Description Logic Unplugged

In [1], Bos introduces a general scheme of underspecification for a wide range of logics. In his approach he breaks down a formula into its parts and introduces variables, called holes, on argument positions. Each part gets a unique handle, called label. The possible connections of holes to labeled parts are represented with a new type of scope constraint. A plugging is a function that assigns a label to every hole. A plugging is consistent if it does not contradict the set of scope constraints. The semantics of an underspecified formula is defined to be the set of tuples of pluggings and denotations. The main motivation for Bos’ work is to represent the scope ambiguity of natural language(NL) expressions. His example utterance “Do not sleep and pay attention, please!” has two readings differing with respect to negation scope. In one reading, the scope of “not” is “sleep”, and in a second reading, “not” has scope over “sleep and pay attention”. An underspecified representation is an expression denoting the set of both possibilities. This form of underspecification is also realized in the interface data structure VIT [2] of the NL processing system VERBMOBIL [7]. For the task of disambiguation, each VIT is mapped to objects of the Description Logic (DL) FLEX [5]. This mapping and the disambiguation task are described in [4]. However, FLEX does not support representation and reasoning with scope ambiguity. In this paper we fill this gap and adopt the approach of [1] to DL. We only describe the task of representing scope ambiguity here; further investigation is necessary for disambiguation. In general, it would be possible to introduce holes in the TBox as well, resulting in an underspecified terminology. But for now we restrict occurrences of holes to the ABox. In contrast to [1], there is no need for labels, because the relevant target entities already have names, namely those of objects. In the next section we present the syntax and semantics of an Unplugged Description Logic (UDL), more precisely a DL with a partly unplugged ABox. Then we present an algorithm of a constraint solver for solving the consistency problem of UDL. Because all interesting inferences can be reduced to consistency, this yields algorithms for subsumption and instance checking. 2 Syntax of UDL