Filling Information Gaps

To find a unique answer to every question of a
puzzle, background information is required beyond
the literal meaning of the text. In Question
1 of Figure 1, for example, without the constraint
that a sculpture may not be exhibited in
multiple rooms, answers B, D and E are all correct.
Human readers deduce this implicit constraint
from their knowledge that sculptures are
physical objects, rooms are locations, and physical
objects can have only one location at any
given time. In principle, such information could
be derived from ontologies. Existing ontologies,
however, have limited coverage, so we also plan
to leverage information about expected puzzle
structures.
Most puzzles we collected are formalizable
as constraints on possible tuples of objects.
The crucial information includes: (a)
the object classes; (b) the constants naming
the objects; and (c) the relations used to
link objects, together with their arguments’
classes. For the sculptures puzzle, this information
is: (a) the classes are sculpture and
room; (b) the constants are C,D,E, F, G,H for
sculpture and 1, 2, 3 for room; (c) the relation
is exhibit(sculpture, room). This information is
obtainable from the parse trees and SL formulas.
Within this framework, implicit world knowledge
can often be recast as mathematical properties
of relations. The unique location constraint
on sculptures, for example, is equivalent
to constraining the mapping from sculptures to
rooms to be injective (one-to-one); other cases
exist of constraining mappings to be surjective
(onto) and/or total. Such properties can be obtained
from various sources, including cardinality
of object classes, pure lexical semantics, and
even through a systematic search for sets of implicit
constraints that, in combination with the
explicitly stated constraints, yield exactly one
answer per question. Figure 3 shows the num

Figure 3: Effect of explicit and implicit constraints
on constraining the number of possible
models
ber of possible models for the sculptures puzzle
as affected by explicit and implicit constraints
in the preamble.