Denotational semantics

  A primary objective in the development of DATR has been the provision of an explicit, mathematically rigorous semantics. This goal was addressed in one of the first publications on the language (E&G 1989b). The definitions given there deal with a subset of DATR that includes core features of the language such as the notions of local and global inheritance and DATR's default mechanism. However, they exclude some important and widely-used constructs, most notably value sequences and evaluable paths. Moreover, it is by no means clear that the 1989 approach can be generalized appropriately to cover these features. In particular, the formal apparatus introduced in 1989 provides no explicit model of DATR's notion of global context. Rather, local and global inheritance are represented by distinct semantic functions ${\cal L}$ and ${\cal G}$. This approach is possible only on the (overly restrictive) assumption that DATR statements involve either local or global inheritance relations, but never both.

The purpose of the present section is to remedy the deficiencies of the work described in E&G 1989b by furnishing DATR with a transparent, mathematical semantics. It is easy to see DATR as a language for representing a certain class of non-monotonic inheritance networks (`semantic nets'). While this perspective provides an intuitive and appealing way of thinking about the structure and representation of lexical knowledge, it is less clear that it provides an accurate or particularly helpful picture of the DATR language itself. In fact, there are a number of constructs available in DATR that are impossible to visualize in terms of simple inheritance hierarchies. For this reason, the approach adopted here reflects a rather different perspective on DATR, as a language for defining certain kinds of partial functions by cases. In the following pages, this viewpoint is made more precise. Section 3.1.4, above, presented the syntax of the DATR language and introduces the notion of a DATR theory. The semantics of DATR is now covered in two stages. Section 4.1 introduces DATR interpretations and describes the semantics of a restricted version of the language without defaults. The treatment of implicit information is covered in Section 4.2, which provides a definition of a default model for a DATR theory. And, in Section 4.3, we make some concluding comments about the semantics given here.