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COMMONWEALTH OF AUSTRALIA Copyright Regulations 1969 WARNING This material has been reproduced and communicated to you by or on behalf of Monash University pursuant to Part VB of the Copyright Act 1968 (the Act). The material in this communication may be subject to copyright under the Act. Any further reproduction or communication of this material by you may be the subject of copyright protection under the Act. Do not remove this notice.
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Converting Finite Automata into Regular Expressions CSE2303 Formal Methods I Lecture 7
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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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EVEN-EVEN ab + ba bb aa ab + ba
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Revision Know Kleene’s Theorem Be able to convert FAs into Regular Expressions Preparation Read –Text Book Chapter 8
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