Abstract
For a language family \(\mathcal{L}\), a syntactic complexity measure K defined on languages of \(\mathcal{L}\), a number \(n\ge 1\), and an n-ary operation \(\circ \) under which \(\mathcal{L}\) is closed, we define \(g_{\circ }^K(m_1,m_2,\dots ,m_n)\) as the set of all integers r such that there are n languages \(L_i\), \(1\le i\le n\), with
In this paper we study these sets for the operation union, catenation, star, complement, set-subtraction, and intersection and the measure number of accepting states (defined for regular languages) as well as for reversal, union, catenation, and star and the measures number of nonterminals, productions, and symbols (defined for context-free languages).
Moreover, we discuss the change of these sets if one restricts to finite languages, unary languages, and finite unary languages.
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Alur, R., Madhusudan, P.: Visibly pushdown languages. In: Proceedings of ACM Symposium on Theory of Computing, pp. 202–211 (2004)
Champarnaud, J.-M., Coulon, F.: Büchi automata reduction by means of left and right trace inclusion preorder. Manuscript (2004)
Dassow, J.: On the number of accepting states of finite automata. J. Automata Lang. Comb. 21, 55–67 (2016)
Dassow, J., Stiebe, R.: Nonterminal complexity of some operations on context-free languages. Fundam. Inform. 83, 35–49 (2008)
Dassow, J., Harbich, R.: Descriptional complexity of union and star on context-free languages. J. Automata Lang. Comb. 17, 123–143 (2012)
Drewes, F., Holzer, M., Jakobi, S., van der Merwe, B.: Tight bounds for cut-operations on deterministic finite automata. In: Durand-Lose, J., Nagy, B. (eds.) MCU 2015. LNCS, vol. 9288, pp. 45–60. Springer, Cham (2015). doi:10.1007/978-3-319-23111-2_4
Edelkamp, S., Jabbar, S.: Large-scale directed model checking LTL. In: Valmari, A. (ed.) SPIN 2006. LNCS, vol. 3925, pp. 1–18. Springer, Heidelberg (2006). doi:10.1007/11691617_1
Gao, Y., Moreira, N., Reis, R., Yu, S.: A review on state complexity of individual operations. Technical report series DCC-2011-08, Version 1.1, University of Porto, Faculty of Sciences, Department of Computer Science (2012), September 2012
Gao, Y., Salomaa, K., Yu, S.: Transition complexity of incomplete DFAs. Fundam. Inform. 110, 143–158 (2011)
Gruska, J.: Some classifications of context-free languages. Inf. Control 14, 152–179 (1969)
Gruska, J.: On the size of context free grammars. Kybernetika 8, 213–218 (1972)
Han, Y.-S., Salomaa, K.: State complexity of basic operations on suffix-free regular languages. Theoret. Comput. Sci. 410, 2537–2548 (2009)
Harbich, R.: Regel- und Symbolkomplexität kontextfreier Sprachen unter ausgewählten Operationen. Dissertation (2018)
Holzer, M., Kutrib, M.: State complexity of basic operations on nondeterministic finite automata. In: Champarnaud, J.-M., Maurel, D. (eds.) CIAA 2002. LNCS, vol. 2608, pp. 148–157. Springer, Heidelberg (2003). doi:10.1007/3-540-44977-9_14
Holzer, M., Kutrib, M.: Nondeterministic finite automata - recent results on the descriptional and computational complexity. Int. J. Found. Comput. Sci. 20, 563–580 (2009)
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 7th International Workshop of Descriptional Complexity of Formal Systems, University of Milano, pp. 170–181 (2005)
Jirásková, G.: On the state complexity of complements, stars, and reversals of regular languages. In: Ito, M., Toyama, M. (eds.) DLT 2008. LNCS, vol. 5257, pp. 431–442. Springer, Heidelberg (2008). doi:10.1007/978-3-540-85780-8_34
Jirásková, G., Krausová, M.: Complexity in prefix-free regular languages. Electron. Proc. Theor. Comput. Sci. 31, 197–204 (2010)
Maia, E., Moreira, N., Reis, R.: Incomplete operational transition complexity of regular languages. Inf. Comput. 244, 1–22 (2015)
Okhotin, A., Salomaa, K.: State complexity of operations on input-driven pushdown automata. J. Comput. Syst. Sci. 86, 207–228 (2017)
Piao, X., Salomaa, K.: Operational state complexity of nested word automata. Theoret. Comput. Sci. 410, 3250–3260 (2009)
Piao, X., Salomaa, K.: State complexity of the concatenation of regular tree automata. Theoret. Comput. Sci. 429, 273–281 (2012)
Piao, X., Salomaa, K.: State complexity of projection and quotient on unranked trees. In: Kutrib, M., Moreira, N., Reis, R. (eds.) DCFS 2012. LNCS, vol. 7386, pp. 280–293. Springer, Heidelberg (2012). doi:10.1007/978-3-642-31623-4_22
Rozenberg, G., Salomaa, A. (eds.): Handbook of Formal Languages, vol. I–III. Springer, Berlin (1997)
Yu, S.: State complexity of regular languages. J. Automata Lang. Comb. 6, 221–234 (2001)
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Dassow, J. (2017). Descriptional Complexity and Operations – Two Non-classical Cases. In: Pighizzini, G., Câmpeanu, C. (eds) Descriptional Complexity of Formal Systems. DCFS 2017. Lecture Notes in Computer Science(), vol 10316. Springer, Cham. https://doi.org/10.1007/978-3-319-60252-3_3
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