Briefly Noted.

And from this it follows that weak generative capacity, as usually conceived, cannot be the right notion of complexity to study in connection with human language (even if it were to be measured by a yardstick different from the Chomsky hierarchy). Important as the three papers I have mentioned are, my favorite is the previously unpublished---and, this time, brand new-pape r by Savitch on context-sensitive grammars. Most of the mathematics underlying the theory of computation and mathematical linguistics is familiar, but the implications of these results for empirical science are rarely discussed explicitly in the literature, and are not widely known. Savitch's contribution is one of the few places where a humdrum formal result, which has been known for a number of years, comes to life. The point is essentially this: we know that the recursively enumerable (r.e.) languages are accepted by Turing machines, which may use an arbitrary amount of tape before accepting a given string, whereas the CSLs are accepted by (non-deterministic) TMs which have a certain (linear) bound on the amount of tape that may be used. Now, given any r.e. language, we can construct a rather similar CSL, in which the extra tape that was required for the computation is built into the input. To do this, a string of special marker symbols is appended to the end of the real input. A computation proceeds as it would in the original (unbounded) TM except that the marked cells are used instead of blank ones. If the TM runs out of the marked cells to hold intermediate results, then the computat ion is aborted without accepting. Intuitively, then, the CSL accepted is not very different from the underlying r.e. language, and we see that each r.e. language can be encoded as a CSL. Moreover , the process of converting this CSL to the underlying r.e. language is trivial: delete the marker symbols (though there is no algorithm for going from the r.e. language to the CSL). Savitch argues, convincingly to my mind, that all this indicates that CSLs have all the structural complexity of r.e. languages, and hence are not suitable as a model of human language---or anything else. We have all been taught to think of the r.e. languages as including "everything", but in a real sense, so do the CSLs. For years, there was a feeling that recursiveness was something to strive for, but now we see that this was much too modest a goal. If human languages are characterized by certain structural universals (e.g., they all use reduplication but not prime length of a string as a grammatical device), then CSLs are already much too inclusive, for they contain all languages that can be characterized in such structural terms. S a v i t c h ' s contribution should help open up the heavily fortified border between the mathematical theory of computation and empirical science. Perhaps it will b e the beginning of a beautiful friendship. In sum, this book is well worth careful study. It is by no means the last word on the subject, but for many it may well be the first word, and Savitch et al. have done an excellent job, both in their selections and in their commentaries, of giving a solid introduction to a sparsely cultivated but already complex field. They have also done much to foster the dissemination and the comprehension of formal complexity results in linguistics and to encourage accuracy and lucidity in the formulation, presentation, and interpretation of such results.