Abstract
We continue our investigation on the descriptional complexity of the cascade product of finite state devices started in [M. Holzer, C. Rauch: The Range of State Complexities of Languages Resulting from the Cascade Product—The Unary Case (Extended Abstract). Proc. CIAA, 2021]. Here we study the general case, that is, cascade products of reset, permutation-reset, permutation, and finite automata in general, where the left operand automaton has an alphabet of size at least two. In all cases, except for the cascade product of two permutation automata, it is shown that the whole range of state complexities, namely the interval [1, nm], where n is the state complexity of the left operand and m that of the right one, is reachable. The cascade product of two permutation automata produces a lot of non-reachable numbers—numbers of this kind are called magic in the relevant literature—even for arbitrary alphabet sizes. These results are in sharp contrast to the unary case.
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
Similar content being viewed by others
Notes
- 1.
There are three types of automata for the left operand of the cascade product, namely unary reset, unary permutation(-reset), and unary finite automata in general and four types of automata for the right operand, that are reset, permutation, permutation-reset, and finite state device without restrictions.
- 2.
For automata with input alphabet of size at least two we have four types of left operands instead of three as in the unary case. This leads to \(4\cdot 4=16\) cases.
References
Ae, T.: Direct or cascade product of pushdown automata. J. Comput. Syst. Sci. 14(2), 257–263 (1977)
Arbib, M.A.: Algebraic Theory of Machines, Languages, and Semigroups. Academic Press, Cambridge (1968)
Čevorová, K.: Kleene star on unary regular languages. In: Jurgensen, H., Reis, R. (eds.) DCFS 2013. LNCS, vol. 8031, pp. 277–288. Springer, Heidelberg (2013). https://doi.org/10.1007/978-3-642-39310-5_26
Čevorová, K., Jirásková, G., Krajňáková, I.: On the square of regular languages. In: Holzer, M., Kutrib, M. (eds.) CIAA 2014. LNCS, vol. 8587, pp. 136–147. Springer, Cham (2014). https://doi.org/10.1007/978-3-319-08846-4_10
Harrison, M.A.: Introduction to Formal Language Theory. Addison-Wesley, Boston (1978)
Holzer, M., Rauch, C.: The range of state complexities of languages resulting from the cascade product—the unary case (extended abstract). In: Maneth, S. (eds.) Implementation and Application of Automata. CIAA 2021. Lecture Notes in Computer Science, vol. 12803. Springer, Cham (2021). https://doi.org/10.1007/978-3-030-79121-6_8
Holzer, M., Hospodár, M.: The range of state complexities of languages resulting from the cut operation. In: Martín-Vide, C., Okhotin, A., Shapira, D. (eds.) LATA 2019. LNCS, vol. 11417, pp. 190–202. Springer, Cham (2019). https://doi.org/10.1007/978-3-030-13435-8_14
Hricko, M., Jirásková, G., Szabari, A.: Union and intersection of regular languages and descriptional complexity. In: Mereghetti, C., Palano, B., Pighizzini, G., Wotschke, D. (eds.) Proceedings of the 7th Workshop on Descriptional Complexity of Formal Systems, pp. 170–181. Universita degli Studi di Milano, Como (2005)
Iwama, K., Kambayashi, Y., Takaki, K.: Tight bounds on the number of states of DFAs that are equivalent to \(n\)-state NFAs. Theor. Comput. Sci. 237(1–2), 485–494 (2000)
Jirásková, G.: Magic numbers and ternary alphabet. Internat. J. Found. Comput. Sci. 22(2), 331–344 (2011)
Maler, O.: On the Krohn-Rhodes cascaded decomposition theorem. In: Manna, Z., Peled, D.A. (eds.) Time for Verification. LNCS, vol. 6200, pp. 260–278. Springer, Heidelberg (2010). https://doi.org/10.1007/978-3-642-13754-9_12
Maler, O., Pnueli, A.: Tight bounds on the complexity of cascaded decomposition of automata. In: Proceedings of the 31st Annual Symposium on Foundations of Computer Science, pp. 672–682. IEEE Computer Society Press, St. Louis (1990)
Author information
Authors and Affiliations
Corresponding author
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2021 Springer Nature Switzerland AG
About this paper
Cite this paper
Holzer, M., Rauch, C. (2021). The Range of State Complexities of Languages Resulting from the Cascade Product—The General Case (Extended Abstract). In: Moreira, N., Reis, R. (eds) Developments in Language Theory. DLT 2021. Lecture Notes in Computer Science(), vol 12811. Springer, Cham. https://doi.org/10.1007/978-3-030-81508-0_19
Download citation
DOI: https://doi.org/10.1007/978-3-030-81508-0_19
Published:
Publisher Name: Springer, Cham
Print ISBN: 978-3-030-81507-3
Online ISBN: 978-3-030-81508-0
eBook Packages: Computer ScienceComputer Science (R0)