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Dept. of Computer Science & IT, FUUAST Automata Theory 2 Automata Theory VII

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Dept. of Computer Science & IT, FUUAST Automata Theory 3 Automata Theory VII Simplification of CFG Elimination of 1)Useless Symbols or Productions: A variable A is useful if it is generating as : S * xAy * w. A V and x,y in (V T)* otherwise useless. If a variable is not reachable it is useless as B is useless in the following rules. B is not reachable from starting symbol S. S A A aA| B bA 2) -productions (Empty Productions) A , it is called “NULLABLE” 3)Unit Productions A B

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Dept. of Computer Science & IT, FUUAST Automata Theory 4 Automata Theory VII Examples: 1)Given a CFG as: G = ({S, A, B, C, E},{a,b,c}, P, S) P is defined as: S AB A a B b B C E c | New Grammar Ĝ is: Ĝ = ({S, A, B}, {a,b}, P’, S), P’ is given as: S AB A a B b

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Dept. of Computer Science & IT, FUUAST Automata Theory 5 Automata Theory VII Examples: 2)Given a CFG as: P is defined as: S aS | A | C A a B aa C aCb Production Rules P’ for the new grammar Ĝ are: S aS | A A a Draw a dependency graph to eliminate B. Useless symbols eliminated

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Dept. of Computer Science & IT, FUUAST Automata Theory 6 Automata Theory VII Examples: 3) Given a CFG as: P is defined as: S AB A aAA | B bBB | Production Rules P’ for the new grammar Ĝ are: S AB | A | B A aAA | aA | a B bBB | bB | b - Productions are eliminated

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Dept. of Computer Science & IT, FUUAST Automata Theory 7 Automata Theory VII Examples: 4) Given a CFG as: G = ({S, A, B, C, D, E},{a,b}, P, S) P is defined as: S AB A a B C | b C D D E E a New Grammar Ĝ is: Ĝ = ({S, A, B, C, D}, {a,b}, P’, S), P’ is given as: S AB A a B b B a C a D a Eliminating unit production

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Dept. of Computer Science & IT, FUUAST Automata Theory 8 Automata Theory VII Every nonempty CFL without has a grammar G in which all productions are in one of two simpler forms, either: 1)A BC, where A, B and C are each variables, or 2)A a G has no useless symbols Chomsky Normal Form (CNF) “ Every context-free language can be generated by a grammar with no useless symbols and no unit productions”

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Dept. of Computer Science & IT, FUUAST Automata Theory 9 Automata Theory VII Examples: 1) A grammar with production P is given S aAbB A aA | a B bB | b For S aAbB, we have S B a AB b B B a a B b b For A aA, we have A B a A For B bB, we have B B b B New Grammar G 1 is given by G 1 = ({S, A, B, B a, B b }, {a,b},P’,S) P’ has productions S B a AB b B A B a A B B b B B a a B b b A a B b Still not CNF change S B a AB b B

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Dept. of Computer Science & IT, FUUAST Automata Theory 10 Automata Theory VII Examples: CNF Grammar G1 is given by G 1 = ({S, A, B, B a, B b, D1, D2}, {a,b},P’,S) P’ has productions S B a D 1 D 1 AD 2 D 2 B b B A B a A B B b B B a a B b b A a B b Now making S B a AB b B into proper form we assume new variables D 1 and D 2 and the productions S B a D 1 D 1 AD 2 D 2 B b B

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Dept. of Computer Science & IT, FUUAST Automata Theory 11 Automata Theory VII Examples: 2) A grammar with production P is given S ABa A aab B Ac For S ABa, we have S ABB a B a a For A aab we have A B a B a B b B b b For A Ac, we have B AB c B c c For B bB, we have B B b B New Grammar G 1 is given by with P’ G 1 = ({S, A, B, B a, B b,B c }, {a,b,c},P’,S) P’ has productions S ABB a A B a B a B b B AB c B a a B b b B c c S ABB a and A B a B a B b are not in proper form

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Dept. of Computer Science & IT, FUUAST Automata Theory 12 Automata Theory VII Examples: CNF Grammar G1 is given by G 1 = ({S, A, B, B a, B b, D1, D2}, {a,b,c},P’,S) P’ has productions S A D 1 D 1 BB a A B a D 2 D 2 B a B b B AB c B a a B b b B c c Now making S ABB a and A B a B a B b into proper form we assume new variables D 1 and D 2 and the productions S A D 1 D 1 BB a A B a D 2 D 2 B a B b

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Dept. of Computer Science & IT, FUUAST Automata Theory 13 Automata Theory VII End of Chapter 7

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