More on the Size of Higman-Haines Sets: Effective Constructions
A not well-known result [9, Theorem 4.4] in formal language theory is that the Higman-Haines sets for any language are regular, but it is easily seen that these sets cannot be effectively computed in general. Here the Higman-Haines sets are the languages of all scattered subwords of a given language and 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 . We focus on language families whose Higman-Haines sets are effectively constructible. In particular, we study the size of Higman-Haines sets for the lower classes of the Chomsky hierarchy, namely for the families of regular, linear context-free, and context-free languages, and prove upper and lower bounds on the size of these sets.
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- 2.Dassow, J., Păun, G.: Regulated Rewriting in Formal Language Theory. Springer, Berlin (1989)Google Scholar
- 7.Gruber, H., Holzer, M.: Results on the average state and transition complexity of finite automata. Descriptional Complexity of Formal Systems (DCFS 2006), University of New Mexico, Technical Report NMSU-CS-2006-001, pp. 267–275 (2006)Google Scholar
- 10.Gruber, H., Holzer, M., Kutrib, M.: The size of Higman-Haines sets. Theoret. Comput. Sci. (to appear)Google Scholar
- 11.Ilie, L.: Decision problems on orders of words. Ph.D. thesis, Department of Mathematics, University of Turku, Finland (1998)Google Scholar