Computational Complexity in Natural Language

We have become so used to viewing natural language in computational terms that we need occasionally to remind ourselves of the methodological commitment this view entails. That commitment is this: we assume that to understand linguistic tasks—tasks such as recognizing sentences, determining their structure, extracting their meaning, and manipulating the information they contain—is to discover the algorithms required to perform those tasks, and to investigate their computational properties. To be sure, the physical realization of the corresponding processes in humans is a legitimate study too, but one from which the computational investigation of language may be pursued in Splendid Isolation. Complexity Theory is the mathematical study of the resources—both in time and space—required to perform computational tasks. What bounds can we place—from above or below—on the number of steps taken to compute such-and-such a function, or a function belonging to such-and-such a class? What bounds can we place on the amount of memory required? It is not surprising, therefore, that in the study of natural language, complexity-theoretic issues abound. Since any computational task can be the object of complexity-theoretic investigation, it would be hopeless even to attempt a complete survey of Complexity Theory in the study of natural language. We focus therefore on a selection of topics in natural language where there has been a particular accumulation of complexity-theoretic results. Section 2 discusses parsing and recognition; Section 3 discusses the computation of logical form; and Section 4 discusses the problem of determining logical relationships between sentences in natural language. But we begin with a brief review of the Complexity Theory itself.