Context-free languages

Context-free grammars (CFG)

Grammar G = (V,T,S,P) is "context-free" if all production rules are of the form A → u, where A is a variable and u is variable or terminal. i.e., LHS of all rules is a single variable.

A language L is context-free if it has a context-free grammar.

Let G be grammar, consider derivation S ⇒^* w.

• G right linear: w contains at most one variable
• G context-free: w can contain multiple variables

Which variable do you expand? Leftmost, or rightmost. The result doesn't depend on the strategy, but on the choice of rules.

Derivation trees

Derivation tree for a CFG has nodes with labels that can be variables, terminals, or λ.

• root is labeled by start symbol
• if there is production rule:
  • A → x₁ … x_n with n  ≥ 1: node labelled A can have children labelled x₁…x_n.
  • A → λ: node labelled A can have one child with label λ
• every node with label that's a terminal is a leaf (has no children)

Labels of leaves of derivation tree, from left to right, skipping λ, form a word in L(G).

Ambiguous grammars

A CFG is ambiguous if there is a word in the language that can have multiple derivation trees.

An example of ambiguity is the dangling else problem. Joerg gave an example in ALGOL, but it's the same in C (and is solved by always making else bind tightly, i.e. to the nearest if):

if (condition)
if (condition2) printf("Whatever");
else printf("Something else");
// without braces or indentation, you don't know whether the else corresponds to the
//  inner or outer if.

Ambiguity is undecidable – you can't algorithmically determine whether a CFG is ambiguous.

Inherently ambiguous language: if the language is context free, and every grammar for that language is ambiguous.

Normalising

Removal of λ rules

λ-production rule: a rule where a variable can rewrite to λ

Steps:

1. Note which variables rewrite to λ (any number of steps).
2. For all variables B from step 1: for every rule A → xBy, add a new rule A → xy.
   • watch out for transitivity: i.e. if A → B and B → λ, then A → λ.
3. Remove all rules with λ on the right hand side.