CFGs: Formal Definition

1. CFGs: Formal Definition G = (V, S, P, S) V = variables a finite set S = alphabet or terminals a finite set P = productions a finite set S = start variable SV Productions’ form, where AV, a(VS)*: • A  a

2. CFGs: Derivations Derivations in one step: bAgGbag Aa P  xS*, a,b,g(VS)* Can choose any variable for use for derivation step. Derivations in zero-or-more steps: G* is the reflexive and transitive closure of G . Language of a grammar: L(G) = {xS* | S G* x}

3. S A B Root label = start node. A A b B Each interior label = variable. a a b Each parent/child relation = derivation step. Each leaf label = terminal or e. All leaf labels together = derived string = yield. Parse Trees S  A | A B A e | a | A b | A A B b | bc | B c | b B • Sample derivations: • S  AB  AAB aAB aaB aabB aabb • S  AB  AbB  Abb AAbb Aabbaabb These two derivations use same productions, but in different orders.

4. S A B A A b B a a b Left- & Rightmost Derivations S  A | A B A e | a | A b | A A B b | bc | B c | b B • Sample derivations: • S  AB  AAB aAB aaB aabB aabb • S  AB  AbB  Abb AAbb Aabbaabb • These two derivations are special. • 1st derivation is leftmost. • Always picks leftmost variable. • 2nd derivation is rightmost. • Always picks rightmost variable.

5. Left / Rightmost Derivations • In proofs… • Restrict attention to left- or rightmost derivations. • In parsing algorithms… • Restrict attention to left- or rightmost derivations. • E.g., recursive descent uses leftmost; yacc uses rightmost.

6. S  A | A B A e | a | A b | A A B b | bc | B c | b B Other derivation trees for this string? w = aabb S S S A B A B A A A b B A A b A A a a b a A b A A A b a a A b e a Derivation Trees Infinitely many others possible.

7. Ambiguity CFG ambiguous any of following equivalent statements: •  string w with multiple derivation trees. •  string w with multiple leftmost derivations. •  string w with multiple rightmost derivations. Defining ambiguity of grammar, not language.

8. Ambiguity & Disambiguation Given ambiguous grammar, would like an equivalent unambiguous grammar. • Allows more knowledge about structure of derivation. • Simplifies inductive proofs on derivations. • Can lead to more efficient parsing algorithms. • In programming languages, want to impose a canonical structure on derivations. E.g., for 1+23. Strategy: Force an ordering on all derivations.

9. Exp  n | Exp + Exp | Exp  Exp What is an equivalent unambiguous grammar? Exp  Term | Term + Exp Term  n | n Term Uses operator precedence left-associativity Disambiguation Example

10. Parsing Designations • Major parsing algorithm classes are LL and LR • The first letter indicates what order the input is read – L means left to right • Second letter is direction in the “parsing tree” the derivation goes, L = top down, R = bottom up • K of LL(k) or LR(k) is number of symbols lookahead in input during parsing • Power of parsing techniques • LL(k) < LR(k) • LL(n) < LL(n+1), LR(n) < LR(n+1) • Choice of LL or LR largely religious