More on the Size of Higman-Haines Sets: Effective Constructions

A not so well-known result in formal language theory is that the Higman-Haines sets for any language are regular [11, Theorem 4.4]. It is easily seen that these sets cannot be effectively computed in general. The Higman-Haines sets are the languages of all scattered subwords of a given language as well as the sets of all words that contain some word of a given language as a scattered subword. Recently, the exact level of unsolvability of Higman-Haines sets was studied in [8]. Here we focus on language families whose Higman-Haines sets are effectively constructible. In particular, we study the size of descriptions of Higman-Haines sets for the lower classes of the Chomsky hierarchy, namely for the family of regular, linear context-free, and context-free languages. We prove upper and lower bounds on the size of descriptions of these sets for general and unary languages. This paper is completely revised and expanded version of a paper presented at the 5th International Conference on Machines, Computations and Universality (MCU) held in Orléans, France, September 10–13, 2007. 2 H. Gruber, M. Holzer, M. Kutrib / More on the Size of Higman-Haines Sets: Effective Constructions