Complexity in Convex Languages

A language L is prefix-convex if, whenever words u and w are in L with u a prefix of w, then every word v which has u as a prefix and is a prefix of w is also in L. Similarly, we define suffix-, factor-, and subword-convex languages, where by subword we mean subsequence. Together, these languages constitute the class of convex languages which contains interesting subclasses, such as ideals, closed languages (including factorial languages) and free languages (including prefix-, suffix-, and infix-codes, and hypercodes). There are several advantages of studying the class of convex languages and its subclasses together. These classes are all definable by binary relations, in fact, by partial orders. Closure properties of convex languages have been studied in this general framework of binary relations. The problems of deciding whether a language is convex of a particular type have been analyzed together, and have been solved by similar methods. The state complexities of regular operations in subclasses of convex languages have been examined together, with considerable economies of effort. This paper surveys the recent results on convex languages with an emphasis on complexity issues.