The language complexity game

The book under review makes certain claims about natural languages couched in mathematical terms. And so I begin with a caveat (see also Manaster Ramer 1992): as in far too many other books and articles in our field, Ristad consistently and mistakenly uses the words prove and proof when speaking not only about the results concerning welldefined mathematical objects (i.e., formal languages), but also about the claims that (some) NLs are properly modeled by these mathematical objects and hence share their properties. In this review, I will consistently distinguish between proofs of properties of formal languages and arguments about the properties of NLs. With this caveat firmly in mind, we may now specify that Ristad's book presents the argument that the understanding of anaphora in NLs, specifically in English, is NP-complete, i.e., it can be computed in nondeterministic polynomial time and is also NP-hard, that is, as hard as the "hardest" problems in the class of problems solvable in nondeterministic polynomial time (see, e.g., [Hopcroft and Ullman 1979 pp. 320341] for a more formal introduction to these concepts). We are thus dealing with an attempt to characterize the properties of NLs in terms of complexity theory rather than the more familiar formal language theory. Moreover, the problem studied is not that of distinguishing grammatical from ungrammatical sentences, as in most work on mathematical linguistics, but rather that of determining what meanings are possible for a given set of sentences. In particular, Ristad's argumentation revolves around various aspects of the problem of determining which anaphoric elements in a given sentence can refer to which potential antecedents. Ristad takes the reader through five rounds of what he calls a "complexity game," which is a contest between a maximizer, who tries to make natural languages as complex as possible, and a minimizer, who seeks to reduce the complexity to a bare minimum. In the first round, we read an argument purporting to demonstrate the NPhardness of any language whose anaphora are required to agree in features such as number, gender, and so on with their antecedents. In the second round, this argument is refuted by the minimizer, who claims that the standard theory of how agreement works is wrong and proposes a new theory, under which anaphoric agreement is now recognizable in deterministic polynomial time. In the third turn, a new set of data leads to the central argument in the book, according to which the anaphora problem is NP-hard after all. The facts crucial to this argument have to do with obviation--that is, conditions under which coreference is impossible, as in He saw him. Not content with this result, the maximizer goes on to the fourth round, in which he presents yet another body of data (dealing with anaphora in elliptical structures) on the basis of which he argues that the anaphora problem is PSPACE-hard (i.e., as hard as the "hardest" problems in the class of problems solvable in deterministic polynomial space, a class which contains dV'7~). But the minimizer replies, in the fifth and final round,