Scope Resolution

One way of dealing with scope ambiguities is by
using underspecified representations (URs). A
UR is a meta-language construct, describing a
set of object-language formulas.3 It describes
the pieces shared by these formulas, but possibly
underspecifies how they combine with each
other. A UR can then be resolved to the specific
readings it implicitly describes.
We use an extension of Hole Semantics
(Blackburn and Bos, 2000)4 for expressing URs
and calculating them from parse trees (modulo
the modifications in §5). There are several advantages
to this approach. First, it supports
the calculation of just one UR per sentence in
a combinatorial process that visits each node of
the parse tree once. This contrasts with approaches
such as Categorial Grammars (Carpenter,
1998), which produce explicitly all the
scopings by using type raising rules for different
combinations of scope, and require scanning the
entire parse tree once per scoping.
Second, the framework supports the expression
of scoping constraints between different
parts of the final formula. Thus it is possible
to express hierarchical relations that must exist
between certain quantifiers, avoiding the problems
of naive approaches such as Cooper storage
(Cooper, 1983). The expression of scoping
constraints is not limited to quantifiers and is
applicable to all other operators as well. Moreover,
it is possible to express scope islands by
constraining all the parts of a subformula to be
outscoped by a particular node.
Another advantage is that URs support efficient
elimination of logically-equivalent readings.
Enumerating all scopings and using
a theorem-prover to determine logical equivalences
requires O(n2) comparisons for n scopings.
Instead, filtering methods (Chaves, 2003)
can add tests to the UR-resolution process,
disallowing certain combinations of operators.
Thus, only one ordering of identical quantifiers
is allowed, so “A man saw a woman” yields
only one of its two equivalent scopings. We also
filter 8¤ and 9§ combinations, allowing only
the equivalent ¤8 and §9. However, numeric
quantifiers are not filtered (the two scopings of
“Three boys saw three films” are not equivalent).
Such filtering can result in substantial
speed-ups for sentences with a few quantifiers
(see (Chaves, 2003) for some numbers).
Finally, our true goal is determining the correct
relative scoping in context rather than enumerating
all possibilities. We are developing
a probabilistic scope resolution module that
learns from hand-labeled training examples to
predict the most probable scoping, using features
such as the quantifiers’ categories and
their positions and grammatical roles in the sentence.