Mathematical Linguistics

For readers of traditional textbooks such as (Partee, Ter Meulen, and Wall, 1990), the term ‘mathematical linguistics’ denotes a rather narrowly circumscribed set of issues including automata theory, set theory, lambda calculus with maybe a little formal language theory thrown in. Kornai’s contribution is refreshingly different in that he treats, in this relatively compact volume, practically all areas of linguistics, phonetics and speech and language processing. Kornai’s motivation for writing this book is to present “a single entry point to the central methods and concepts of linguistics that are made largely inaccessible to the mathematician, computer scientist, or engineer by the surprisingly adversarial style of argumentation . . . and the proliferation of unmotivated notation and formalism . . . all too often encountered in research papers and monographs in the humanities.” There is no question that much of what passes for rigor (mathematical and scientific) in linguistics is a joke, and that there is clearly a need for any work that can place the field on a more solid footing. It also seems likely that Kornai is the only person who could have written this book. The book is divided into ten chapters, including a short introductory chapter, which lays the groundwork and identifies the potential audience, and a concluding chapter where Kornai reveals his own views on what is important in the field, which in the interests of balance he has largely suppressed throughout the book. Chapter 2 is also introductory in that it presents basic concepts of generation (via a ruleset), axioms and string rewriting. The main chapters (3–9) deal with a variety of topic areas relating to language and speech, starting with phonology in Chapter 3. This chapter introduces the notion of phonemes, distinctive features, autosegmental phonology and computation using finite automata. Kornai offers many details that are of course lacking in most linguistic treatments, such as a proof that the number of well-formed association lines between two tiers of length n is asymptotically (6 + 4 √ 2). This is Theorem 3.3.1 on p. 36 – see the discussion on p. 37 why this matters. Chapter 4 deals with morphology, which for Kornai includes not only word formation, but also prosody (including stress assignment and moraic structure), as well as Optimality Theory and Zipf’s law. The fifth chapter treats syntax, including categorial grammar, phrase structure, dependency frameworks, valency and weighted models of grammar, ending with a discussion of weighted finite automata and hidden Markov models. In the context of weighted models Kornai implies that Chomsky’s original notion of degree of grammaticality fits naturally as an instance of a weighted model with a particular semiring; of course, exactly what the ⊕ and ⊗ operators of that semiring map to remain to be seen insofar as the notion ‘degree of grammaticality’ has never been rigorously defined, a point Kornai should have made himself.