Abstract
In formal language theory, if a grammar consists of rules of the type \((r,F_r)\), where r is a context-free rule and \(F_r\) is an associated finite set of strings called the forbidding set, such that r can be applied to a string only if none of the strings in \(F_r\) is present in the string, then the grammar is said to be a generalized forbidding (GF) grammar. There are four main parameters that describe the size of a GF grammar, namely, (i) d, the maximum length of strings in the forbidding sets, (ii) i, the maximum cardinality of the forbidding sets, (iii) n, the number of nonterminals used in the grammar, and (iv) c, the number of rules with nonempty forbidding set. The family of languages described by a GF grammar of size (at most) (d, i, n, c) is denoted by GF(d, i, n, c). We study for what sizes of generalized forbidding grammars one can obtain the computational power of Turing machines. We specifically show this result for sizes (2, 6, 8, 6), (2, 5, 8, 8), (2, 4, 9, 9) and (2, 3, 20, 18). These results improve on previous results on the descriptional complexity of generalized forbidding grammars.
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Csuhaj-Varjú, E., Kelemenová, A.: Descriptional complexity of context-free grammar forms. Theoret. Comput. Sci. 112(2), 277–289 (1993)
Csuhaj-Varjú, E., Păun, G., Vaszil, G.: PC grammar systems with five context-free components generate all recursively enumerable languages. Theoret. Comput. Sci. 299(1–3), 785–794 (2003)
Dassow, J., Păun, G.: Regulated Rewriting in Formal Language Theory. EATCS Monographs in Theoretical Computer Science. Springer, Heidelberg (1989)
Fernau, H.: Closure properties of ordered languages. EATCS Bull. 58, 159–162 (1996)
Fernau, H., Freund, R., Oswald, M., Reinhardt, K.: Refining the nonterminal complexity of graph-controlled, programmed, and matrix grammars. J. Autom. Lang. Combin. 12(1/2), 117–138 (2007)
Fernau, H., Kuppusamy, L., Oladele, R.O.: New nonterminal complexity results for semi-conditional grammars. In: Manea, F., Miller, R.G., Nowotka, D. (eds.) CiE 2018. LNCS, vol. 10936, pp. 172–182. Springer, Cham (2018). https://doi.org/10.1007/978-3-319-94418-0_18
Fernau, H., Kuppusamy, L., Oladele, R.O., Raman, I.: Minimizing rules and nonterminals in semi-conditional grammars: non-trivial for the simple case. In: Durand-Lose, J., Verlan, S. (eds.) MCU 2018. LNCS, vol. 10881, pp. 88–104. Springer, Cham (2018). https://doi.org/10.1007/978-3-319-92402-1_5
Geffert, V.: Normal forms for phrase-structure grammars. RAIRO Informatique théorique et Applications 25, 473–498 (1991)
Masopust, T.: Simple restriction in context-free rewriting. J. Comput. Syst. Sci. 76(8), 837–846 (2010)
Masopust, T., Meduna, A.: Descriptional complexity of generalized forbidding grammars. In: Geert, V., Pighizzini, G.: (eds.) 9th International Workshop on Descriptional Complexity of Formal Systems - DCFS, pp. 170–177. University of Kosice, Slovakia (2007)
Masopust, T., Meduna, A.: Descriptional complexity of grammars regulated by context conditions. In: Loos, R., Fazekas, S.Z., Martín-Vide, C. (eds.) LATA 2007. Proceedings of the 1st International Conference on Language and Automata Theory and Applications, volume report 35/07, pp. 403–412. Research Group on Mathematical Linguistics, Universitat Rovira i Virgili, Tarragona (2007)
Meduna, A.: Generalized forbidding grammars. Intern. J. Comput. Math. 36, 31–39 (1990)
Meduna, A., Svec, M.: Descriptional complexity of generalized forbidding grammars. Intern. J. Comput. Math. 80(1), 11–17 (2003)
Meduna, A., Zemek, P.: Regulated Grammars and Automata. Springer, New York (2014). https://doi.org/10.1007/978-1-4939-0369-6
Okubo, F.: A note on the descriptional complexity of semi-conditional grammars. Inform. Process. Lett. 110(1), 36–40 (2009)
Rogers, J., Lambert, D.: Extracting forbidden factors from regular stringsets. In: Kanazawa, M., de Groote, P., Sadrzadeh, M. (eds.) Proceedings of the 15th Meeting on the Mathematics of Language, MOL, pp. 36–46. ACL (2017)
Shannon, C.E.: A universal Turing machine with two internal states. In: Shannon, C.E., McCarthy, J. (eds.) Automata Studies. Annals of Mathematics Studies, vol. 34, pp. 157–165. Princeton University Press, Princeton (1956)
van der Walt, A.P.J., Ewert, S.: A shrinking lemma for random forbidding context languages. Theoret. Comput. Sci. 237(1–2), 149–158 (2000)
Vaszil, G.: On the descriptional complexity of some rewriting mechanisms regulated by context conditions. Theoret. Comput. Sci. 330, 361–373 (2005)
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Fernau, H., Kuppusamy, L., Oladele, R.O., Raman, I. (2019). Improved Descriptional Complexity Results on Generalized Forbidding Grammars. In: Pal, S., Vijayakumar, A. (eds) Algorithms and Discrete Applied Mathematics. CALDAM 2019. Lecture Notes in Computer Science(), vol 11394. Springer, Cham. https://doi.org/10.1007/978-3-030-11509-8_15
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