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

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

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

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

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

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

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”

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

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

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

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

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