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Converting Finite Automata into Regular Expressions CS402
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Kleene’s Theorem Regular Expression Finite Automaton NFA- GTG TGNFA
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First Three Proofs Every Finite Automaton is a NFA Every NFA is a Transition Graph. Every Transition Graph is a Generalised Transition Graph.
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How to convert a Generalised Transition Graph into a Regular Expression
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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
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Make a unique Start State - -...
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Make a unique Start State - -... -
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Make a unique Start State... -
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Make a unique Final State + +...
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Make a unique Final State + +... +
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Make a unique Final State... +
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Eliminate multiple loops... R1R1 R2R2 R3R3 R 1 + R 2 + R 3
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Eliminate multiple edges... R1R1 R2R2 R3R3 R 1 + R 2 + R 3...
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State Elimination
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... R1R1 R2R2 12 R 1 R 2 1 2
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... R 1 (S)* R 2 1 2... R1R1 R2R2 12 S
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R1R1 R2R2 13 S 2 4 R3R3 R4R4 R 1 (S )* R 3 1 3 4 2... R 1 (S )* R 2 R 1 (S )* R 4
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... R1R1 R2R2 13 S 2 R3R3 R4R4 R 1 (S )* R 3 1 3 2... R 1 (S )* R 2 R 1 (S )* R 4
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... R1R1 R2R2 1 3 S 2 4 R3R3 R4R4 5 R5R5 R 1 (S )* R 3 1 3 4 2... R 1 (S )* R 4... 5 R 1 (S )* R 2 R 5 (S )* R 2 R 5 (S )* R 4 R 5 (S )* R 3
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... R1R1 R2R2 1 3 S 2 4 R3R3 R4R4 5 R5R5 1 3 2 4 5
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R1R1 R2R2 1 3 S 2 4 R3R3 R4R4 5 R5R5 1 3 2 4 5
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Revision Know Kleene’s Theorem Be able to convert FAs into Regular Expressions Preparation Read –Lectures 11 and 12, Practice on Example
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