Notes on computational linguistics.pdf - UCLA Department of ...

Stabler - Lx 185/209 2003
6 Context free parsing: stack-based strategies
6.1 LL parsing
(1) Recall the definition of TD, which uses an “expansion” rule that we will now call “LL,” because this
method consumes the input string from Left to right, and it constructs a Leftmost parse:
G, Γ ,S ⊢ G [axiom] for definite clauses Γ ,goalG, S ⊆ Σ ∗
G, Γ ,S ⊢ (?-p, C)
G, Γ ,S ⊢ (?-q1,...,qn,C)
G, Γ ,wS ⊢ (?-w,C) [scan]
G, Γ ,S ⊢ (?-C)
[ll] if (p:-q1,...,qn) ∈ Γ
(2) As discussed in §1 on page 6, the rule ll is sound. To review that basic idea from a different perspective,
notice, for example, that [ll] licenses inference steps like the following:
G, Γ ,S ⊢ (?-p, q)
G, Γ ,S ⊢ (?-r,s,q)
[ll] if (p:-r,s) ∈ Γ
In standard logic, this reasoning might be represented this way:
¬(p ∧ q) ∧ ((r ∧ s) → p)
¬(r ∧ s ∧ q)
Is this inference sound in the propositional calculus? Yes. This could be shown with truth tables, or
we could, for example, use simple propositional reasoning to deduce the conclusion from the premise
using tautologies and modus ponens.
¬(p ∧ q) ∧ ((r ∧ s) → p)
¬(A∧B)↔(¬A∨¬B)
(¬p ∨¬q) ∧ ((r ∧ s) → p) (¬A∨B)↔(A→B)
(p →¬q) ∧ ((r ∧ s) → p) (A∧B)↔(B∧A)
((r ∧ s) → p) ∧ (p →¬q) ((A→B)∧(B→C))→(A→C)
(r ∧ s) →¬q (A→B)↔(¬A∨B)
¬(r ∧ s) ∨¬q ¬(A∧B)↔(¬A∨¬B)
(¬r ∨¬s ∨¬q) (¬A∨¬B∨¬C)↔¬(A∧B∧C)
¬(r ∧ s ∧ q)
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