Conversions

1
Conversions

• Regular Expression to FA
• FA to Regular Expression

2
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(?)
3
Regular Languages
Definition. A language is regular if there exists
a regular expression that describes all its
strings
4
Examples
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 ?
5
Example
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
6
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)
7
Algorithm A (from text)
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

8
Recall Construction on NFA-ls
l
l
M(R1)
l
l
M(R)
M(R2)
l
l
M(R)
M(R1 U R2)
l
l
l
M(R1)
M(R2)
M(R1 R2)
9
Exercise

• Obtain a finite automaton accepting a regular
  expression
• One or two in class
• How about
• (a ? ab) a (ba)
• (a ? b) aba
• Make up several!
• Use the union, concatenation, and operations

10
Algorithm B (from text)
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
• If it doesnt, can this be taken care of ?
• There are no transitions directed to the initial
  state
• If there is one, can this be taken care of ?
• There are no transitions starting at the
  favorable state

Definition. An expression diagram is a labeled
directed graph like a FSA, but with transitions
labeled by regular expressions. as opposed to
single elements of ?.
11
Algorithm B States to RE Unions
For every pair of nodes such that there is more
than one transition from one to the other one
expr1
expr2

exprn
expr1 ? expr2 ? ? exprn
12
Algorithm B States to RE Kleene
For every pair of nodes such that there is an
intermediate node connecting them
expr1
expr3
expr2
13
Simple Example
14
Delete node q1
b
q1
a
ab
15
Delete node q2
(abab)
a
ab
q2
b
16
Detailed Example
17
Delete node q1
18
Delete node q2
abab
ab
19
Finally
abab
(abab)a ((b ? bb) (abab)a)