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

Stabler - Lx 185/209 2003
also has nested dependencies just like {a n b n | n ≥ 0}, but this time the number of a’s in words of the language
is
0, 2, 4, 6,...
Plotting position in the sequence against value, these sets are both linear.
Let’s write scalar product of an integer k and a pair (x, y) this way:
k(x, y) = (kx, ky)
, and we add pairs in the usual way (x, y) + (z, w) = (x + y,z + w). ThenasetS of pairs (or tuples of higher
arity) is said to be linear iff there are finitely many pairs (tuples) v0,v1,...,vk such that
S ={v0 +
k
nvi| n ∈ N, 1 ≤ i ≤ k}.
Asetissemilinear iff it is the union of finitely many linear sets.
Theorem: Finite state and context free languages are semilinear
Semilinearity Hypothesis: Human languages are semilinear (Joshi, 1985)
Theorem: Many unification grammar languages are not semilinear!
Here is a unification grammar that accepts {a2n| n>0}.
% apowtwon.pl
’S’(0) :˜ [a,a].
’S’(s(X)) :˜ [’S’(X),’S’(X)].
i=1
Michaelis and Kracht (1997) argue against Joshi’s semilinearity hypothesis on the basis of the case markings
in Old Georgian, 18 which we see in examples like these (cf also Boeder 1995, Bhatt&Joshi 2003):
(51) saidumlo-j igi sasupevel-isa m-is γmrt-isa-jsa-j
mystery-nom the-nom kingdom-gen the-gen God-gen-gen-nom
‘the mystery of the kingdom of God’
(52) govel-i igi sisxl-i saxl-isa-j m-is Saul-is-isa-j
all-nom the-nom blood-nom house-gen-nom the-nom Saul-gen-gen-nom
‘all the blood of the house of Saul’
Michaelis and Kracht infer from examples like these that in this kind of possessive, Old Georgian requires
the embedded nouns to repeat the case markers on all the heads that dominate them, yielding the following
pattern (writing K for each case marker):
[N1 − K1[N2 − K2 − K1[N3 − K3 − K2 − K1 ...[Nn − Kn − ...− K1]]]]
It is easy to calculate that in this pattern, when there are n nouns, there are n(n+1)
2 case markers. Such a
language is not semilinear.
18A Kartevelian language with translations of the Gospel from the 5th century. Modern Georgian does not show the phenomenon
noted here.
61