what types of
languages are regular. Languages of this
type are amenable to super-fast recognition
of their elements
Would be nice to know for example, which of
the following are regular:
3
Language Examples
   Unary prime numbers:
{ 11, 111, 11111, 1111111, 11111111111, … }
= {12, 13, 15, 17, 111, 113, … }
= { 1p | p is a prime number }
   Unary squares:
{, 1, 14, 19, 116, 125, 136, … }
= { 1n | n is a perfect square }
   Palindromic bit strings:
{, 0, 1, 00, 11, 000, 010, 101, 111, …}
= {x  {0,1}* | x = xR } o
Will explore whether or not these are regular in
future.
4
Finite Languages
All the previous examples had the following
property in common: infinite cardinality
NOTE: The strings which made up the
language were finite (as they always will be
in this course); however, the collection of
such strings was infinite.
Before looking at infinite languages, should
definitely look at finite languages.

5
Languages of Cardinality 1
Q: Is the singleton language containing
one string regular? For example, is
{ banana }
regular?

6
Languages of Cardinality 1
A: Yes.

Q: What’s, wrong with this example?

7
Languages of Cardinality 1
A: Nothing, really. This an example of a
nondeterministic FA. This turns out to be the
most concise way to encapsulate the
language { banana }

But we will deal with nondeterminism in coming
lectures. So:
Q: Is there a way of fixing this and making it
deterministic?
8
Languages of Cardinality 1
A: Yes, just add a fail state q7; I.e., put a
state that sucks in all strings different from
“banana” for all eternity –unless they happen
to be the “banana” prefixes {, b, ba, ban,
bana, banan}.

9
Two Strings
Q: How about two strings? For example
{ banana, nab } ?

11
Two Strings
A: Just add another route:

12
Arbitrary Finite Number of
Strings
Q1: How about more? For example
{ banana, nab, ban, babba } ?

Q2: Or less (the empty set):
Ø = {} ?

13
Arbitrary Finite Number of
A1:          Strings

14
Arbitrary Finite Number of
Strings: Empty Language
A2: Build a 1-state automaton whose
accept states set F is empty!

15
Arbitrary Finite Number of
Strings
THM: All finite languages are regular.
Proof : Can always construct a tree whose
leaves are word-ending. In our example the
tree is:                  n          b
a           a
n       b           b
a               b
n                   a
a

Now make word endings into accept states, add
a fail sink-state and add links to the fail state
to finish the construction.              •
16
Infinite Cardinality
Q: Are all regular languages finite?

17
Infinite Cardinality
A: No! Many infinite languages are regular.
Common Mistake 1: The strings of regular
languages are finite, therefore the regular
languages must be finite.
Common Mistake 2: Regular languages are –by
definition– accepted by finite automata,
therefore regular languages are finite.
Q: Give an example of a infinite but regular
language.

18
Infinite Cardinality
   bit strings with an even number of b’s

   Simplest example is S*

many, many more
Home exercise: think of a criterion for non-
finiteness

19
Regular Operations
You may have come across the regular
operations when doing advanced searches
utilizing programs such as emacs, egrep,
perl, python, etc. There are three basic
operations we will work with:
1. Union
2. Concatenation
3. Kleene-star
And a fourth definable in terms of the previous:
4. Kleene-plus

20
Regular Operations –
Summarizing Table
Operation Symbol UNIX version       Meaning
match one of
Union                 |         the patterns
implicit in   match patterns
Concatenation   
UNIX         in sequence
Kleene-                          Match pattern 0
*       *
star                            or more times
Kleene-                          Match pattern 1
+       +
plus                            or more times

21
Regular operations - Union
UNIX: to search for all lines containing
vowels in a text one could use the
command
egrep -i `a|e|i|o|u’
Here the pattern “vowel ” is matched by
any line containing one of a, e, i, o or u.
Q: What is a string pattern?

