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Formal Grammars

Formal Grammars. Denning, Sections 3.3 to 3.6. Formal Grammar, Defined. A formal grammar G is a four-tuple G = (N,T,P, ), where N is a finite nonempty set of nonterminal symbols T is a finite nonempty set of terminal symbols disjoint from N P is a finite nonempty set of productions

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Formal Grammars

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  1. Formal Grammars Denning, Sections 3.3 to 3.6

  2. Formal Grammar, Defined • A formal grammar G is a four-tupleG = (N,T,P,), where • N is a finite nonempty set of nonterminal symbols • T is a finite nonempty set of terminal symbols disjoint from N • P is a finite nonempty set of productions •  is the sentential symbol or start symbol not in N or T

  3. Formal Grammar, cont. • Each production in P has the formA  where , , and  are in (N  T)* and may possibly be empty, and A=, or A is in N. • Read: If A has  to its left and  to its right, then A can be replaced by , in that context. • If  and  are empty, then the production is A  , and A can be replaced by , context-free.

  4. Example • G = (N,T,P,), whereN = { A }T = { 0, 1 }P = {   ,   A, A  0A1, A  01 }

  5. Some Terminologies • Formal grammar – phrase structure grammar • Generating a sentence – successive rewriting of sentential forms through the use of productions of the grammar, starting with . • Derivation of a sentence – the sequence of sentential forms needed to generate the sentence from .

  6. Sentential Form • Let G be a formal grammar. A string of symbols in (N  T)*  {} is called a sentential form. • Examples00A11 is a sentential form0A100A is another sentential form is a sentential form is a sentential form

  7. Immediately Derived • Let G be a formal grammar. If    is a production of G, and  =  and ’ =  are sentential forms, then we say that ’ is immediately derived from  by the rule   , and we write   . • Example: From 0A1 we can immediately derive 00A11 by the rule A  0A1. We write 0A1  00A11.

  8. Derivation • If w1, w2, w3, … , wn is a sequence of sentential forms such thatw1  w2  w3  …  wnthen we say that wn is derivable from w1, and we write w1 * wn. The above sequence is called a derivation of wn from w1.

  9. Example derivation •   A  0A1  00A11  000111. Thus the sentential form 000111 is derivable from . •   . Thus the empty string is derivable from .

  10. Language of a Grammar • The language L(G) generated by a formal grammar G is the set of terminal strings derivable from .L(G) = { w  T* |  * w } • If w  L(G) we say that w is a string, sentence or word in the language generated by G.

  11. Example • In the previous example G,L(G) = { , 01, 0011, 000111, … }= { 0k1k | k  0 }.

  12. Example 2 • G = (N, T, P, )N = {A, B}T = {0, 1}P = {   | A | B; A  1 | 0A ; B  0 | 1B; } • L(G) = { , 0k1, 1k0 | k  0 }

  13. Types of Grammars T0 unrestricted T1 context sensitive W set ofall strings T2 context free T3 regular

  14. T0 Unrestricted Grammars • Each production in P has the formA  where , , and  are in (N  T)* and may possibly be empty, and A=, or A  N. • If  = , then |A| > ||, and grammar G is called contracting.

  15. T1 Context Sensitive Grammar • Each production in P has the formA    where , ,   (N  T)* and  and  may possibly be empty,   , and A=, or A  N.

  16. T1 Example •   AA  aABC ; A  abCCB * BCbB  bbbC  bccC  cc • L(G) = {akbkck | k  1}

  17. T2 Context-Free Grammar • Each production in P has the formA    where   (N  T)* - {}, and A=, or A  N.

  18. T2 Examples • Double parentheses language  A ;A  AA ;A  ( ) | (A) | [ ] | [A] ;L(G) = ? • Matching pair language  A ;A  aAb | ab ;L(G) = { akbk | k  1 }

  19. T3 Regular Grammar • Left Linear: Each production in P has the form:A  Ba | a ;    • Right Linear: Each production in P has the form:A  aB | a ;    • Here A  N or A=, B  N, a  T

  20. T3 Example • Even parity grammar:  A | ;A  0 | 0A | 1B ;B  1 | 0B | 1A ; • L(G) = Strings with even number of 1s

  21. T3 Example • Strings of 1s followed by zeroes:  A | B | ;A  1 | 1A | 1B ;B  0 | 0B ; • L(G) = Prefix of 1s (possibly empty) followed by suffix of 0s (possibly empty)

  22. Derivation Trees • The derivation tree T corresponding to a derivation = w1 w2  w3  …  wnis an ordered tree whose root is , and whose leaves are terminal symbols. If a step in the derivation is:  12 … kthen the subtree rooted at  will have children 1, 2, … , and k.

  23. Derivation Tree Example •   A  0A1  00A11  A 0 A 1 0 A 1

  24. Leftmost derivations • A leftmost derivation is a derivation in which at each step, only the left-most nonterminal symbol is replaced by a production of the grammar.

  25. Ambiguity • A grammar G is ambiguous if for some string w in L(G), w has more than one distinct leftmost derivation. • If for each string w in L(G), w has exactly one leftmost derivation, then the grammar is called unambiguous.

  26. Example of Ambiguous Grammar • N = { A } ;T = { a, x, y } ;  A ;A  a | AyA | AxA ;

  27. Example ofUnambiguous Grammar • N = { A, B } ;T = { a, x, y } ;  A ;A  B | AyB ;B  a | Bxa ;

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