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

Stabler - Lx 185/209 2003
3.3 Recognizers: time and space
Given a recognizer, a (propositional) grammar Γ ,andastrings ∈ Σ ∗ ,
(7) a proof that s has category c ∈ N has space complexity k iff the goals on the right side of the deduction
(“the workspace”) never have more than k conjuncts.
(8) For any string s ∈ Σ∗ , we will say that s has space complexity k iff for every category A, every proof
that s has category A has space complexity k.
(9) Where S is a set of strings, we will say S has space complexity k iff every s ∈ S has space complexity k.
(10) Set S has (finitely) bounded memory requirements iffthereissomefiniteksuch that S has space
complexity k.
(11) The proof that s has category c has time complexity k iff the number of proof steps that can be taken
from c is no more than k.
(12) For any string s ∈ Σ∗ , we will say that s has time complexity k iff for every category A, every proof that
s has category A has complexity k.
3.3.1 Basic properties of the top-down recognizer
(13) The recognition method introduced last time has these derivation rules:
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 ,pS ⊢ (?-p, C) [scan]
G, Γ ,S ⊢ (?-C)
if (p:-q1,...,qn) ∈ Γ
To prove that a string s ∈ Σ ∗ has category a given grammar Γ , we attempt to find a deduction of the
following form, where [] is the empty string:
goal theory resources workspace
?-a , Γ , s ⊢ ?-a
…
?-a , Γ , [] ⊢
Since this defines a top-down recognizer, let’s call this logic TD.
(14) There is exactly one TD deduction for each derivation tree. That is: s ∈ yield(G,A) has n leftmost
derivations from A iff there n TD proofs that s has category A.
(15) Every right branching RS ⊆ yield(G,A) has bounded memory requirements in TD.
(16) No infinite left branching LS ⊆ yield(G,A) has bounded memory requirements in TD.
(17) If there is any left recursive derivation of s from A, then the problem of showing that s has category A
has infinite space requirements in TD, and prolog may not terminate.
44