Recap: Transformation NFA ? DFA

1
Recap Transformation NFA ? DFA
p1
s
p2
s1
?
gt
pm
sn
2
Closure Under Set Operations
Theorem. Let L and M two regular languages, then
the following languages are also regular

• L ? M
• LC
• L ? M
• L M
• LM
• L

3
Regular Expressions
A regular expression is defined inductively over
the alphabet ? ? (, ), e, ?, ?, as
follows

• ?, e, and each character in ? are regular
  expressions (step 0)
• If ? and ? are regular expressions then
• (? ? ?)
• (??)
• ?
• are also regular expressions (inductive step)
• No other string over the alphabet ? ? (, ), e,
  ?, ?, is a regular expression
• Exceptions ?, ?, ?, ?k (for a fixed k gt 0)

4
Language Representing Regular Expressions

• We define a mapping from a regular expression ?
  to a language L(?) as follows
• Step 0
• L(?)
• L(e)
• L(?) for each ? ? ?
• Inductive step
• L((? ? ?))
• L((??))
• L(?)

e
?
L(?) ? L(?)
L(?)L(?)
L(?)
5
Example (1)
Example 1. Find L((ab)a)
L(aba)
L(a)L(b)L(a) w w is of the form abna
with n 0, 1, 2,
Example 2. Find L((a(a ?b))). L(a(a
?b))
L(a)(L(a) ? L(b)) aw w is a word in ?
6
Example (2)
Find the regular expression for the language L in
the alphabet a,b such that the words in L
contains the substring aaa
Find the regular expression for the language L in
the alphabet a,b such that the words in L
contains the substring aaa or bbb
7
Main Theorem About Finite Automata (Kleene)
(1) Given a finite automata A, there is a regular
expression expr such that L(A) L(expr)
(2) Given a regular expression expr, there is a
finite automata A such that L(A) L(expr)
8
Theorem 1.54
Input a regular expression expr Output a
finite automaton accepting L(expr)

• Convert any step-0 element (any of the characters
  in ?, ? or e) occurring in expr to a finite
  automaton accepting this element
• Apply the theorem about closure under set
  operations to every
• Union
• Concatenation
• Kleene star

9
Example
Obtain a finite automaton accepting the regular
expression ((a ?ab)ba).
The impact of this Algorithm is a dream (or a
nightmare?) of programmers given the
specification of a language there is an algorithm
that recognizes it automatically. Example Yak
and Lex tools in Unix.
10
Lemma 1.60
Input a finite automaton A Output a
regular expression expr such that L(expr) L(A)

• Assumptions about the automaton A
• A has a single favorable state
• There are no transitions directed to the initial
  state
• There are no transitions starting at the
  favorable state
• Nodes will be labeled 1, 2, , n, where 1 is the
  initial state and n is the favorable state

Definition. An generalized NFA is am NFA in which
the arrows are labeled by regular expressions.
(Unlike regular NFAs in which transitions are
only labeled by characters in ?)
11
Construction (1)
for every pair of nodes such that there is more
than one transition from one to the other one
12
Construction (2)
for every pair of nodes such that there is an
intermediate node connecting them
13
Example
a
a
r
r
s
q
q
gt
b
b
b
b
14
Homework Friday Sep. 21

• 1.12
• 1.21 (a)
• 1.28 (c)
• 1.43 (either picture and an accompanying
  explanation or formal proof are fine)
• 1.60 (either state diagram or formal definition
  are fine)