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Converting Finite Automata into Regular Expressions CSE2303 Formal Methods I Lecture 7

Kleene’s Theorem Regular Expression Finite Automaton NFA-  GTG TGNFA

First Three Proofs Every Finite Automaton is a NFA Every NFA is a Transition Graph. Every Transition Graph is a Generalised Transition Graph.

How to convert a Generalised Transition Graph into a Regular Expression

Make a unique Start State with no input transitions Are there any Final States ? Make a unique Final State with no output transitions Eliminate multiple loops Eliminate multiple edges Is the number of states > 2? Eliminate a state which is NOT the Start state or the Final state Is the GTG connected? Write  Write the label NO YES

Make a unique Start State

Make a unique Start State  

Make a unique Start State... -  

Make a unique Final State

Make a unique Final State  

Make a unique Final State... +  

Eliminate multiple loops... R1R1 R2R2 R3R3 R 1 + R 2 + R 3

Eliminate multiple edges... R1R1 R2R2 R3R3 R 1 + R 2 + R 3...

State Elimination

... R1R1 R2R2 12 R 1 R 2 1 2

... R 1 (S)* R R1R1 R2R2 12 S

R1R1 R2R2 13 S 2 4 R3R3 R4R4 R 1 (S )* R R 1 (S )* R 2 R 1 (S )* R 4

... R1R1 R2R2 13 S 2 R3R3 R4R4 R 1 (S )* R R 1 (S )* R 2 R 1 (S )* R 4

... R1R1 R2R2 1 3 S 2 4 R3R3 R4R4 5 R5R5 R 1 (S )* R R 1 (S )* R R 1 (S )* R 2 R 5 (S )* R 2 R 5 (S )* R 4 R 5 (S )* R 3

... R1R1 R2R2 1 3 S 2 4 R3R3 R4R4 5 R5R

R1R1 R2R2 1 3 S 2 4 R3R3 R4R4 5 R5R

EVEN-EVEN ab + ba bb  aa ab + ba

Revision Know Kleene’s Theorem Be able to convert FAs into Regular Expressions Preparation Read –Text Book Chapter 8