22
String Patterns
A: A good way to define a pattern is as a
set of strings, i.e. a language. The
language for a given pattern is the set
of all strings satisfying the predicate of
the pattern.
EG: vowel-pattern =
{ the set of strings which
contain at least one of: a e i o u }

23
UNIX patterns vs.
Computability patterns
In UNIX, a pattern is implicitly assumed
to occur as a substring of the matched
strings.
In our course, however, a pattern needs
to specify the whole string, and not just
a substring.

24
Regular operations - Union
Computability: union is exactly what we
expect. If you have patterns
A = {aardvark}, B = {bobcat},
C = {chimpanzee}
union the patterns together to get
AB C = {aardvark, bobcat,
chimpanzee}

25
Regular operations -
Concatenation
UNIX: to search for all consecutive
double occurrences of vowels, use:
egrep -i `(a|e|i|o|u)(a|e|i|o|u)’
Here the pattern “vowel ” has been
repeated. Parentheses have been
introduced to specify where exactly in
the pattern the concatenation is
occurring.
26
Regular operations -
Concatenation
Computability. Consider the previous
result:
L = {aardvark, bobcat, chimpanzee}

Q: What language results when we
concatenate L with itself obtaining
LL ?

27
Regular operations -
Concatenation
A: LL =
{aardvark, bobcat, chimpanzee}{aardvark, bobcat, chimpanzee}

=
{aardvarkaardvark, aardvarkbobcat, aardvarkchimpanzee,
bobcataardvark, bobcatbobcat, bobcatchimpanzee,
chimpanzeeaardvark, chimpanzeebobcat, chimpanzeechimpanzee}

Q1: What is L{} ?
Q2: What is LØ ?
28
Algebra of Languages
A1: L{} = L. In general, {} is the identity
in the “algebra” of languages. I.e., if we
think of concatenation as being like
multiplication, {} acts like the number 1.
A2: LØ = Ø. Opposite to {}, Ø acts like the
number zero obliterating everything it is
concatenated with.
Note: We can carry on the analogy between
numbers and languages. Addition becomes
union, multiplication becomes concatenation.
This forms a so-called “algebra”.
29
Regular operations – Kleene-*
UNIX: search for lines consisting purely of
vowels (including the empty line):
egrep -i `^(a|e|i|o|u)*\$’
NOTE: ^ and \$ are special symbols in UNIX
regular expressions which respectively anchor
the pattern at the beginning and end of a
line. The trick above can be used to convert
any Computability regular expression into an
equivalent UNIX form.

30
Regular operations – Kleene-*
Computability: Suppose we have a
language
B = { ba, na }

Q: What is the language B * ?

31
Regular operations – Kleene-*
A:
B * = { ba, na }*=
{ ,
ba, na,
baba, bana, naba, nana,
bababa, babana, banaba, banana,
nababa, nabana, nanaba, nanana,
babababa, bababana, … }

32
Regular operations – Kleene-+
Kleene-+ is just like Kleene-* except that the
pattern is forced to occur at least once.
UNIX: search for lines consisting purely of
vowels (not including the empty line):
egrep -i `^(a|e|i|o|u)+\$’
Computability: B+ = { ba, na }+=
{ ba, na,
baba, bana, naba, nana,
bababa, babana, banaba, banana,
nababa, nabana, nanaba, nanana,
babababa, bababana, … }
33
Generating the Regular
Languages
The real reason that regular languages are
called regular is the following:
THM: The regular languages are all those
languages which can be generated starting
from the finite languages by applying the
regular operations.
This will be proved in the coming lectures.
Q: Can we start with even more basic
languages than arbitrary finite languages?

34
Generating the Regular
Languages
A: Yes. We can start with languages consisting
of single strings which are themselves just a
single character. These are the “atomic”
regular languages.
EG: To generate the finite language
L = { banana, nab }
we can start with the atomic languages
A = {a}, B = {b}, N = {n}.
Then we can express L as:
L = (B A N A N A)  (N A B )
35
Blackboard Exercises
Express the DFA patterns from the
previous board-exercises using regular
operations in both UNIX-style and
Computability-style.

36

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