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Context-free Grammars

Context-free Grammars. Recursive definition Grammar Derivation. What is the syntax of the description (meta-language)? External form of the grammar How do we associate a language with such a description? Semantics or use of the grammar

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Context-free Grammars

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  1. Context-free Grammars L4CFG

  2. Recursive definition • Grammar • Derivation L4CFG

  3. What is the syntax of the description (meta-language)? • External form of the grammar • How do we associate a language with such a description? • Semantics or use of the grammar • Generators: Grammars specify languages by generating strings in the language. • Recognizers: Grammars can be viewed as a notation for describing a family of recognition algorithms. L4CFG

  4. CFG = (V, ,P,S) • V : Finite set of variables/non-terminals • : Alphabet/Finite set of terminals • P : Finite set of rules/productions • S : Start symbol L4CFG

  5. a* represents a context-free language because we can write a CFG for it. • A-rule, null-rule, -rule. • Context-freeness: An A-rule can be applied whenever A occurs in a string, irrespective of the context (that is, non-terminals and terminals around A). • Cf. context-sensitive grammar (“declare-use”) L4CFG

  6. From CFG to Language • One-step Derivation (Defn: 3.1.2) • w is derivable from v in CFG, if there is a finite sequence of rule applications such that: L4CFG

  7. Let G=(V, S, P, S) be a CFG. • is a sentential form, if . • is a sentence, if . • The language of G, L(G) = L4CFG

  8. S S a a b b • Derivation of aabb • Derivation tree l L4CFG

  9. L4CFG

  10. Derivation Tree (abstracts derivation) S A B A B b a l l L4CFG

  11. Examples: CFGs and CFLs L4CFG

  12. L4CFG

  13. Note that parentheses are part of the meta-language. L4CFG

  14. L4CFG

  15. L4CFG

  16. Left to right generation of string. L4CFG

  17. Regular Grammar • A regular grammar is a CFG in which each of the rules has the form: (Right Linear Grammar) L4CFG

  18. Equivalent Regular Grammar • A regular grammar is a CFG in which each of the rules has the form: (Left Linear Grammar) L4CFG

  19. L4CFG

  20. L4CFG

  21. Derivations Note that the above result does not hold for sentential forms. L4CFG

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