Abstract
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 [10]. 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.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
References
Buntrock, G., Otto, F.: Growing context-sensitive languages and Church-Rosser languages. Inform. Comput. 141, 1–36 (1998)
Dassow, J., Păun, G.: Regulated Rewriting in Formal Language Theory. Springer, Berlin (1989)
Ehrenfeucht, A., Haussler, D., Rozenberg, G.: On regularity of context-free languages. Theoret. Comput. Sci. 27, 311–332 (1983)
Fernau, H., Stephan, F.: Characterizations of recursively enumerable sets by programmed grammars with unconditional transfer. J. Autom., Lang. Comb. 4, 117–152 (1999)
Gilman, R.H.: A shrinking lemma for indexed languages. Theoret. Comput. Sci. 163, 277–281 (1996)
Glaister, I., Shallit, J.: A lower bound technique for the size of nondeterministic finite automata. Inform. Process. Lett. 59, 75–77 (1996)
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)
Haines, L.H.: On free monoids partially ordered by embedding. J. Combinatorial Theory 6, 94–98 (1969)
Higman, G.: Ordering by divisibility in abstract algebras. Proc. London Math. Soc. 2, 326–336 (1952)
Gruber, H., Holzer, M., Kutrib, M.: The size of Higman-Haines sets. Theoret. Comput. Sci. (to appear)
Ilie, L.: Decision problems on orders of words. Ph.D. thesis, Department of Mathematics, University of Turku, Finland (1998)
Kruskal, J.B.: The theory of well-quasi-ordering: A frequently discovered concept. J. Combinatorial Theory 13, 297–305 (1972)
van Leeuwen, J.: A regularity condition for parallel rewriting systems. SIGACT News 8, 24–27 (1976)
van Leeuwen, J.: Effective constructions in well-partially-ordered free monoids. Discrete Mathematics 21, 237–252 (1978)
Author information
Authors and Affiliations
Editor information
Rights and permissions
Copyright information
© 2007 Springer-Verlag Berlin Heidelberg
About this paper
Cite this paper
Gruber, H., Holzer, M., Kutrib, M. (2007). More on the Size of Higman-Haines Sets: Effective Constructions. In: Durand-Lose, J., Margenstern, M. (eds) Machines, Computations, and Universality. MCU 2007. Lecture Notes in Computer Science, vol 4664. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-540-74593-8_17
Download citation
DOI: https://doi.org/10.1007/978-3-540-74593-8_17
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-540-74592-1
Online ISBN: 978-3-540-74593-8
eBook Packages: Computer ScienceComputer Science (R0)