1. CSCI 2670 Introduction to Theory of Computing
September 7, 2005

2. Agenda Yesterday Today Tomorrow
Proved every RE has a corresponding DFA
I.e., every RE describes a regular language
Showed how to convert GNFA to RE
Today
Prove every DFA has a corresponding RE
Example of conversion from GNFA to RE
Tomorrow
Proving a language is not regular

3. GNFA’s A GNFA is an NFA with the following properties:
The start state has transition arrows going to every other state, but no arrows coming in from any other state
There is exactly one accept state and there is an arrow from every other state to this state, but no arrows to any other state from the accept state
The start state is not the accept state

4. GNFA’s (continued)
Except for the start and accept states, one arrow goes from every state to every other state and also from each state to itself
Instead of being labeled with symbols from the alphabet, transitions are labeled with regular expressions

5. Converting a GNFA to a RE
If the GNFA has two states, then the label connecting the states is the RE
Otherwise, remove one state at a time without changing the language accepted by the machine until the GNFA has two states

6. Removing one state from a GNFAq2 q1 q3
a12a13a33*a32
q1’
q2’
These two portions of GNFA’s recognize the same strings

7. Accounting for loops a22a23a33*a32 a11a13a33*a31 a12a13a33*a32 q1’