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Context-free grammars, derivation trees, and ambiguity

Context-free grammars, derivation trees, and ambiguity. Module 06.1 COP4020 – Programming Language Concepts Dr. Manuel E. Bermudez. Context-free Grammars Derivation Trees Ambiguity. Topics. Definition : A context-free grammar (CFG)

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Context-free grammars, derivation trees, and ambiguity

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  1. Context-free grammars, derivation trees, and ambiguity Module 06.1COP4020 – Programming Language Concepts Dr. Manuel E. Bermudez

  2. Context-free Grammars Derivation Trees Ambiguity Topics

  3. Definition: A context-free grammar (CFG) is a quadrupleG = (, , P, S),where all productions are of the formA →, for A ∊ and  ∊ (∪ )*. Re-writing using grammar rules: βAγ⇒βγif A → (derivation). Left-most derivation: At each step, the left-most nonterminal is re-written. Right-most derivation: At each step, the right-most nonterminal is re-written. Context-free grammars

  4. Sample grammar and derivations

  5. Derivation trees • Describe re-writes, independently of the order (left-most or right-most). • Each tree branch matches a production rule in the grammar. • Leaves are terminals. • Bottom contour is the sentence. • Left recursion causes left branching. • Right recursion causes right branching.

  6. Goal of parsing: Examine input string, determine whether it's legal. Same as: try to build derivation tree. Therefore, tree should be unique. Definition: A CFG is ambiguous if there exist two different right-most (or left-most, but not both) derivations for some sentence z. (Equivalent) Definition: A CFG is ambiguous if there exist two different derivation trees for some sentence z. Ambiguous grammars

  7. Classic ambiguities • Simultaneous left/right recursion: E → E + E → i • Dangling else problem: S → if E then S → if E then S else S → … Ambiguity is undecidable: no algorithm exists.

  8. summary • Context-free Grammars • Derivation Trees • Ambiguity

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