We examine the problem of generating definite noun phrases that are appropriate referring expressions; that is, noun phrases that (a) successfully identify the intended referent to the hearer whilst (b) not conveying to him or her any false conversational implicatures (Grice, 1975). We review several possible computational interpretations of the conversational implicature maxims, with different computational costs, and argue that the simplest may be the best, because it seems to be closest to what human speakers do. We describe our recommended (...) algorithm in detail, along with a specification of the resources a host system must provide in order to make use of the algorithm, and an implementation used in the natural language generation component of the IDAS system. (shrink)