Regular Languages, Regular Operations

1
Regular Languages, Regular Operations

• September 11, 2001

2
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

3
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

4
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.

5
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.

6
Languages of Cardinality 1

• Q Is the singleton language containing one
  string regular? For example, is
• banana
• regular?

7
Languages of Cardinality 1

• A Yes.
• Q Whats, wrong with this example?

8
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?

9
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.

10
ABCEZ Goes Bananas

• Show how ABCEZ works on this example.

11
Two Strings

• Q How about two strings? For example
• banana, nab ?

12
Two Strings

• A Just add another route

13
Arbitrary Finite Number of Strings

• Q1 How about more? For example
• banana, nab, ban, babba ?
• Q2 Or less (the empty set)
• Ø ?

14
Arbitrary Finite Number of Strings

• A1

15
Arbitrary Finite Number of Strings Empty Language

• A2 Build a 1-state automaton whose accept
  states set F is empty!

16
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
• Now make word endings into accept states, add a
  fail sink-state and add links to the fail state
  to finish the construction.

17
Infinite Cardinality

• Q Are all regular languages finite?

18
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.

19
Infinite Cardinality

• bit strings with an even number of bs
• Simplest example is S
• many, many more
• Home exercise think of a criterion for
  non-finiteness

20
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
• Union
• Concatenation
• Kleene-star
• And a fourth definable in terms of the previous
• Kleene-plus

21
Regular Operations Summarizing Table
22
Regular operations - Union

• UNIX to search for all lines containing vowels
  in a text one could use the command
• egrep -i aeiou
• Here the pattern vowel is matched by any line
  containing one of a, e, i, o or u.
• Q What is a string pattern?

23
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

24
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.

25
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

26
Regular operations - Concatenation

• UNIX to search for all consecutive double
  occurrences of vowels, use
• egrep -i (aeiou)(aeiou)
• Here the pattern vowel has been repeated.
  Parentheses have been introduced to specify where
  exactly in the pattern the concatenation is
  occurring.

27
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 ?

28
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?e ?
• Q2 What is L?Ø ?

29
Algebra of Languages

• A1 L?e L. In general, e is the identity
  in the algebra of languages. I.e., if we think
  of concatenation as being like multiplication,
  e acts like the number 1.
• A2 L?Ø Ø. Opposite to e, 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.

30
Regular operations Kleene-

• UNIX search for lines consisting purely of
  vowels (including the empty line)
• egrep -i (aeiou)
• 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.

31
Regular operations Kleene-

• Computability Suppose we have a language
• B ba, na
• Q What is the language B ?

32
Regular operations Kleene-

• A
• B ba, na
• e,
• ba, na,
• baba, bana, naba, nana,
• bababa, babana, banaba, banana,
• nababa, nabana, nanaba, nanana,
• babababa, bababana,

33
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 (aeiou)
• Computability B ba, na
• ba, na,
• baba, bana, naba, nana,
• bababa, babana, banaba, banana,
• nababa, nabana, nanaba, nanana,
• babababa, bababana,

34
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?

35
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 )

36
Blackboard Exercises

• Express the DFA patterns from the previous
  board-exercises using regular operations in both
  UNIX-style and Computability-style.