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

Stabler - Lx 185/209 2003
17.2 A simple phonology, orthography
Phonological analysis of an acoustic input, and orthographic analysis of an written input, will commonly yield
morethanonepossibleanalysisoftheinputtobeparsed. In fact, the relation the input and the morpheme
sequence to be parsed will typically be many-many: the definite articles a,an will get mapped to the same
syntactic article, and an input element like read will get mapped to the bare verb, the bare noun, the verb +
present, and the verb + past.
Sometimes it is assumed that the set of possible analyses can be represented with a regular grammar or
finite state machine. Let’s explore this idea first, before considering reasons for thinking that it cannot be right.
(10) For any set S, letSɛ = (S ∪{ɛ}). Thenasusual,afinite state machine(FSM) A =〈Q, Σ,δ,I,F〉 where
Q is a finite set of states (= ∅);
Σ1 is a finite set of input symbols (= ∅);
δ ⊆ Q × Σɛ × Q,
I ⊆ Q, the initial states;
F ⊆ Q, the final states.
(11) Intuitively, a finite transducer is an acceptor where the transitions between states are labeled by pairs.
Formally, we let the pairs come from different alphabets: T =〈Q, Σ1, Σ2,δ,I,F〉 where
Q is a finite set of states (= ∅);
Σ1 is a finite set of input symbols (= ∅);
Σ2 is a finite set of output symbols (= ∅);
δ ⊆ Q × Σ ɛ 1 × Σɛ2 × Q,
I ⊆ Q, the initial states;
F ⊆ Q, the final states.
(12) And as usual, we assume that for any state q and any transition function δ, 〈q, ɛ, ɛ, q〉 ∈δ.
(13) For any transducers T = 〈Q, Σ1, Σ2,δ1,I,F〉 and T ′ = 〈Q ′ , Σ ′ 1 , Σ′ 2 ,δ2,I ′ ,F ′ 〉,definethecomposition
T ◦ T ′ = 〈Q × Q ′ , Σ1, Σ ′ 2 ,δ,I × I′ ,F × F ′ 〉 where δ = {〈〈qi,q ′
′
i 〉,a,b,〈qj,q j 〉〉| for some c ∈ (Σɛ2 ∩
Σ ′ɛ
1 ), 〈qi,a,c,qj〉 ∈δ1 and 〈q ′
i ,c,b,q′ j 〉∈δ2} (Kaplan and Kay, 1994, for example).
(14) And finally, for any transducer T =〈Q, Σ1, Σ2,δ,I,F〉 let its second projection 2(T ) be the FSM A =
〈Q, Σ1,δ ′ ,I,F〉, whereδ ′ ={〈qi,a,qj〉| for some b ∈ Σ ɛ 2 , 〈qi,a,b,qj〉 ∈δ}.
(15) Now for any input s ∈ V ∗ where s = w1w2 ...wn for some n ≥ 0, let string(s) be the transducer
〈{0, 1,...,n}, Σ, Σ,δ0, {0}, {n}〉, whereδ ={〈i−1,wi,wi,i〉| 0 ≤ i}.
(16) Let a (finite state) orthography be a transducer M = 〈Q, V, Σ, δ,I,F〉 such that for any s ∈ V ∗ ,
2(str ing(s) ◦ M) represents the sequences of syntactic atoms to be parsed with a grammar whose
vocabulary is Σ. For any morphology M, let the function inputM from V ∗ to Σ∗ be such that for any
s ∈ V ∗ , input(s) = 2(str ing(s) ◦ M).
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